Tese de Doutorado
Me´todos Estoca´sticos e de Teoria de Campos
Aplicados a Problemas Motivados na Ecologia
e Oncologia
Renato Vieira dos Santos
26 de Julho 2013
Universidade Federal de Minas Gerais,
Instituto de Cieˆncias Exatas,
Departamento de Fı´sica
Orientador: Ronald Dickman

Resumo
Esta tese de doutorado utiliza me´todos estoca´sticos e de teoria de campos para tratar dois temas diferentes:
• A dinaˆmica populacional das ce´lulas tronco do caˆncer nos tumores em geral.
• Ana´lises de processos ecologicamente motivados formulados na rede, utilizando te´cnicas de mecaˆnica
estatı´stica de na˜o equilı´brio.
No primeiro caso sa˜o aplicadas te´cnicas estoca´sticas para obtermos as distribuic¸o˜es de probabilidade
para a densidade populacional das chamadas ce´lulas tronco do caˆncer, com o objetivo de propor uma
explicac¸a˜o para uma controve´rsia relacionada a` frequeˆncia com que estas ce´lulas aparecem nos tumores.
Utiliza-se tambe´m, em um outro trabalho e para o mesmo fim, a teoria de campo me´dio de Weiss, bem
como te´cnicas matema´ticas para a obtenc¸a˜o do que a`s vezes e´ chamado de tamanho crı´tico de Kierstead-
Skellam-Slobodkin∗. Os resultados destes estudos foram aceitos para publicac¸a˜o em [1] e em [2].
No segundo caso utiliza-se te´cnicas de teoria estatı´stica de campos em modelos inspirados em problemas
de ecologia teo´rica. Primeiramente estudamos um modelo onde dois processos de contato sa˜o acoplados
por um mecanismo do tipo simbio´tico e os expoentes crı´ticos foram obtidos. Este trabalho esta´ publicado
em [3]. Posteriormente estudamos um modelo onde, sob certas circunstaˆncias, populac¸o˜es escassas podem
apresentar maiores chances de sobreviveˆncia no longo prazo quando comparadas a`s chances de populac¸o˜es
mais numerosas. A este fenoˆmeno demos o nome de sobreviveˆncia do mais escasso no espac¸o. Este artigo
e´ uma extensa˜o que leva em conta a distribuic¸a˜o espacial para um modelo previamente proposto e seus
resultados foram aceitos para publicac¸a˜o em [4].
Temos em seguida um modelo que propo˜e uma explicac¸a˜o para o problema da existeˆncia da reproduc¸a˜o
sexuada na natureza, apesar de todos os custos, quando se compara com o me´todo rival de reproduc¸a˜o, a
reproduc¸a˜o assexuada. Este artigo foi submetido para publicac¸a˜o em [5].
Finalmente temos um modelo onde o fenoˆmeno da discreteza induzindo coexisteˆncia ocorre. Neste caso
ha´ a induc¸a˜o de coexisteˆncia entre duas espe´cies quando se leva em considerac¸a˜o o cara´ter discreto das
interac¸o˜es e as subsequentes flutuac¸o˜es estatı´sticas, como modeladas pela equac¸a˜o mestra. Caso contra´rio,
as espe´cies seriam extintas. Estuda-se tambe´m os efeitos das constantes de difusa˜o das espe´cies. Este
trabalho foi submetido para publicac¸a˜o em [6].
Esta tese e´ organizada como segue: No primeiro capı´tulo descrevemos de forma geral o me´todo do
grupo de renormalizac¸a˜o dinaˆmico utilizado em capı´tulos posteriores. Utilizamos para isso um exemplo
tı´pico que demonstra muitas das caracterı´sticas passı´veis de serem descritas pelo grupo de renormalizac¸a˜o.
Utilizamos como exemplo o processo de aniquilac¸a˜o de pares. A sequeˆncia de capı´tulos subsequente
trata dos diversos artigos publicados e submetidos para publicac¸a˜o. No apeˆndice foram colocados diversos
ca´lculos detalhadamente feitos no que se refere ao procedimento de Doi-Peliti, que e´ um procedimento hoje
padra˜o para se mapear a equac¸a˜o mestra de um processo estoca´stico na rede d−dimensional em uma teoria
de campos equivalente. Com isso os poderosos me´todos analı´ticos da teoria de campos ficam disponı´veis.
∗Ou KISS no acroˆnimo em ingleˆs para Kierstead-Skellam-Slobodkin size.
i
ii
Abstract
This thesis uses stochastic and field theory methods to address two different issues:
• The population dynamics of cancer stem cells in tumors in general.
• Analyses of ecologically motivated processes formulated on lattices, using techniques of nonequili-
brium statistical mechanics.
In the first case stochastic techniques are applied to obtain the probability distributions for the population
density of the so-called cancer stem cells, with the aim to propose an explanation for a controversy related
to the frequency with which these cells appear in tumors. In another paper mean-field theory is used to
obtain the critical size defined by Kierstead-Skellam-Slobodkin †. The results of these studies have been
accepted for publication in [1] and [2].
In the second case we use techniques from statistical field theory in models inspired by problems of
theoretical ecology. First we study a model where two contact processes are coupled by a symbiotic
mechanism and critical exponents were obtained. This work is published in [3]. Subsequently we study
a model in which, under certain circumstances, scarce populations may have higher chances of survival
in the long run when compared to the chances of the larger populations. This phenomenon was called
survival of scarcer space. This article is an extension that takes into account the spatial distribution of a
model previously proposed and their results were accepted for publication in [4].
We next discuss a model which proposes an explanation for the problem of the existence of sexual
reproduction in nature, despite all its costs when compared with the rival method of asexual reproduction.
This article was submitted for publication in [5].
Finally we have a model where the phenomenon of discreteness inducing coexistence occurs. In this
case there is the induction of coexistence between two species when one takes into account the discrete
character of the interactions and subsequent statistical fluctuations, as modeled by the master equation.
Otherwise, one of the species would be extinct. We also studied the effects of varying diffusion rates of the
species. This work was submitted for publication in [6].
This thesis is organized as follows: In the first chapter we describe the method of dynamic renorma-
lization group used in later chapters. We use a typical example, the process of annihilation of pairs that
demonstrates many of the characteristics that can be described by the renormalization group. The subse-
quent chapters deals with several articles published and submitted for publication. The Appendix contains
details of various calculations using the Doi-Peliti procedure, which today is a standard procedure for map
the master equation of a stochastic d−dimensional lattice into of a field theory. In this way the powerful
analytical methods of field theory are made available.
†Or KISS, the acronym for Kierstead-Skellam-Slobodkin size in tumor growth.
iii
iv
Agradecimentos
Uma das alegrias da conclusa˜o foi olhar para tra´s e lembrar de todos os amigos e familiares que me ajuda-
ram e me apoiaram ao longo desta longa, mas gratificante estrada.
Eu gostaria de expressar minha sincera gratida˜o ao Professor Ronald Dickman, que na˜o e´ apenas meu
mentor, sempre solida´rio e paciente, mas tambe´m um querido amigo. Eu na˜o poderia ter pedido por
melhores exemplos de competeˆncia, profissionalismo e de pessoa, bem como de fonte de inspirac¸a˜o. Eu na˜o
poderia estar mais orgulhoso das minhas raı´zes acadeˆmicas e espero que eu possa, por sua vez, transmitir
os valores da pesquisa que me foram passados.
Gostaria tambe´m de agradecer aos meus examinadores, Professor Ricardo Schwartz Schor, Professor
Joa˜o Antoˆnio Plascak, Professor Itzhak Roditi, Professor Allbens Atman Picardi Faria e Professor Welles
Antoˆnio Martinez Morgado, que forneceram incentivos e um feedback construtivo. Na˜o e´ uma tarefa fa´cil
a revisa˜o de uma tese, e eu sou grato por seus comenta´rios inteligentes e detalhados.
Esta tese foi financiada pelo Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico (CNPq),
e eu gostaria de agradecer a esta organizac¸a˜o por seu apoio generoso.
Para os funciona´rios, em particular para Shirley Maciel, para os colegas da Universidade Federal de
Minas Gerais, e para o nosso grupo de mecaˆnica estatı´stica, Adailton, Alexandre, Bruno, Loba˜o, Luciana,
Marcos e Ricardo: Obrigado por me acolher como um amigo e ajudar a desenvolver as ide´ias nesta tese.
Eu sou grato pela oportunidade de ser parte do grupo.
E por u´ltimo, mas na˜o menos importante, agradec¸o imensamente a` Linaena, que compartilha minhas
paixo˜es, e cujo apoio foi fundamental para que a realizac¸a˜o desta tese fosse possı´vel.
Renato Vieira dos Santos
Belo Horizonte, Julho de 2013.
v
vi
Suma´rio
1. Introduc¸a˜o 1
1.1. Processo de reac¸a˜o-difusa˜o de aniquilac¸a˜o de pares . . . . . . . . . . . . . . . . . . . . . 2
1.2. Campo me´dio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3. Descric¸a˜o em termos da teoria estatı´stica de campos . . . . . . . . . . . . . . . . . . . . . 3
1.4. Observac¸o˜es referentes a` teoria de campos . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4.1. Equac¸o˜es de campo cla´ssicas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5. Expansa˜o diagrama´tica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5.1. Teoria de campo livre: difusa˜o pura . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.6. Perturbac¸a˜o em torno da difusa˜o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6.1. Diagramas em a´rvore correspondem a` teoria de campo me´dio . . . . . . . . . . . 9
1.7. Renormalizac¸a˜o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7.1. Divergeˆncias UV primitivas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.8. O procedimento de renormalizac¸a˜o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.8.1. A ide´ia da renormalizac¸a˜o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.8.2. Ana´lise dimensional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.8.3. Equac¸a˜o de Callan-Symanzik e a func¸a˜o Beta . . . . . . . . . . . . . . . . . . . . 14
1.9. Heurı´stica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2. Processo de contato com simbiose 19
3. Sobreviveˆncia do mais escasso no espac¸o 31
4. A importaˆncia de ser discreto no sexo 53
5. Uma possı´vel explicac¸a˜o para a frequeˆncia varia´vel das ce´lulas tronco do caˆncer nos tumores 83
6. O ruı´do e o KISS no nicho das ce´lulas tronco do caˆncer 101
7. Discreteza induzindo coexisteˆncia 111
8. Conclusa˜o 123
A. Mapeamento de Doi 125
A.1. Introduc¸a˜o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
A.1.1. Caracterı´sticas ba´sicas dos sistemas de reac¸a˜o-difusa˜o . . . . . . . . . . . . . . . 125
A.1.2. Teoria de Doi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
A.2. A representac¸a˜o de Doi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
A.2.1. Comparac¸a˜o com o me´todo da segunda quantizac¸a˜o da mecaˆnica quaˆntica . . . . . 131
A.2.2. O deslocamento de Doi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
vii
Suma´rio
B. Formulac¸a˜o contı´nua em uma teoria de campos 133
B.1. Representac¸a˜o em estados coerentes e integrais funcionais . . . . . . . . . . . . . . . . . 133
B.1.1. Representac¸a˜o em estados coerentes . . . . . . . . . . . . . . . . . . . . . . . . . 133
B.1.2. Formulac¸a˜o em integral funcional . . . . . . . . . . . . . . . . . . . . . . . . . . 135
B.1.3. Processos estoca´sticos de uma u´nica espe´cie em d−dimenso˜es . . . . . . . . . . . 140
B.2. Estados coerentes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
B.2.1. Estados coerentes na representac¸a˜o n . . . . . . . . . . . . . . . . . . . . . . . . 145
B.2.2. Ortogonalidade e relac¸o˜es de completeza . . . . . . . . . . . . . . . . . . . . . . 146
B.3. Prova propriedade de 〈P | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
B.4. Generalizac¸a˜o da formulac¸a˜o em integral funcional para d dimenso˜es . . . . . . . . . . . 148
Refereˆncias Bibliogra´ficas 153
viii
Capı´tulo 1
Introduc¸a˜o
A mecaˆnica estatı´stica e´ um dos mais antigos brac¸os da fı´sica teo´rica e ainda um atual campo de pesquisa
da fı´sica moderna. Lida com sistemas fı´sicos em que as propriedades macrosco´picas podem ser inferidas
de uma descric¸a˜o microsco´pica [7, 8, 9, 10, 11, 12, 13]. Ela e´ dividida em mecaˆnica estatı´stica de equilı´brio
e mecaˆnica estatı´stica de na˜o equilı´brio. Apesar de muitas te´cnicas matema´ticas terem sido desenvolvidas
para estudar sistemas em equilı´brio, o exame de sistemas de muitos corpos em na˜o equilı´brio ainda e´ uma
tarefa desafiadora para a fı´sica teo´rica, pois conceitos mais gerais e mais elaborados sa˜o necessa´rios. A ne-
cessidade de se estudar fı´sica estatı´stica de na˜o equilı´brio surge da sua relevaˆncia em descrever os processos
da natureza. A maioria dos sistemas sa˜o expostos a uma interac¸a˜o com outros sistemas resultando em um
fluxo de energia e mate´ria e portanto na˜o pode ser descrita em termos da termodinaˆmica de equilı´brio.
O foco deste capı´tulo sera´:
• Estudar os sistemas de muitas partı´culas que se difundem reagindo umas com as outras de uma
maneira definida e prover a modelagem correta de tais sistemas com o objetivo de explicar as
observac¸o˜es experimentais ou computacionais. Tal estudo fornecera´ a base teo´rica e conceitual ne-
cessa´ria ao entendimento de capı´tulos posteriores.
Em geral estaremos interessados em duas questo˜es relacionadas a estes processos de reac¸a˜o e difusa˜o:
O primeiro ponto se ocupa da existeˆncia de um estado estaciona´rio, ou seja, do estado onde a densidade de
partı´culas na˜o muda com o tempo e, em caso de existeˆncia, sua dependeˆncia espacial. A segunda questa˜o,
sendo naturalmente muito mais difı´cil de responder, examina a forma com que o sistema se aproxima do
estado estaciona´rio. Como estaremos interessados no comportamento assinto´tico de tais sistemas, pode-
remos sempre assumir que estes sistemas se encontram no regime limitado por difusa˜o, onde densidades
baixas de partı´culas esta˜o presentes e a dinaˆmica sera´ dominada pela taxa de difusa˜o.
Macroscopicamente, a evoluc¸a˜o temporal destes sistemas e´ a`s vezes modelada por equac¸o˜es diferenciais
ordina´rias (EDOs) para a quantidade observa´vel de interesse. Essas varia´veis podem ser, por exemplo, a
densidade populacional me´dia, concentrac¸a˜o quı´mica ou a magnetizac¸a˜o.
Incluindo tambe´m os graus de liberdade espaciais, as varia´veis se tornam func¸o˜es locais e devemos
utilizar equac¸o˜es diferenciais parciais (EDPs). Desta forma, comportamentos macrosco´picos emergentes
podem ser revelados tais como a formac¸a˜o de padro˜es e a propagac¸a˜o de frentes de onda [14].
Entretanto, devido a`s interac¸o˜es subjacentes entre os constituintes do sistema, correlac¸o˜es entre as
partı´culas podem surgir, possivelmente induzindo fenoˆmenos coletivos, que envolvem oscilac¸o˜es tempo-
rais denominadas quasi-ciclos [15], formac¸a˜o de padro˜es [16], gerac¸a˜o [17] e exacerbac¸a˜o [18] do efeito
Allee [19, 20], etc. A importaˆncia deste ruı´do interno na dinaˆmica macrosco´pica destes sistemas tem sido
estudada extensivamente nos anos recentes [21], causando o surgimento de uma enxurrada de artigos nesta
1
Capı´tulo 1. Introduc¸a˜o
a´rea [22]. Este fato indica uma grande efervesceˆncia na pesquisa com fenoˆmenos estoca´sticos relacionados
ao inevita´vel cara´ter discreto das interac¸o˜es.
Muitos destes sistemas exibem comportamento de escala universal em sua evoluc¸a˜o temporal para o
estado estaciona´rio. Este comportamento crı´tico pode ser descrito por leis de poteˆncia, com expoentes
crı´ticos para o paraˆmetro de ordem que caracterizam as propriedades assinto´ticas do sistema. Por exemplo,
a taxa efetiva de reac¸a˜o do processo de reac¸a˜o-difusa˜o pode mudar na presenc¸a de flutuac¸o˜es e correlac¸o˜es.
Entretanto, a descric¸a˜o da dependeˆncia temporal de sistemas estoca´sticos de muitas partı´culas realizando
processos de reac¸a˜o e difusa˜o em termos de EDOs e EDPs, negligenciam inteiramente as correlac¸o˜es e
variac¸o˜es espaciais, e sa˜o denominadas de descric¸o˜es de campo me´dio. A`s vezes na˜o e´ possı´vel explicar
o comportamento de escala correto para os expoentes crı´ticos aplicando a teoria de campo me´dio, mas o
surgimento destes fenoˆmenos sugere o uso de me´todos conhecidos da teoria estatı´stica de campos [12].
Um dos grandes sucessos da fı´sica teo´rica nas u´ltimas de´cadas foi a explicac¸a˜o das classes universalidade
e leis de escala na fı´sica da mate´ria condensada e fı´sica de partı´culas. A teoria estatı´stica de campos
fornece o ferramental matema´tico para estudar estes sistemas. Tais ferramentas sa˜o as te´cnicas do grupo
de renormalizac¸a˜o e teoria de perturbac¸a˜o.
A primeira parte desta tese lida exatamente com a aplicac¸a˜o destas te´cnicas aos sistemas de reac¸a˜o-
difusa˜o de modo a ir ale´m dos me´todos baseados em teoria de campo me´dio. Entretanto, por questo˜es de
simplicidade e clareza, iremos nos restringir neste capı´tulo ao processo de aniquilac¸a˜o de pares A+A λ→ /0,
um processo de reac¸a˜o-difusa˜o especı´fico que sera´ explicado com mais detalhes depois.
Primeiramente, a teoria de campo me´dio para o processo de aniquilac¸a˜o de pares sera´ utilizada e as suas
caracterı´sticas sera˜o comparadas aos experimentos e simulac¸o˜es nume´ricas. Para que possamos generalizar
o modelo e aplicar os me´todos de teoria estatı´stica de campos, utilizaremos uma abordagem baseada em
redes regulares d−dimensionais, que sera˜o os ambientes onde os processos de reac¸a˜o-difusa˜o ocorrera˜o.
Baseados na descric¸a˜o dos processos de reac¸a˜o-difusa˜o na rede, a equac¸a˜o mestra sera´ reformulada em
termos de operadores de criac¸a˜o e eniquilac¸a˜o, em analogia com a mecaˆnica quaˆntica no espac¸o de Fock
[23]. Esta formulac¸a˜o fornece uma descric¸a˜o adequada para sistemas com nu´mero de partı´culas varia´veis.
Esta descric¸a˜o permite uma formulac¸a˜o em termos de integrais funcionais e uma teoria de campo e´ obtida.
Renormalizando a teoria de campo, o comportamento universal de escala e os expoentes crı´ticos corretos
para a forma como o sistema se aproxima do estado estaciona´rio podem ser extraı´dos.
1.1. Processo de reac¸a˜o-difusa˜o de aniquilac¸a˜o de pares
Vamos examinar e exemplificar as caracterı´sticas dos problemas que surgem na ana´lise de sistemas de
reac¸a˜o-difusa˜o, utilizando o processo de um u´nico tipo de partı´cula denominado processo de aniquilac¸a˜o
de pares
A+A λ→ /0 + difusa˜o. (1.1)
As partı´culas da espe´cie A difundem no espac¸o e se aniquilam mutuamente com uma taxa de reac¸a˜o
λ. Vamos investigar este sistema simples de reac¸a˜o-difusa˜o porque todas as principais caracterı´sticas do
comportamento crı´tico podem ser extraı´das analiticamente.
O estado estaciona´rio deste processo e´ muito simples. Dependendo da condic¸a˜o inicial, se comec¸amos
com um nu´mero par ou ı´mpar de partı´culas, o estado estaciona´rio ira´ conter 0 partı´culas (espac¸o vazio) ou
somente uma partı´cula ira´ sobrar. O foco da nossa ana´lise estara´ na questa˜o de como o sistema se aproxima
do estado estaciona´rio.
2
1.2. Campo me´dio
1.2. Campo me´dio
Sistemas de reac¸a˜o-difusa˜o sa˜o geralmente estudados em termos de equac¸o˜es diferenciais [24, 25]. Em
um espac¸o d−dimensional com uma dimensa˜o temporal, elas sa˜o equac¸o˜es para a densidade me´dia de
partı´culas e para o processo de aniquilac¸a˜o de pares (em d = 1) teremos:
∂
∂t
ρ(x, t) = D∇2ρ(x, t)︸ ︷︷ ︸
difusa˜o
−λρ2(x, t)︸ ︷︷ ︸
reac¸a˜o
(1.2)
com a condic¸a˜o inicial ρ(x,0) = f (x). O primeiro termo no lado direito esta´ relacionada a` difusa˜o e o se-
gundo termo, a` aniquilac¸a˜o. Se negligenciarmos momentaˆneamente o termo difusivo, obteremos a equac¸a˜o
cine´tica
d
dt
ρ(t) =−λρ2(t) (1.3)
que e´ uma equac¸a˜o diferencial ordina´ria com soluc¸a˜o dada por
ρ(t) =
1
1
ρ(0) +λt
, (1.4)
onde ρ(0) denota a densidade inicial de partı´culas no tempo t0 = 0. Esta soluc¸a˜o (1.4) se comporta assin-
toticamente como ρ∼ t−1.
Se retornarmos ao sistema na˜o homogeˆneo, os efeitos da difusa˜o devera˜o ser levados em conta. Mas
os efeitos da difusa˜o tendem a uniformizar a distribuic¸a˜o de partı´culas e a soluc¸a˜o da equac¸a˜o (1.2) ira´
realizar o mesmo comportamento de escala assinto´tico para o decaimento temporal na equac¸a˜o (1.4). Afi-
nal, para ρ 1 temos ρ2  ρ de modo que a difusa˜o domina a dinaˆmica, ate´ o momento em que ρ seja
essencialmente uniforme.
Entretanto, resultados rigorosos [26] mostram que o comportamento assinto´tico depende da dimensa˜o d
da rede e portanto tal abordagem via EDPs e´ uma descric¸a˜o inerentemente inadequada. Para o processo de
aniquilac¸a˜o de pares, encontramos
ρ(x, t)∼
 t
−1/2 se d = 1 (descric¸a˜o em termos de EDPs e´ incorreta)
ln t · t−1 se d = 2 (descric¸a˜o em termos de EDPs e´ incorreta)
t−1 se d > 2 (descric¸a˜o em termos de EDPs e´ correta)
(1.5)
De fato, tal abordagem de campo me´dio negligencia quaisquer flutuac¸o˜es e correlac¸o˜es espaciais no
sistema. Iremos incluir as flutuac¸o˜es estatı´sticas na nossa ana´lise e iremos reescrever o processo de
aniquilac¸a˜o de pares em termos da teoria estatı´stica de campos. Aplicando os me´todos de grupo de
renormalizac¸a˜o dinaˆmico [27], obteremos o comportamento assinto´tico temporal correto.
1.3. Descric¸a˜o em termos da teoria estatı´stica de campos
Seguindo os passos detalhadamente descritos nos apeˆndices (A) e (B), a ac¸a˜o para o processo de aniquilac¸a˜o
de pares pode ser escrita como
S[φ˜(x, t),φ(x, t)] =
∫
ddx
{∫ t
0
dτ L
[
φ˜(x,τ),φ(x,τ)
]−ρ0φ˜(x,0)} , (1.6)
com
L [φ˜(x, t),φ(x, t)] = φ˜(∂t −D∇2)φ︸ ︷︷ ︸
difusa˜o pura
+2λφ˜φ2+λφ˜2φ2︸ ︷︷ ︸
reac¸a˜o A+A λ→ /0
, (1.7)
3
Capı´tulo 1. Introduc¸a˜o
onde a parte bilinear do Lagrangeano L corresponde ao processo de difusa˜o pura e as interac¸o˜es de maior
ordem representam o termo de reac¸a˜o A+ A λ→ /0. As equac¸o˜es (1.5), (1.6) e (1.7) sera˜o os pontos de
partida para a ana´lise posterior. A integral funcional junto com a ac¸a˜o constroem a base da ana´lise via
teoria estatı´stica de campos para o processo de aniquilac¸a˜o de pares.
O nu´mero me´dio de partı´culas n¯ no limite contı´nuo se torna a densidade me´dia de partı´culas ρ e e´ igual
a` me´dia do campo ρ(x, t) = 〈φ(x, t)〉,
ρ(x, t) =
∫
D[φ˜(x, t)]D[φ(x, t)] φ(x, t) e−S[φ˜(x,t),φ(x,t)]∫
D[φ˜(x, t)]D[φ(x, t)] e−S[φ˜(x,t),φ(x,t)]
= 〈φ(x, t)〉 ≡ G1(x, t). (1.8)
Mostraremos posteriormente como calcular (1.8) perturbativamente.
Podemos generalizar este procedimento para outras reac¸o˜es. A u´nica mudanc¸a necessa´ria e´ o estabe-
lecimento do Lagrangeano da teoria, o que nos leva a ter de encontrar o Hamiltoniano microsco´pico na
rede.
1.4. Observac¸o˜es referentes a` teoria de campos
1.4.1. Equac¸o˜es de campo cla´ssicas
Vamos obter as equac¸o˜es cla´ssicas de campo referentes a` ac¸a˜o (1.6). Para obtermos as soluc¸o˜es esta-
ciona´rias da ac¸a˜o, devemos calcular
∂S
∂φ
=
∂S
∂φ˜
≡ 0, (1.9)
que sa˜o resolvidas por φ˜ = 0 e (∂t −D∇2)φ+ 2λφ2 = 0. Tomando a esperanc¸a matema´tica desta u´ltima,
temos
(∂t −D∇2)〈φ〉+2λ〈φ2〉= 0,
⇒ ∂
∂t
ρ(x, t) = D∇2ρ(x, t)−2λ〈φ2〉,
que tem a forma parecida com a equac¸a˜o (1.2). A diferenc¸a entre estas equac¸o˜es esta´ na diferenc¸a entre os
termos de reac¸a˜o. Se assumirmos que podemos fatorar o termo 〈φ2〉 na forma 〈φ2〉= 〈φ〉2 = ρ2, obtemos a
equac¸a˜o (1.2). Mas assumir 〈φ2〉= 〈φ〉2 e´ correto somente na auseˆncia de flutuac¸o˜es e correlac¸o˜es espaciais.
1.5. Expansa˜o diagrama´tica
Vamos resumir os resultados obtidos na u´ltima sec¸a˜o. Reescrevemos a descric¸a˜o estoca´stica microsco´pica
da reac¸a˜o de aniquilac¸a˜o de pares (1.1) em termos da equac¸a˜o mestra em uma teoria de campos (1.6, 1.7)
utilizando o formalismo da segunda quantizac¸a˜o no limite contı´nuo. Estamos agora na posic¸a˜o de calcular
a func¸a˜o de um ponto 〈φ(x, t)〉 via equac¸a˜o (1.8), que inclui os efeitos das flutuac¸o˜es estatı´sticas. Para
isso, vamos aplicar os me´todos da teoria de campos, em particular, me´todos perturbativos e de grupo de
renormalizac¸a˜o para calcular 〈φ(x, t)〉.
Para calcular 〈φ(x, t)〉, definimos o funcional gerador Z[h˜,h] introduzindo fontes de campo externas h˜,h,
linearmente acopladas aos campos φ˜,φ na ac¸a˜o:
Z[h˜,h]≡
∫
D[φ˜]D[φ] exp
{
−S[φ˜,φ]−
∫
x,τ
(
h˜φ˜+hφ
)}
, (1.10)
4
1.5. Expansa˜o diagrama´tica
onde a notac¸a˜o
∫
x,τ · · · ≡
∫
ddx
∫ t
0 dτ · · · foi usada. Enta˜o, a func¸a˜o de um ponto pode ser computada to-
mando a primeira derivada funcional do funcional gerador com relac¸a˜o ao campo de fonte h e fazendo
h˜ = h = 0,
G1 = 〈φ〉= 1Z[0,0]
δ
δh
Z[h˜,h]
∣∣∣
h˜=h=0
. (1.11)
O objetivo e´ agora calcular o funcional gerador Z[h˜,h], o que pode ser feito perturbativamente.
1.5.1. Teoria de campo livre: difusa˜o pura
Para calcular o funcional gerador Z[h˜,h] e enta˜o calcular ρ via (1.8) ou (1.11) perturbativamente, temos
que estabelecer uma teoria, normalmente denominada teoria de campo livre, que pode ser resolvida exata-
mente. Usualmente, ela consistira´ da parte bilinear na ac¸a˜o, cuja integrac¸a˜o funcional pode ser realizada
exatamente (pelo menos formalmente) realizando integrac¸o˜es funcionais Gaussianas em torno da qual uma
expansa˜o perturbativa pode ser formulada.
Para o processo de aniquilac¸a˜o de pares, a parte bilinear da ac¸a˜o (S0[φ˜,φ]) e´ a seguinte:
S0[φ˜,φ] =
∫
ddx
∫ t
0
dτ φ˜(∂t −D∇2)︸ ︷︷ ︸
≡O(τ,x)
φ. (1.12)
Na equac¸a˜o acima definimos o operador O(τ,x) ≡ ∂t −D∇2. O funcional gerador relacionado a esta ac¸a˜o
da teoria livre, Z0[h˜,h], pode ser escrito como
Z0[h˜,h]≡
∫
D[φ˜]D[φ] exp
{
−
∫
x,τ
(
φ˜Oφ+ h˜φ˜+hφ
)}
. (1.13)
Se definirmos o operador inverso O−1 do operador difusa˜o como segue,
O(x, t)O−1(x′, t ′) = δ(x− x′)δ(t− t ′), (1.14)
podemos completar o quadrado na equac¸a˜o (1.13) com a substituic¸a˜o φ → φ+ ∫x,τO−1h˜ e realizar a
integrac¸a˜o Gaussiana, obtendo
Z0[h˜,h] =
∫
D[φ˜]D[φ] exp
{
−
∫
x,τ
φ˜Oφ+
∫
x,τ
∫
x′,τ′
hO−1h˜
}
= Z0[0,0] exp
{∫
x,τ
∫
x′,τ′
hO−1h˜
}
, (1.15)
que e´ o resultado exato da teoria de campo livre.
Propagador: o inverso do operador difusa˜o
A forma explı´cita do operador inverso O−1 precisa ser especificada. Isto pode ser feito aplicando a trans-
formada de Fourier a` equac¸a˜o (1.14). Este procedimento e´ comum em teorias de campos [28]. No espac¸o
de Fourier,
O−1(q,ω) =
1
Dq2− iω ≡ G0(q,ω). (1.16)
5
Capı´tulo 1. Introduc¸a˜o
Convenientemente, ao operador inverso da teoria de campo livre e´ dado um ro´tulo pro´prio, aqui G0(q,ω).
G0(q,ω) e´ o propagador da nossa teoria de campo livre,
〈φ(q,ω)φ˜(q′,ω′)〉0 = (2pi)d+1δ(q+q′)δ(ω+ω′)G0(q,ω), (1.17)
onde as me´dias com relac¸a˜o ao campo livre sa˜o denotadas como 〈〉0.
Se retornarmos ao domı´nio do tempo aplicando a transformada inversa de Fourier apenas na coordenada
temporal, obteremos
G0(q, t) =
1
2pi
∫
dω e−iωtG0(q,ω) =Θ(t) e−Dq
2t , (1.18)
com a func¸a˜o de Heaviside definida por
Θ(t) =
{
1 se t ≥ 0,
0 se t < 0.
Enta˜o o propagador G0 inerentemente representa uma ordem causal dos campos φ e φ˜, ja´ que somente
campos anteriores φ˜ sa˜o conectados a campos posteriores φ,
〈φ(q, t)φ˜(q′, t ′)〉0 ∝ Θ(t− t ′) e−D(q−q′)2(t−t ′).
Esta e´ uma caracterı´stica crucial do propagador da difusa˜o sob a qual a estrutura dos diagramas de
Feynman na expansa˜o perturbativa e´ restringida. Veremos suas consequeˆncias na pro´xima sec¸a˜o.
1.6. Perturbac¸a˜o em torno da difusa˜o
Vamos retornar ao nosso objetivo inicial, o ca´lculo do funcional gerador Z[h˜,h] para a ac¸a˜o (1.6), (1.7) e o
respectivo ca´lculo da func¸a˜o de um ponto 〈φ(x, t)〉 via (1.11). O funcional gerador pode ser escrito como
segue,
Z[h˜,h] =
∫
D[φ˜]D[φ] exp
{
−S[φ˜,φ]−
∫
x,τ
(h˜φ˜+hφ)
}
=
∫
D[φ˜]D[φ] e−S0 exp
{
−
∫
x,τ
(
2λφ˜φ2+λφ˜2φ2
)}
e−
∫
x,τ(h˜φ˜+hφ)
com condic¸a˜o inicial (c.i.) φ(x,0) = ρ0(x)
= exp
{
−
∫
x,τ
(
2λ
δ
δh˜
δ2
δh2
+λ
δ2
δh˜2
δ2
δh2
)}
Z0[h˜,h]+ c.i.
1.15
= Z0[0,0] exp
{
−
∫
x,τ
(
2λ
δ
δh˜
δ2
δh2
+λ
δ2
δh˜2
δ2
δh2
)}
exp
{∫
x,τ
∫
x′,τ′
hG0h˜
}
+ c.i. (1.19)
A ide´ia agora e´ expandir a exponencial com os termos de reac¸a˜o no paraˆmetro de acoplamento λ de
acordo com eλx = ∑n(λx)n/n!. Fazendo isso, teremos uma expansa˜o perturbativa “em torno” do termo de
difusa˜o, onde as reac¸o˜es constituem as perturbac¸o˜es.
Fisicamente, esta forma de expandir o funcional gerador faz sentido, ja´ que estamos interessados no
comportamento temporal assinto´tico da func¸a˜o de um ponto 〈φ〉, ou seja, em um regime dominado por
difusa˜o. As reac¸o˜es sa˜o consideradas pequenas perturbac¸o˜es ao caso da pura difusa˜o.
6
1.6. Perturbac¸a˜o em torno da difusa˜o
Figura 1.1.: Propagador e ve´rtices para a reac¸a˜o de aniquilac¸a˜o de pares.
Por razo˜es de clareza e simplicidade, usualmente denotamos esta se´rie perturbativa em termos dos cha-
mados diagramas de Feynman [29]. Cada diagrama nesta notac¸a˜o gra´fica corresponde a uma expressa˜o em
termos de integrais de acordo com a seguinte identificac¸a˜o:
Por exemplo, o termo 2λφ˜φ2 corresponde a um ve´rtice que conecta dois propagadores que entram no
ve´rtice com um propagador que sai do ve´rtice. A estrutura causal do propagador do processo de reac¸a˜o-
difusa˜o assegura ordenamento temporal, ja´ que o tempo aumenta da direita para a esquerda em um diagrama
de Feynman.
As regras de Feynman para se construir todos os diagramas que contribuem para 〈φ(x, t)〉 sa˜o mais
convenientemente formuladas no espac¸o de momentos, ja´ que estamos lidando com um sistema translaci-
onalmente invariante no espac¸o e no tempo. Elas podem ser resumidas como segue [28]:
• Desenhe todos os diagramas com uma perna externa a` esquerda e com pernas iniciais a` direita.
• Cada linha corresponde ao propagador G0 (figura (1.1)).
• Os dois ve´rtices (figura (1.1)) conectam linhas internas. Um 3-ve´rtice carrega um fator de −λ1 =
−2λ, e um 4-ve´rtice contribui com um fator −λ2 =−λ.
• Integramos sobre cada lac¸o de momento p com medida 1
(2pi)d
∫
dd p · · · e sobre cada tempo indeter-
minado t0 com medida
∫ t ′′
t ′ dt0 · · · .
• Impomos conservac¸a˜o de momento em cada ve´rtice. O propagador final deve ter q= 0 (uniformidade
espacial) e todos os propagadores conectados com uma perna inicial (figura (1.1)) tambe´m deve ter
q = 0, por causa da condic¸a˜o inicial no espac¸o de Fourier. Cada perna inicial carrega um fator ρ0.
• Incluir um fator de simetria para cada diagrama sendo igual ao nu´mero de possibilidades de conectar
um propagador interno a um ve´rtice.
7
Capı´tulo 1. Introduc¸a˜o
Figura 1.2.: Lac¸os em um instante retornam zero devido a` estrutura causal do propagador.
Note que lac¸os em um tempo especı´fico t ′ na˜o sa˜o considerados (1.2), devido ao ordenamento temporal
e estrutura causal do propagador G0(t) = Θ(t) e−Dq
2t . Uma integrac¸a˜o sobre cada lac¸o e´ proporcional a`
integral
∫ t ′′
t ′
dt G0(q,0) = 0.
Para a func¸a˜o de um ponto, a expansa˜o diagrama´tica, de acordo com as regras de Feynman, retorna
Figura 1.3.: Expansa˜o diagrama´tica para a func¸a˜o de um ponto.
Para exemplificar a correspondeˆncia entre os diagramas e as expresso˜es em termos de integrais, vamos
olhar mais detalhadamente duas delas, identificadas como (1) e (2) na figura (1.3):
(1) =
∫ t
0
dt1 G0(0, t− t1)(−λ1)G0(0, t1)2 ρ20 ∝−λ1ρ20t,
(2) =
∫ t
0
dt2
∫ t2
0
dt1 G0(0, t− t2)(−λ1)×
×
∫ dd p
(2pi)d
G0(p, t2− t1)(−λ2)G0(−p, t2− t1)2G0(0, t1)2ρ20.
A primeira integral (1) corresponde a um decre´scimo linear da densidade inicial no tempo, mas a segunda
integral (2) demonstra um problema no nosso procedimento perturbativo, ja´ que ela diverge. Para ver isso,
considere um acoplamento efetivo λef, sendo parte da expressa˜o integral acima, dado pela expressa˜o
I(t2)≡−λef =
∫ t2
0
dt1
∫ dd p
(2pi)d
G0(p, t2− t1)(−λ2)G0(−p, t2− t1)2 ∝−λ2 t1−d/22 , (d 6= 2,4). (1.20)
Temos que distinguir entre dois casos dependendo da dimensa˜o d. O comportamento qualitativo muda
na dimensa˜o crı´tica dc = 2 [27].
8
1.7. Renormalizac¸a˜o
• d < dc = 2 : Para tempos pequenos t → 0, I(t) e´ finito e a expansa˜o perturbativa e´ finita. Este
limite e´ tambe´m denominado de limite ultravioleta (UV) (q,ω→ ∞). Para tempos longos t → ∞,
(I(t)) diverge e a teoria de perturbac¸a˜o falha. Este limite e´ denominado limite infravermelho (IV)
(q,ω→ 0).
• d > dc = 2 : Teoria de perturbac¸a˜o falha para t→ 0, mas funciona para t→ ∞.
As divergeˆncias UV para d ≥ 2 sa˜o causadas pelo limite contı´nuo (ver apeˆndices), onde o espac¸amento
da rede a→ 0. Fisicamente, sempre existira´ um corte de curta distaˆncia Λ que pode ser introduzido “a`
ma˜o” na ana´lise que resolve a divergeˆncia UV.
Em contraste, a divergeˆncia IV para d ≤ 2, que e´ o caso de interesse, e´ mais problema´tica, ja´ que a
se´rie perturbativa na˜o pode ser obtida por um argumento fı´sico como feito para o caso UV. Apesar disso, e´
possı´vel extrair o comportamento de escala correto para a densidade me´dia de partı´culas mesmo para d ≤ 2
utilizando os me´todos do grupo de renormalizac¸a˜o que sera˜o descritos posteriormente.
1.6.1. Diagramas em a´rvore correspondem a` teoria de campo me´dio
Vamos olhar mais de perto para os diagramas em a´rvore, ou seja, os diagramas de Feynman sem lac¸os.
Figura 1.4.: Expressa˜o iterativa para os diagramas em a´rvore.
Para a soma de todos os diagramas em a´rvore ρtree, obtemos a seguinte expressa˜o:
ρtree(t) = ρ0−λ1
∫ t
0
dt1 ρtree(t1)2. (1.21)
Tomando a derivada temporal desta equac¸a˜o, obteremos a equac¸a˜o da teoria de campo me´dio obtida
anteriormente para o processo de aniquilac¸a˜o de pares com a condic¸a˜o inicial correta (1.2). Deste modo,
toda a ana´lise baseada nos diagramas em a´rvore corresponde a equac¸a˜o diferencial parcial da teoria de
campo me´dio e as flutuac¸o˜es e correlac¸o˜es sa˜o representadas pelas expresso˜es integrais representadas pelos
lac¸os nos diagramas. A ana´lise das correc¸o˜es devidas a`s flutuac¸o˜es tera˜o um importante papel na sec¸a˜o
seguinte.
1.7. Renormalizac¸a˜o
Nas u´ltimas sec¸o˜es vimos como calcular 〈φ(x, t)〉 perturbativamente. Entretanto, esta abordagem deu ori-
gem a divergeˆncias no limite IV para tempos assinto´ticos abaixo da dimensa˜o crı´tica dc = 2. Me´todos de
renormalizac¸a˜o conseguira˜o tornar finitos os resultados para os diagramas em lac¸o que em princı´pio sa˜o
divergentes. O custo desta operac¸a˜o e´ que alguns paraˆmetros da teoria ira˜o depender de um corte Λ in-
troduzido na ana´lise. O comportamento de escala e´ extraido pelos me´todos do grupo de renormalizac¸a˜o
dinaˆmico [27].
9
Capı´tulo 1. Introduc¸a˜o
1.7.1. Divergeˆncias UV primitivas
Queremos investigar as divergeˆncias UV primitivas que encontramos na expansa˜o diagrama´tica de 〈φ(x, t)〉
em d = 2. Estas divergeˆncias sa˜o chamadas de primitivas por serem superficiais e poderem ser removidas
por um corte de curta distaˆncia Λ−1. A dependeˆncia em Λ−1 sera´ reescrita em termos da regularizac¸a˜o
dimensional [28]. Apenas os diagramas conectados amputados sera˜o examinados, ja´ que sa˜o os lac¸os
contidos neles e´ que causam as divergeˆncias. O “amputado” se refere ao fato que os propagadores das
linhas externas sa˜o desprezados. A soma de todos os diagramas conectados amputados para a func¸a˜o de
n−pontos e´ denotada por Fˆn(q,ω).
Divergeˆncias primitivas tem sua origem nas integrais associadas aos lac¸os, enta˜o Fˆ pode ser escrito como
Fˆ ∝
∫
d pdL
1
(Dp2− iω)I ∝
∫ ∞
0
d|p| |p|dL−1 1
(Dp2− iω)I , (1.22)
onde L denota o nu´mero de lac¸os no diagrama de Feynman e I o nu´mero de linhas internas. O diagrama
corresponde a uma integral UV divergente se o grau superficial de divergeˆncia GSD≡ dL−2I ≥ 0. Apenas
por razo˜es topolo´gicas, pode ser inferido que o grau superficial de divergeˆncia pode ser calculado como
[28]
GSD = (d−4)L+∑
n
(n−4)Vn−E +4. (1.23)
Vn denota o nu´mero de ve´rtices de valeˆncia n e E o nu´mero de pernas externas. Entretanto, GSD ≤ 0
na˜o garante que um diagrama e´ finito, ja´ que subdivergeˆncias na˜o foram levadas em conta, significando
que diagramas com GSD ≤ 0 podem possuir subgra´ficos divergentes. Apesar disso, o grau superficial de
divergeˆncia e´ uma ferramenta u´til para analisar e resolver divergeˆncias via te´cnicas de renormalizac¸a˜o.
No nosso caso, temos d = 2 e GSD ≥ 0⇔ −V3−E + 4 ≥ 2L. Vemos que quanto mais lac¸oes esta˜o
envolvidos, melhores as chances para um diagrama ser bem definido. Para o pior caso com L = 0, somente
duas situac¸o˜es devem ser consideradas:
• V3 = 0 : No caso de auseˆncia de interac¸o˜es do tipo 3-ve´rtice, segue a condic¸a˜o para divergeˆncia
superficial: E ≤ 4. Como a interac¸a˜o envolve ve´rtices sempre com dois campos entrantes, somente
diagramas com um ou dois campos que saem sa˜o permitidos.
• V3 = 1 : Agora, a condic¸a˜o se torna E ≤ 3 a apenas diagramas com um campo que sai sa˜o permitidos.
Assim somente dois conjuntos de diagramas de Feynman carregam divergeˆncias UV, a saber, Fˆ3(q,ω) e
Fˆ4(q,ω) (figura (1.5))
Figura 1.5.: Fˆ4(q,ω) (esquerda) e Fˆ3(q,ω) (direita) conte´m todos os diagramas de Feynman que da˜o origem
aos infinitos na expansa˜o diagrama´tica.
Note que nenhuma divergeˆncia superficial pode aparecer em Fˆ2(q,ω) = Dq2− iω (figura (1.6)). Isto e´
devido a` estrutura de interac¸a˜o do processo de aniquilac¸a˜o de pares e a` propriedade de ordenamento tem-
poral do propagador com a consequeˆncia de que divergeˆncias na˜o podem ocorrer. Portanto, o propagador
10
1.7. Renormalizac¸a˜o
permanece inalterado durante o processo de renormalizac¸a˜o e a constante de difusa˜o na˜o precisa ser renor-
malizada. O mesmo vale para os campos φ˜, φ e, na linguagem da renormalizac¸a˜o, temos Zφ˜ = Zφ = 1 e
ZD = 1. Esta e´ uma caracterı´stica do processo de aniquilac¸a˜o de pares e esta´ em contraste com a teoria de
campo escalar com interac¸a˜o φ4, por exemplo, onde o termo de massa precisa ser renormalizado. E´ enta˜o
possı´vel fazer D = 1 em todas as ordens na teoria de perturbac¸a˜o.
Figura 1.6.: Na˜o e´ necessa´ria renormalizac¸a˜o para o propagador.
Podemos imediatamente escrever as expanso˜es diagrama´ticas inteiras para Fˆ3 e Fˆ4, novamente por causa
da estrutura dos ve´rtices junto com a estrutura causal do propagador.
Figura 1.7.: Expansa˜o diagrama´tica para Fˆ3.
Figura 1.8.: Expansa˜o diagrama´tica para Fˆ4.
Denominando de J(q,ω) a expressa˜o integral para o lac¸o, podemos escrever Fˆ3(q,ω) como uma se´rie
geome´trica
11
Capı´tulo 1. Introduc¸a˜o
Fˆ3 = −2(−2λ)− (−2λ)J(−λ)22− (−2λ)J(−λ)J(−λ)22
= 4λ(1−2Jλ+22J2λ2∓·· ·)
= 4
λ
1+2λJ
≡ 4λR, (1.24)
que e´ um resultado exato em todas as ordens na expansa˜o em lac¸o. Este na˜o e´ o caso na teoria escalar φ4,
onde os ca´lculos podem ser feitos apenas ordem a ordem na expansa˜o perturbativa.
A integral J(q,ω) pode ser calculada por regularizac¸a˜o dimensional, onde as divergeˆncias sa˜o transfor-
madas em po´los de func¸o˜es que dependem de um paraˆmetro pequeno ε ≡ 2− d, que descreve a distaˆncia
da dimensa˜o crı´tica.
Podemos calcular J(q,ω) da seguinte forma:
J(q,ω) =
∫ ddq′
(2pi)d
∫ dω′
2pi
1
q′2− iω
1
(q′−q)2− i(ω′−ω)
∝
∫ ddq′
q′2+(q′−q)2− iω
q′→q′/√2
∝
∫ ddq′
q′2−2q′ · q√
2
+q′− iω
=
Γ(1−d/2)
(4pi)d/2Γ(1)
· 1
(q2/2− iω)1−d/2 , (1.25)
onde no u´ltimo passo a func¸a˜o Gamma foi introduzida (Γ(α) = 1/α ·Γ(α+1),Γ(1) = 1,Γ(1/2) =√pi) e
a seguinte identidade [30]
∫ ddk
(k2+2k · p+m2)s =
Γ(s−d/2)
(4pi)d/2Γ(s)
· 1
(m2− p2)s−d/2 .
Com ε= 2−d, obtemos finalmente
J(q,ω) =
1
(8pi)d/2
·Γ
( ε
2
)
·
(
q2
2
− iω
)−ε/2
, (1.26)
Fˆ3 = 4 · λ
1+ 2λ
(8pi)d/2
·Γ( ε2) ·( q22 − iω)−ε/2 . (1.27)
Podemos agora reconhecer a divergeˆncia olhando para o paraˆmetro ε. Se q= 0, a expressa˜o para Fˆ3 para
ε > 0 diverge quando ω→ 0, representando a singularidade IV. A divergeˆncia UV e´ agora expressa pelos
po´los da func¸a˜o Γ para ε= 0,−2,−4, · · · , pois Γ( ε2)∼ 2ε para ε→ 0.
1.8. O procedimento de renormalizac¸a˜o
1.8.1. A ide´ia da renormalizac¸a˜o
Comec¸amos com uma teoria de campos cujo Lagrangeano pode ser descrito essencialmente como
12
1.8. O procedimento de renormalizac¸a˜o
L = L0+Lint = φ˜(∂t −D∇2)φ+2λφ˜φ2+λφ˜2φ2.
Foi visto que em duas dimenso˜es, as func¸o˜es de n−pontos conectadas amputadas Fˆn(q,ω;λ,Λ) apre-
sentam divergeˆncias IV (para q→ 0 e t → ∞) e divergeˆncias UV devido ao limite contı´nuo para n = 3,4.
As boas notı´cias sa˜o que Fˆ2 e´ bem definida e finita e portanto os campos e a constante de difusa˜o na˜o
necessitam ser renormalizados. Ale´m disso, e´ possı´vel calcular Fˆ3, Fˆ4 em todas as ordens na expansa˜o em
lac¸os.
Foi mostrado nas equac¸o˜es (1.27, 1.25) que se o limite contı´nuo for tomado (correspondento a fazer Λ→
0 ou equivalentemente ε→ 0 na regularizac¸a˜o dimensional), o acoplamento efetivo diverge. Entretanto, na
teoria, deve existir um acoplamento fı´sico λfis, que pode ser medido em um experimento e corresponde a
uma interac¸a˜o efetiva que esta´ representada na figura (1.9).
Figura 1.9.: Constante de acoplamento efetiva λfis.
Enta˜o consideramos o ansatz λfis = λ+ λ¯ = λ
(
1+ λ¯/λ
)
, em que λ e´ o acoplamento que aparece no
Lagrangeano e λ˜ e´ um paraˆmetro introduzido que depende do corte Λ. Enta˜o, a ide´ia e´ permitir λ depender
do corte Λ, λ→ λˆ(Λ). Queremos realizar um limite mais sofisticado que e´ fazer Λ→∞ com λˆ(Λ) variando,
mas com λfis mantido fixo. Devemos obter respostas finitas para todas as quantidades observa´veis neste
limite. A existeˆncia de tal limite na˜o e´ trivial, mas pode ser atingido para o processo de aniquilac¸a˜o de
pares introduzindo um nu´mero finito de termos aditivos no Lagrangeano, tambe´m conhecidos como contra
termos, e tal teoria e´ dita sob esta condic¸a˜o ser renormaliza´vel. Note que, em geral, a constante de difusa˜o
D assim como os campos φ˜ e φ devem depender do corte Λ, mas isso na˜o ocorre para o processo de
aniquilac¸a˜o de pares como mencionado anteriormente.
Veremos que o procedimento mencionado acima remove as divergeˆncias, mas ao custo da variac¸a˜o nos
paraˆmetros (e dos campos, quando for o caso) com o corte Λ. Na pra´tica, as divergeˆncias podem ser
canceladas adicionando contra termos ao Lagrangeano L → L +Lc.t.. Normalmente isto deve ser feito
ordem a ordem na expansa˜o perturbativa, mas aqui no´s sabemos que as divergeˆncias em todas as ordens e
podemos imediatamente escrever os contra termos necessa´rios
Lc.t. = Aφ˜φ2+Bφ˜2φ2,
de modo que novos ve´rtices sa˜o introduzidos na expansa˜o de L+Lc.t., como mostrados na figura (1.10)
Estes novos ve´rtices sa˜o construı´dos e definidos de modo a subtraı´rem exatamente as divergeˆncias que
ocorrem no Lagrangeano original L . Assim, levando-se em conta o resultado de (1.20) e (1.22), obtemos
A = λR =
λ
1+ 2λ
(8pi)d/2
·Γ( ε2) ·( q22 − iω)−ε/2 = λˆ.
1.8.2. Ana´lise dimensional
Antes de realizar a ana´lise de renormalizac¸a˜o, uma ana´lise dimensional mais precisa sera´ feita, pois a
expansa˜o perturbativa apenas faz sentido em uma quantidade adimensional. Veremos que, a na˜o ser em
13
Capı´tulo 1. Introduc¸a˜o
Figura 1.10.: Novos ve´rtices devidos aos contra termos.
duas dimenso˜es, o acoplamento λ carrega a dimensa˜o ε e enta˜o, uma quantidade adimensional precisa ser
introduzida. Portanto, vamos examinar novamente a ac¸a˜o
S[φ˜(x, t),φ(x, t)] =
∫
ddx
{∫ t
0
dτ
[
φ˜(∂t −D∇2)φ+2λφ˜φ2+λφ˜2φ2
]−ρ0φ˜.(x,0)}
A ac¸a˜o S aparece na exponencial na integrac¸a˜o funcional e portanto possui dimensa˜o 0, onde as quanti-
dades sa˜o medidas em unidades de momento, por exemplo, Λ. Assim, [S] = 0, [p] = 1, [x] =−1 e estamos
livres para escolher [D] = 0, uma constante de difusa˜o adimensional. Segue do operador de difusa˜o que
[1/x2] = [1/t]⇒ [t] =−2. Ale´m disso, temos que [φ˜] = 0, implicando em [φ] = d e [λ] = 2−d = ε. Uma
constante de acoplamento adimensional g ([g] = 0) e´ definida em d = 2− ε dimensio˜es pela introduc¸a˜o de
uma escala arbitra´ria µ com [µ] = 1, sendo tambe´m denominada escala RG,
λ= g ·µε e λˆ= gˆ · µˆε.
1.8.3. Equac¸a˜o de Callan-Symanzik e a func¸a˜o Beta
Nesta subsec¸a˜o, a ide´ia de renormalizac¸a˜o e a ana´lise dimensional sera˜o colocadas juntas e as consequeˆncias
fı´sicas sera˜o inferidas.
Definimos o Lagrangeano despido LB como sendo o Lagrangeano L+Lc.t. escrito agora em termos dos
acoplamentos adimensionais,
LB = L+Lc.t.
= φ˜
(
∂t −D∇2
)
φ+2gˆ · µˆεφ˜φ2+ gˆ · µˆεφ˜2φ2+ Aˆ ·µεφ˜φ2+ Bˆ ·µεφ˜2φ2
= φ˜B(∂t −DB∇2)φB+2gˆB ·µεφ˜Bφ2B+ gˆB ·µεφ˜2Bφ2B. (1.28)
Na segunda linha, os termos dos contra termos A, B foram expressos em termos dos acoplamentos
adimensionais. Na terceira linha, reescrevemos o Lagrangeano despido na mesma forma do Lagrangeano
original definindo os campos despidos (φ˜B ≡ φ˜, φB ≡ φ) e os paraˆmetros despidos (DB ≡ D, gˆB ≡ gˆ+ Aˆ).
Note que os paraˆmetros despidos e os campos despidos dependem do corte ε, mas sa˜o independentes da
escala RG µ. Em constraste, o paraˆmetro renormalizado gˆ = gˆ(µ) depende da escala µ, mas na˜o de ε.
14
1.8. O procedimento de renormalizac¸a˜o
Entretanto, a arbitrariedade de µ ira´ restringir a teoria. Para vermos como, podemos reescrever a integral
funcional em termos das quantidades despidas como segue,
SB[φ˜B,φB] =
∫
ddx
{∫ t
0
dτ LB[φ˜B,φB]−ρ0φ˜B(x,0)
}
e D[φB] =D[φ]
⇒ 〈φB(x, t)〉=
∫
D[φ˜B]D[φB] φB e−SB[φ˜B,φB]∫
D[φ˜B]D[φB] e−SB[φ˜B,φB]
≡ ρ(x, t). (1.29)
O u´ltimo passo reflete o fato que predic¸o˜es na˜o devem depender da escala RG µ, ja´ que este paraˆmetro
foi introduzido como uma escala arbitra´ria. Matematicamente, o vı´nculo se escreve da seguinte forma para
a func¸a˜o de um ponto:
GB1 (x, t; gˆB,ε)≡ G1(x, t; gˆ,µ). (1.30)
Como o lado esquerdo da equac¸a˜o (1.30) na˜o depende de µ, a dependeˆncia explı´cita de G1 em µ no lado
direito deve ser cancelada pela dependeˆncia implı´cita de gˆ(µ), expressa por
µ
d
dµ
G1(x, t,D, gˆ,µ)
∣∣∣
PN
= 0.
O ponto de normalizac¸a˜o PN e´ escolhido por convenieˆncia em q2 = 0 e −iω2 = µ2 e corresponde a
manter o paraˆmetro despido gˆB fixo. Aplicando a u´ltima equac¸a˜o, temos
µ
d
dµ
∣∣∣
PN
= µ
∂
∂µ
+µ
dgˆ
dµ
∣∣∣
PN︸ ︷︷ ︸
≡βg
· ∂
∂gˆ
,
onde a func¸a˜o beta βg foi definida. A func¸a˜o beta pode ser calculada como
βg(gˆ) =−εgˆ+ εgˆ2 2
(8pi)d/2
·Γ
( ε
2
)
. (1.31)
Finalmente, a restric¸a˜o para a func¸a˜o de um ponto pode ser escrita como
(
µ
∂
∂µ
+βg(gˆ)
∂
∂gˆ
)
G1 = 0. (1.32)
A equac¸a˜o acima e´ conhecida como equac¸a˜o de Callan-Symanzik.
Como a func¸a˜o beta captura a dependeˆncia da constante de acoplamento gˆ com a escala µ, suas raı´zes
correspondem a`s invariaˆncias de escala da teoria. Se o paraˆmetro na˜o varia com a mudanc¸a de escala,
a teoria definida pelo paraˆmetro gˆ se torna invariante de escala. As raı´zes de βg(gˆ) esta˜o em gˆ1 = 0 e
gˆ2 =
(8pi)d/2
2·Γ( ε2 )
. A primeira raiz em gˆ1 corresponde a λ = 0, o que significa que existe somente difusa˜o. A
segunda raiz, entretanto, na˜o e´ trivial e corresponde ao ponto fixo esta´vel para o acoplamento. Como
Γ
( ε
2
) ∼ 1ε , obtemos gˆ2 ∼ ε quando ε→ 0. O ponto fixo gˆ2 e´ crucial para o comportamento de escala e
determina o comportamento assinto´tico da func¸a˜o de um ponto como mencionado anteriormente.
Estamos interessados na dependeˆncia temporal da func¸a˜o de um ponto 〈φ(x, t)〉. Da equac¸a˜o (1.8) e da
ana´lise dimensional [φ] = d, segue que [G1] = [φ] = d. Consequentemente, a func¸a˜o de um ponto pode ser
reescrita como segue
G1(t, gˆ,µ) = µd · Gˆ1(µ2t, gˆ).
15
Capı´tulo 1. Introduc¸a˜o
Figura 1.11.: Func¸a˜o beta β(gˆ) =−εgˆ
(
1− gˆgˆ2
)
.
Assim, podemos transformar uma derivada em µ na equac¸a˜o de Callan-Symanzik (1.29) em uma derivada
temporal,
µ
∂
∂µ
G1(t, gˆ,µ) = µ
∂
∂µ
(
µd · Gˆ1(µ2t, gˆ)
)
=
(
d+2t
∂
∂t
)
G1(t, gˆ,µ).
Entretanto, uma sutileza surge por causa do termo de fronteira ρ0x˜(x,0) na ac¸a˜o. Uma ana´lise matema-
ticamente rigorosa [27] mostra que um termo adicional aparece na equac¸a˜o de Callan-Symanzik devido a`
densidade inicial. Finalmente, obtemos a equac¸a˜o completa de Callan-Symanzik,
(
d+2t
∂
∂t
+βg(gˆ)
∂
∂gˆ
−dφ0 ∂∂ρ0
)
ρ(t, gˆ,ρ0,µ) = 0, (1.33)
uma equac¸a˜o diferencial parcial de primeira ordem para a densidade de partı´culas renormalizada ρ que
pode ser resolvida pelos me´todos das caracterı´sticas, um me´todo padra˜o de ana´lise de equac¸o˜es diferenciais
parciais [31]. Introduzimos um paraˆmetro de fluxo l e escolhemos as curvas corretas (caracterı´sticas) onde
a equac¸a˜o diferencial parcial pode ser reduzida a um conjunto de equac¸o˜es diferenciais ordina´rias. As
soluc¸o˜es das equac¸o˜es diferenciais ordina´rias podem ser obtidas ao longo das caracterı´sticas e resolvem a
equac¸a˜o diferencial parcial.
O resultado importante deste procedimento padra˜o e´ ρ(t, gˆ,ρ0,µ)∼ t−d/2 para ε= dd−d = 2−d > 0 e
uma ana´lise completa (incluindo d = 2) revela,
ρ(t, gˆ,ρ0,µ)∼ t−d/2 =
 t
−1/2 se d = 1,
ln t · t−1 se d = 2,
t−1 se d > 2,
(1.34)
que e´ exatamente o resultado encontrado em (1.5). A abordagem do grupo de renormalizac¸a˜o para o
processo de aniquilac¸a˜o de pares reproduz exatamente o comportamento temporal assinto´tico sugerido por
simulac¸o˜es nume´ricas. O comportamento de longo prazo para a teoria de campo me´dio se torna exato
acima da dimensa˜o crı´tica dc = 2. Em dimenso˜es menores (d = 1), a hipo´tese de campo me´dio (fatorizac¸a˜o
da densidade de probabilidade) na˜o descreve o comportamento de escala acuradamente, a soluc¸a˜o ρ∼ t−1
deve ser renormalizada para ρ∼ t−1/2.
16
1.9. Heurı´stica
1.9. Heurı´stica
Apontamos anteriormente que a expansa˜o em lac¸os vai ale´m da descric¸a˜o de campo me´dio para a reac¸a˜o
de aniquilac¸a˜o. E´ possı´vel entender fisicamente por que a descric¸a˜o de campo me´dio se torna errada para
tempos longos em dimensa˜o um?
Considere uma rede uni-dimensional, onde quase todos os sı´tios da rede sa˜o ocupados por partı´culas da
espe´cie A, de modo que a reac¸a˜o de aniquilac¸a˜o A+A λ→ /0 pode em princı´pio acontecer em todo lugar.
Enta˜o, a dinaˆmica inicial e´ acuradamente descrita pela soluc¸a˜o da equac¸a˜o de campo me´dio com ρ−1.
A` medida que o tempo avanc¸a, os sı´tios da rede se tornam mais e mais diluidos e a taxa de reac¸a˜o
fica limitada pelo tempo de primeira passagem (para o encontro de dois agentes na rede) de um passeio
aleato´rio em d = 1, que escala exatamente com t−1/2. Enta˜o, o comportamento para tempos longos de ρ e´
renormalizado para ρ∼ t−1/t−1/2 = t−1/2.
Fica claro que a dimensionalidade do sistema e´ crucial, ja´ que ela determina a dimensionalidade do
processo de difusa˜o e portanto o processo de passeio aleato´rio subjacente. E´ conhecido que para d = 1, um
passeio aleato´rio retorna a um ponto especı´fico no espac¸o em um tempo finito quase certamente, enquanto
que para d > 2 isto na˜o ocorre [32]. Portanto, para d > 2, as correc¸o˜es devido a`s correlac¸o˜es e flutuac¸o˜es
estatı´sticas ao comportamento de campo me´dio ρ ∼ t−1 se tornam irrelevantes. Estes resultados sera˜o
importantes em alguns capı´tulos posteriores desta tese, em particular no capı´tulo referente ao artigo de
tı´tulo A importaˆncia de ser discreto no sexo.
17
18
Capı´tulo 2
Processo de contato com simbiose
Esta introduc¸a˜o e´ baseada nos resultados do artigo [3], publicado em Physical Review E.
Neste modelo estuda-se um processo de contato com duas espe´cies que interagem de maneira simbio´tica.
Cada sı´tio da rede pode estar vazio ou hospedar indivı´duos de espe´cie A e/ou B. Ocupac¸a˜o mu´ltipla da
mesma espe´cie e´ proibida. Simbiose e´ representada por uma taxa de morte reduzida µ < 1 para indivı´duos
em sı´tios com ambas as espe´cies presentes. Caso contra´rio, a dinaˆmica e´ igual a` do processo de contato
ba´sico, com criac¸a˜o (em sı´tios vizinhos vazios) a` taxa λ e morte de indivı´duos (isolados) a` taxa 1.
Sejam σi e ηi as varia´veis indicadoras de ocupac¸a˜o do sı´tio i pelas espe´cies A e B respectivamente. Os
estados permitidos para um sı´tio σi,ηi sa˜o (0,0), (0,1), (1,0), e (1,1). As transic¸o˜es (0,0)→ (1,0) e
(0,1)→ (1,1) ocorrem a taxa λrA, onde rA e´ a frac¸a˜o de primeiros vizinhos ocupados pela espe´cie A. De
forma semelhante, a taxa para as transic¸o˜es (0,0)→ (0,1) e (1,0)→ (1,1) e´ λrB, com rB sendo a frac¸a˜o
de primeiros vizinhos ocupados pela partı´cula B. As transic¸o˜es (0,1)→ (0,0) e (1,0)→ (0,0) ocorrem
a taxa 1, enquanto (1,1)→ (1,0) e (1,1)→ (0,1) ocorrem a` taxa µ. Este conjunto de taxas de transic¸a˜o
descrevem um par de processos de contato habitando a mesma rede. Para µ = 1 os dois processos evoluem
independentemente, mas para µ < 1 eles interagem simbioticamente ja´ que as taxas de aniquilac¸a˜o sa˜o
reduzidas em sı´tios onde ambas as espe´cies esta˜o presentes.
Para facilitar a construc¸a˜o da teoria de campos pertinente, vamos aqui formular o modelo de processo
de contato com simbiose em termos de reac¸o˜es envolvendo processos de Gribov. No processo de Gribov,
a aniquilac¸a˜o A+A→ A, a divisa˜o A→ A+A e o decaimento A→ /0 podem ser intepretados no contexto
da teoria de populac¸o˜es com representando nascimentos e mortes de indivı´duos da populac¸a˜o de interesse.
No processo de contato simbio´tico, permitimos que outra espe´cie B participe das interac¸o˜es na rede com a
espe´cie A. A maneira com que essas interac¸o˜es se da˜o deve induzir algum tipo de simbiose entre as enti-
dades que interagem. Nesta sec¸a˜o extenderemos as ide´ias relacionadas ao cla´ssico processo de percolac¸a˜o
direcionada para uma situac¸a˜o onde duas espe´cies A e B coexistem na rede, de modo que as interac¸o˜es
entre espe´cies diferentes induzam uma diminuic¸a˜o na taxa de morte dos agentes. Uma possibilidade para
tal e´ acoplarmos dois processos de Gribov de modo que as taxas de morte das populac¸o˜es A e B sejam
diminuidas com os encontros entre as partı´culas na rede, de modo que o processo microsco´pico possa ser
descrito por A
α

α′
A+A B
β


β′
B+B
A
µe−γB
⇀ /0 B µe
−γA
⇀ /0
(2.1)
Desta forma, temos dois processos de Gribov conectados pelas taxas de morte das partı´culas sendo da forma
µe−γB para as partı´culas A e da forma µe−γA para as partı´culas B. A presenc¸a da partı´cula ‘parceira’ faz com
que a taxa de morte seja reduzida. A ide´ia aqui e´ seguir o roteiro em que se mapeia o processo estoca´stico
descrito acima em uma formulac¸a˜o que permite a aplicac¸a˜o das ide´ias e me´todos da teoria quaˆntica de
19
Capı´tulo 2. Processo de contato com simbiose
campos [33]. Nesta formulac¸a˜o fica manifesta a presenc¸a de uma ac¸a˜o efetiva que permite desenvolver tais
me´todos. Partindo desta ac¸a˜o efetiva, me´todos perturbativos permitem estimativas dos expoentes crı´ticos
com a consequente determinac¸a˜o da classe de universalidade do modelo. Este foi o procedimento realizado
em [3], que contemplou tambe´m resultados simulacionais obtidos por coautores. Resultados simulacionais
para o modelo formulado na rede como descrito acima revelaram que a transic¸a˜o de fase entre estados ativos
e absorventes e´ contı´nua e que a taxa de criac¸a˜o crı´tica λc e´ reduzida na presenc¸a de simbiose. Isto significa
que a perda de uma espe´cie podera´ rapidamente levar a extinc¸a˜o ja´ que enta˜o o sistema constituira´ um
processo de contato simples que estara´ operando com λ < λc. A expansa˜o em ε oriunda do procedimento
GR indicou que o modelo se apresenta na classe de universalidade PD, fato consistente com os resultados
obtidos numericamente para o comportamento crı´tico, com algumas anomalias aparecendo no expoente z
associadas a`s flutuac¸o˜es do paraˆmetro de ordem para tempos longos, onde se espera um desenvolvimento
do tipo ∼ t1/z. Este fenoˆmeno fica mais pronunciado para valores fracos de simbiose, representados por
valores de µ perto de 1. Pode-se conjecturar que com simbiose forte o sistema e´ atraido rapidamente para
o ponto fixo da PD enquanto que para simbiose fraca o sistema executa longas excurso˜es em regimes nos
quais o comportamento tipo PD na˜o e´ evidente, antes de finalmente retornar para as vizinhanc¸as do ponto
fixo PD.
20
PHYSICAL REVIEW E 86, 011121 (2012)
Symbiotic two-species contact process
Marcelo Martins de Oliveira,1,* Renato Vieira Dos Santos,2 and Ronald Dickman2,†
1Departamento de Fı´sica e Matema´tica, Campus Alto Parapeba, Universidade Federal de Sa˜o Joa˜o del Rei,
36420-000 Ouro Branco, Minas Gerais, Brazil
2Departamento de Fı´sica and National Institute of Science and Technology for Complex Systems, Instituto de Cieˆncias Exatas,
Universidade Federal de Minas Gerais, 30123-970 Belo Horizonte, Minas Gerais, Brazil
(Received 24 May 2012; published 19 July 2012)
We study a contact process (CP) with two species that interact in a symbiotic manner. In our model, each site
of a lattice may be vacant or host individuals of species A and/or B; multiple occupancy by the same species
is prohibited. Symbiosis is represented by a reduced death rate μ < 1 for individuals at sites with both species
present. Otherwise, the dynamics is that of the basic CP, with creation (at vacant neighbor sites) at rate λ and
death of (isolated) individuals at a rate of unity. Mean-field theory and Monte Carlo simulation show that the
critical creation rate λc(μ) is a decreasing function of μ, even though a single-species population must go extinct
for λ < λc(1), the critical point of the basic CP. Extensive simulations yield results for critical behavior that are
compatible with the directed percolation (DP) universality class, but with unusually strong corrections to scaling.
A field-theoretic argument supports the conclusion of DP critical behavior. We obtain similar results for a CP
with creation at second-neighbor sites and enhanced survival at first neighbors in the form of an annihilation rate
that decreases with the number of occupied first neighbors.
DOI: 10.1103/PhysRevE.86.011121 PACS number(s): 05.50.+q, 05.70.Ln, 05.70.Jk, 02.50.Ey
I. INTRODUCTION
Absorbing-state phase transitions have attracted much
interest in recent decades, as they appear in a wide variety
of problems such as population dynamics, heterogeneous
catalysis, interface growth, and epidemic spreading [1–5].
Interest in such transitions has been further stimulated by
recent experimental realizations [6,7].
The absorbing-state universality class associated with
directed percolation (DP) has proven to be particularly robust.
The DP-like behavior appears to be generic for absorbing-
state transitions in models with short-range interactions and
lacking a conserved density or symmetry beyond translational
invariance [8,9]. In contrast, models possessing two absorbing
states linked by particle-hole symmetry belong to the voter
model universality class [10].
The contact process (CP) [11] is probably the best under-
stood model exhibiting an absorbing-state phase transition; it
has been known for many years to belong to the DP class.
The CP can be interpreted as a stochastic birth-and-death
process with a spatial structure. As a control parameter (the
reproduction rate λ) is varied, the system undergoes a phase
transition between extinction and survival. In this context it is
natural to seek a manner to include symbiotic interactions in
the CP. In the present work, this is done by allowing two
CPs (designated as species A and B) to inhabit the same
lattice. The two species interact via a reduced death rate μ
at sites occupied by individuals of both species. (Aside from
this interaction, the two populations evolve independently.)We
find, using mean-field theory andMonte Carlo simulation, that
the symbiotic interaction favors survival of amixed population,
in that the critical reproduction rate λc decreases as we reduce
*mmdeoliveira@ufsj.edu.br
†dickman@fisica.ufmg.br
μ. Note that for λ(μ) < λ < λ(1), only mixed populations
survive; in isolation, either species must go extinct.
In addition to its interest as a simple model of symbiosis,
the critical behavior of the two-species CP is intriguing in
the context of nonequilibrium universality classes. By analogy
with the (equilibrium) n-vector model, in which the critical
exponents depend on the number of spin components n, one
might imagine that the presence of two species would modify
the critical behavior. Using extensive simulations, we find that
the critical behavior is consistent with that of DP, althoughwith
surprisingly strong corrections to scaling. An argument based
on field theory supports the conclusion of DP scaling. We note
that our result agrees with that of Janssen, who studied general
multispecies DP processes [12]. Similar conclusions apply to
a related model, a CP with creation at second-neighbor sites
and enhanced survival at first neighbors, in the form of an
annihilation rate that decreases with the number of occupied
first neighbors. (In this case the two species inhabit distinct
sublattices.)
The balance of this paper is organized as follows. In Sec. II
we define the models and analyze them using mean-field
theory. In Sec. III we present our simulation results and in
Sec. IV we discuss a field-theoretic approach. Section V is
devoted to discussion and conclusions.
II. MODELS AND MEAN-FIELD THEORY
To begin we review the definition of the basic contact
process. Following the usual nomenclature, we refer to an
active site as being occupied by a particle and an inactive
one as vacant. The CP [11] is a stochastic interacting particle
system defined on a lattice, with each site i either occupied
by a particle [σi(t) = 1] or vacant [σi(t) = 0]. Transitions
from σi = 1 to 0 occur at a rate of unity, independent of
the neighboring sites. The reverse transition, a vacant site
becoming occupied, is possible only if at least one of its nearest
neighbors (NNs) is occupied: The transition from σi = 0 to 1
011121-11539-3755/2012/86(1)/011121(9) ©2012 American Physical Society
DE OLIVEIRA, DOS SANTOS, AND DICKMAN PHYSICAL REVIEW E 86, 011121 (2012)
occurs at rate λr , where r is the fraction of NNs of site i
that are occupied. Thus the state σi = 0 for all i is absorbing.
At a certain critical value λc the system undergoes a phase
transition between the active and the absorbing state [11].
The CP has been studied intensively via series expansion and
Monte Carlo simulation and its critical properties are known
to high precision [1,3–5,13].
We now define a two-species symbiotic contact process
(2SCP). Let the indicator variables for occupation of site i
by species A and B be σi and ηi , respectively. The allowed
states for a site (σi,ηi) are (0,0), (0,1), (1,0), and (1,1). The
transitions (0,0) → (1,0) and (0,1) → (1,1) occur at rate λrA,
with rA the fraction of NNs bearing a particle of species
A. Similarly, the rate for the transitions (0,0) → (0,1) and
(1,0) → (1,1) is λrB , with rB the fraction of NNs bearing
a particle of species B. The transitions (0,1) → (0,0) and
(1,0) → (0,0) occur at a rate of unity, whereas (1,1) → (1,0)
and (1,1) → (0,1) occur at rate μ. This set of transition rates
describes a pair of contact processes inhabiting the same
lattice. For μ = 1 the two processes evolve independently, but
for μ < 1 they interact symbiotically since the annihilation
rates are reduced at sites with both species present. We note
that the rates are symmetric under exchange of species labels
A and B.
We also study a CP with creation at second-neighbor sites.
In Ref. [14] a modified CP was defined as follows.
(i) In addition to creation at NNs, at rate λ1, we allow
creation at second neighbors, at rate λ2. For bipartite lattices
such as the ring or the square lattice, λ1 is the rate of creation
in the opposite sublattice, while λ2 is the rate in the same
sublattice as the replicating particle.
(ii) The annihilation rate at a given site is 1 + νn2, with n
denoting the number of occupied NNs.
For ν > 0, the presence of particles in one sublattice tends
to suppress their survival in the other, leading to the possibility
of sublattice ordering, as discussed in Ref. [14].
Suppose now that λ1 = 0, and let λ2 ≡ λ. Then the
populations in the two sublattices constitute distinct species
since creation is always in the same sublattice. For ν < 0
moreover, the two species interact in a symbiotic manner,
analogous to that in the two-species CP defined above. (For
ν = 0 the two sublattices evolve independently.) We call this
process the symbiotic sublattice contact process (SSLCP).
Both the 2SCP and SSLCP possess four phases: the
fully active phase (nonzero populations of both species), a
symmetric pair of partly active phases (only one species
present), and the inactive phase (all sites inactive). The latter
is absorbing while the partly active phases represent absorbing
subspaces of the dynamics. (That is, a species cannot reappear
once it goes extinct.) Let λc,0 denote the critical creation rate
of the basic CP. In the 2SCP with μ = 1 (or the SSLCP
with ν = 0), the critical creation rate must be λc,0. The same
applies for the transitions from the partly active phases to the
absorbing one, regardless of the value of μ or ν. Intuitively,
in the presence of symbiotic interactions, one expects the
transition from the fully active to the absorbing phase to
occur at some λc < λc,0 since the annihilation rate is reduced.
Since this expectation is borne out numerically, the partly
active phases are of little interest as they are not viable in the
vicinity of the fully active-absorbing phase transition. Under-
standing the latter transition is the principal objective of this
study.
As a first step in characterizing the phase diagrams of the
models, we develop mean-field approaches. The derivation
of a dynamic mean-field theory (MFT) for an interacting
particle system begins with the equations of motion for the
set of one-site probabilities (or, more generally, the n-site
joint probability distribution) [1]. In this equation, the n-site
probability distribution is inevitably coupled to the distribution
for n + 1 or more sites. An n-site MFT is obtained by
estimating the latter distribution(s) in terms of that for n sites.
Here we consider the simplest cases, n = 1 and 2.
Consider the 2SCP in the one-site approximation. Denoting
the probabilities for a given site to be vacant, occupied
by species A only, by species B only, and doubly oc-
cupied by p0, pA, pB , and pAB , respectively, assuming
spatial homogeneity, and factorizing two-site joint probabil-
ities (p[(σi,ηi),(σj ,ηj )] = p[(σi,ηi)]p[(σj ,ηj )]) one readily
obtains the equations
dp0
dt
= −λp0(ρA + ρB) + pA + pB,
dpA
dt
= λp0ρA + μpAB − (1 + λρB)pA, (1)
dpB
dt
= λp0ρB + μpAB − (1 + λρA)pB,
dpAB
dt
= λ(pAρB + pBρA) − 2μpAB,
where ρA = pA + pAB and ρB = pB + pAB . If one species is
absent (so that, say, pB = pAB = 0) this system reduces to the
MFT for the basic contact process p˙A = λpA(1 − pA) − pA
with a critical point at λ = 1. To study the effect of symbiosis
we seek a symmetric solution pA = pB = p. In this case one
readily finds the stationary solution
p = μ
2λ(1 − μ) [2(1 − μ) − λ +
√
λ2 − 4μ(1 − μ)] (2)
and
pAB =
λp2
μ − λp . (3)
For μ > 1/2, p grows continuously from zero at λ = 1,
marking the latter value as the critical point. The activity
grows linearly, p  [μ/(2μ − 1)](λ − 1), in this regime. For
μ < 1/2, however, the expression is already positive for
λ = √4μ(1 − μ) < 1 and there is a discontinuous transition
at this point. The value μ = 1/2 may be viewed as a tricritical
point; here p ∼ √λ − 1 for λ > 1. Numerical integration of
the MFT equations confirms the above results. For μ < 1/2,
MFT in fact furnishes the spinodal values of λ. For a given set
of initial probabilities, the numerical integration converges to
the active stationary solution for λ > λ∗ and to the absorbing
state for smaller values of λ. For the most favorable initial
condition, i.e., pAB(0) → 1, λ∗ → λ(−) =
√
4μ(1 − μ), the
lower spinodal, while for a vanishing initial activity ρA,
ρB → 0, λ∗ → λ(+) = 1. The stationary activity at λ∗ is
nonzero. Figure 1 shows the stationary probabilities versus
λ for μ = 1/4.
011121-2
SYMBIOTIC TWO-SPECIES CONTACT PROCESS PHYSICAL REVIEW E 86, 011121 (2012)
FIG. 1. (Color online) Density p of species A (bottom curve) and
of doubly occupied sitespAB (top curve) in the one-site approximation
for the 2SCP, with μ = 0.25.
The two-site MFT for the one-dimensional 2SCP involves
ten pair probabilities and a set of 32 transitions. The resulting
phase diagram is qualitatively similar to that of the one-site
MFT. For μ > 0.75, the transition is continuous and occurs
at λ = 2, the same value as for the basic CP at this level of
approximation. There is a tricritical point at μ = 0.75, below
which the transition is discontinuous; Fig. 2 shows the phase
boundary.
The one-siteMFT for the SLCPwas developed in Ref. [14].
Adapted to the present case (creation only in the same
sublattice, symbiotic interaction), the equation is
dρA
dt
= −(1 − νq2ρ2B)ρA + λ2ρA(1 − ρA) (4)
and similarly for ρA ® ρB , on a lattice of coordination number
q. (Here ρj denotes the fraction of occupied sites in sublattice
j .) As we seek a symmetric solution, we set ρA = ρB . The
resulting equation yields a continuous phase transition at
λ = 1, independent of ν. (Note that ν must be greater than
−1/16; smaller values correspond to a negative annihilation
rate, for ρ near unity.) The two-site approximation is likely
FIG. 2. (Color online) Phase boundary in the λ-μ plane as given
by two-site MFT for the 2SCP on the line. The curved portion
represents the lower spinodal λ(−)(μ).
to provide a better description of the SSLCP since in this
case the nearest-neighbor double occupancy probability is an
independent variable, analogous to pAB in the one-site MFT
of the 2SCP. Since such an analysis is unlikely to result in
additional insights, we shall not pursue it here.
Although MFT predicts a discontinuous phase transition
in the 2SCP in any number of dimensions, such a transition
is not possible in one-dimensional systems with short-range
interactions and free of boundary fields [15]. In one dimension
the active-absorbing transition should be continuous, as we
have indeed verified in simulations. Although our simulations
show no evidence of a discontinuous transition in two
dimensions (d = 2), such a transition remains a possibility
for d > 2, for small values of μ. A discontinuous transition
might also arise under rapid particle diffusion, as this generally
favors mean-field-like behavior.
III. SIMULATIONS
We performed extensive Monte Carlo simulations of the
2SCP on rings and the square lattice (with periodic boundaries)
and of the SSLCP on rings. A general observation is that
both models appear to be more strongly affected by finite-size
corrections than is the basic CP.
In the simulation algorithm for the two-species CP, we
maintain two lists, i.e., of singly and doubly occupied sites.
Let Ns and Nd denote, respectively, the numbers of such sites,
so that Np = Ns + 2Nd is the number of particles. The total
rate of (attempted) transitions is λNp + Ns + 2μNd ≡ 1/t ,
where t is the time increment associated with a given step in
the simulation. At each such step, we choose among the events:
(i) creation attempt by an isolated particle, with probability
λNst ; (ii) creation attempt by a particle at a doubly occupied
site, with probability 2λNdt ; (iii) annihilation of an isolated
particle, with probability Nst ; and (iv) annihilation of a
particle at a doubly occupied site, with probability 2μNdt .
Once the event type is selected we choose a site i from the
appropriate list. In the case of annihilation, a particle is simply
removed, while creation requires the choice of a neighbor j of
site i and can proceed only if j is not already occupied by a
particle of the species to be created. For creation by a particle
at a doubly occupied site, the species of the daughter particle
is chosen to be A or B with equal probability and similarly for
annihilation at a doubly occupied site.
In simulations of the SSLCP we maintain a list of occupied
sites. At each step a site is selected from the list; an attempt
to create a new particle, at one of the second-neighbor sites,
is chosen with probability p = λ/(1 + λ2 + μn21); the site is
vacated with the complementary probability 1 − p. The time
increment associated with each event is t = 1/Np, with Np
the number of particles just prior to the event.
A. Results: The 2SCP in one dimension
We studied the 2SCP using three values of μ: 0.9, 0.75,
and 0.25. While the first case may be seen as a relatively small
perturbation of the usual CP (μ = 1), the third represents a
very strong departure from the original model. We perform
three kinds of studies: quasistationary (QS) [16], initial
decay (starting from a maximally active configuration), and
011121-3
DE OLIVEIRA, DOS SANTOS, AND DICKMAN PHYSICAL REVIEW E 86, 011121 (2012)
FIG. 3. (Color online) Quasistationary simulation of the one-
dimensional 2SCP: moment ratios mρ (solid symbols) and mq (open
symbols) versus 1/L for the one-dimensional model with μ = 0.75.
The top curve in each pair is for λ = 3.0336 and the bottom is for
λ = 3.0337.
spreading, in which the initial condition is a doubly occupied
site in an otherwise empty lattice. Although the critical
value λc(μ) can be estimated using each method, spreading
simulations proved the most effective in this regard.
In the QS simulations, we study system sizes 800, 1600,
3200, 6400, and 12800, with each run lasting 107 time units;
averages and uncertainties are calculated over 10–80 runs. We
use three well established criteria to estimate the critical value:
(i) power-law dependence of the order parameter on system
size ρ ∼ L−β/ν⊥ , (ii) power-law dependence of the lifetime
τ ∼ Lz, and (iii) convergence of the moment ratio mρ(L) to
a finite limit mc as L → ∞ [17]. Here mρ ≡ 〈ρ2〉/〈ρ〉2. The
order parameter is defined as the density of individuals, i.e.,
ρ = (NA + NB)/L. A related quantity of interest is the density
q of doubly occupied sites; the moment ratiomq is defined in a
manner analogous to mρ . Two further quantities of interest are
the scaled variances of ρ and q; we define χρ ≡ Ld var(ρ) and
similarly for χq . The expected critical behavior is χ ∼ Lγ/ν⊥ ,
where the critical exponent γ satisfies the hyperscaling relation
γ = dν⊥ − 2β [1].
A preliminary estimate of λc is obtained from the crossings
of mρ for successive system sizes L and 2L. For μ = 0.75, for
example, this yields λc = 3.0337. The plot of mρ and mq (see
Fig. 3) indicates that λc > 3.0336 (since mρ curves upward),
while the slight downward curvature for λ = 3.0037 suggests
that this value may be slightly above critical. This graph also
suggests that mρ and mq approach the same limiting value,
despite marked differences for smaller system sizes. Table I
FIG. 4. (Color online) Initial-decay simulation of the 2SCP in
one dimension: decay of the particle density ρ (upper curve) and
the density q of doubly occupied sites in initial-decay studies with
μ = 0.9, λ = 3.2273, and system size L = 51 200. The slopes of the
regression lines are −0.161 (ρ) and −0.162 (q).
summarizes our findings for the critical parameters obtained
from QS simulations.
The initial-decay studies use, as noted above, an initial
configuration with all sites doubly occupied. The activity then
decays, following a power law ρ ∼ t−δ at the critical point [18]
until it saturates at its QS value. The larger the system size,
the longer the period of power-law decay and the more precise
the resulting estimate for the critical exponent δ; here we use
L = 25 600 or 51 200. Averages are calculated over 500–3000
realizations. As the order parameter decays, its fluctuations
build up; at the critical point, the moment ratio is expected
to follow m − 1 ∼ t1/z [19]. Since we expect ρ and q to
scale in the same manner, we define exponents δρ and δq ,
and similarly zρ and zq , based on the behavior of mρ and mq ,
respectively. Figure 4 shows, for μ = 0.9, that ρ and q decay
in an analogous manner and follow power laws at long times,
although there are significant deviations from a simple power
law at short times; the decay exponents are consistent with the
value of δ for directed percolation in one space and one time
dimension (see Table II). The growth of fluctuations follows
a more complicated pattern, as shown in Fig. 5. At relatively
short times, mρ − 1 ∼ t1/zρ , with zρ = 1.63(2), which is not
very different from the DP value; mq − 1 also grows as a
power law in this regime, but with an apparent exponent of
zq = 2.06(1). At longer times zρ appears to take a smaller
value [1.31(1) for 7.5 < ln t < 10.5], while zq shifts to a value
close to that of DP [1.61(1) for 10 < ln t < 14]. The reason
for the distinct behaviors of mρ and mq , in marked contrast
TABLE I. Two-species symbiotic CP in one dimension: results from QS simulations, with L = 800, 1600, 3200, 6400, and 12800. For
μ = 0.25 the maximum size is 6400.
μ λc β/ν⊥ z mρ mq (γ /ν⊥)ρ (γ /ν⊥)q
0.9 3.2273(1) 0.25(2) 1.50(5) 1.168(12) 1.164(4) 0.627(20) 0.474(7)
0.75 3.03370(5) 0.241(6) 1.64(5) 1.163(10) 1.166(2) 0.528(6) 0.486(1)
0.25 1.76297(1) 0.248(3) 1.56(4) 1.168(3) 1.169(3) 0.500(1) 0.492(2)
CP or DP 3.29785 0.25208(5) 1.5807(1) 1.1736(1) 0.49584(9)
011121-4
SYMBIOTIC TWO-SPECIES CONTACT PROCESS PHYSICAL REVIEW E 86, 011121 (2012)
TABLE II. Two-species symbiotic CP in one dimension: results
from initial-decay studies.
μ L λc δ zρ zq
0.9 51200 3.2273 0.161(1) 1.44(1) 1.54(2)
0.75 25600 3.0337 0.1625(10) 1.48(4) 1.55(4)
0.25 51200 1.76297 0.1581(3) 1.56(1) 1.58(1)
DP 0.1599 1.5807(1)
with the similar scaling of ρ(t) and q(t), is unclear. While
scaling anomalies are observed in the initial-decay studies for
μ = 0.9 and 0.75, for strong symbiosis (μ = 0.25) they are
absent, as seen in Table II, which summarizes the results of
the initial-decay studies. (In this table, the values listed for zρ
and zq reflect the latter part of the evolution, during which the
order parameter decays in the expected manner.)
In the spreading studies, each realization runs to amaximum
time of tm (unless it falls into the absorbing state prior to this).
The system size is taken large enough so that activity never
reaches the boundary. Here we use tm = 2 × 106 andL = 105;
averages are calculated over 104 or 2 × 104 realizations. At
the critical point, one expects to observe power-law behavior
of the survival probability P (t) ∼ t−δ , the mean number of
particles n(t) ∼ tη, and the mean-square distance of particles
from the initial seed R2(t) ∼ t zs [18]. Here δ is the same
exponent as governs the initial decay of the activity and zs
is related to the dynamic exponent z via zs = 2/z. Deviations
from asymptotic power laws, indicating off-critical values of
the control parameter λ, are readily identified in spreading
simulations, leading to precise estimates for λc.
The spreading behavior is characterized by clean power
laws, as illustrated in Fig. 6. As this plot makes clear, the
mean particle number np and the mean number of doubly
occupied sites n2 growwith the same critical exponent. Precise
estimates of the spreading exponents are obtained via analysis
of local slopes such as δ(t), defined as the inclination of a
FIG. 5. (Color online) Initial-decay simulation of the 2SCP in
one dimension: growth of fluctuations in ρ (bottom curve) and q
(top curve) for the same parameters as in Fig. 4. The slopes of the
regression lines are (from left to right) 0.613, 0.694, and 0.649.
FIG. 6. (Color online) Spreading simulation of the 2SCP in one
dimension: survival probability P (t), total particle number np(t),
number of doubly occupied sites n2(t), and mean-square distance
from seed R2(t). The parameters are μ = 0.25 and λ = 1.76 297.
least-square linear fit to the data (on logarithmic scales), on
the interval [t/a, at]. (The choice of the factor a represents
a compromise between high resolution, for smaller a, and
insensitivity to fluctuations, for larger values; here we use
a = 4.59.) Curvature in a plot of a local slope versus 1/t
signals an off-critical value. Figure 7 shows the behavior of
δ(t) for μ = 0.25. The spreading exponents, summarized in
Table III, are in good agreement with the values for DP in 1+1
dimensions. (We note that in all three cases, ηp = η2 to within
uncertainty.)
B. Contact process with creation at second neighbors
We studied the SSLCP using QS and initial-decay simula-
tions. The results from the former, based on finite-size scaling
analysis of studies using L = 800, 1600, 3200, 6400, and
12 800, are summarized in Table IV. The value of ν⊥ was
estimated (for ν = −0.1 only) via analysis of the derivatives
|dm/dλ|, d ln τ/dλ, and d ln ρp/dλ in the neighborhood of
the critical point. Finite-size scaling implies that the derivatives
follow |dx/dp| ∝ L1/ν⊥ (here x stands for any of the quantities
FIG. 7. (Color online) Spreading simulation of the 2SCP in one
dimension: local slope δ(t) versus 1/t forμ = 0.25 and (from bottom
to top) λ = 1.7629, 1.762 95, 1.762 97, and 1.7630.
011121-5
DE OLIVEIRA, DOS SANTOS, AND DICKMAN PHYSICAL REVIEW E 86, 011121 (2012)
TABLE III. One-dimensional 2SCP: results from spreading
simulations.
μ λc δ η zs
0.9 3.2273 0.165(1) 0.310(1) 1.257(2)
0.75 3.0337 0.1595(5) 0.3180(5) 1.265(5)
0.25 1.76297 0.158(1) 0.315(3) 1.265(10)
DP 3.29785 0.15947(5) 0.31368(4) 1.26523(3)
mentioned). We estimate the derivatives via least-squares
linear fits to the data on an interval that includes λc. (The
intervals are small enough that the graphs show no significant
curvature.) Linear fits to the data for m, ln ρp, and ln τ yield
1/ν⊥ = 0.94(2), 0.92(3), and (again) 0.92(3), respectively,
leading to the estimate ν⊥ = 1.08(3).
Results of the initial-decay studies are summarized in
Table V. As in the two-species CP, the value of z obtained
from analysis of m(t) appears to be smaller than the DP
value, whereas the result obtained from the QS simulations
is consistent with that of DP.
C. Two-species contact process in two dimensions
We performed extensive Monte Carlo simulations of the
2SCP on square lattices using both initial-decay and QS
simulations. In order to locate the critical point with good
precision, we study the initial decay of the particle density,
starting from a maximally active initial condition (all sites
doubly occupied). We use lattices of linear size L = 4000 and
average over at least 20 different realizations. Figure 8 shows
the decay of ρ(t) for μ = 0.1. After an initial transient, during
which the density evolves slowly, the particle density follows
a power law with δ = 0.46(1), which is compatible with the
value [δ = 0.4523(10)] for the DP class in 2+ 1 dimensions.
The transient behavior lasts longer the larger is μ, as shown in
Fig. 9. However, the relaxation is seen to cross over to DP-like
behavior for all values studied, except for μ = 0.9, for which
the transient regime persists throughout the entire simulation.
Having determined λc to good precision in the initial-
decay studies, we perform QS simulations of the model on
square lattices of linear size L = 20,40, . . . ,320 with periodic
boundaries. Figure 10 shows moment-ratio crossings and
the finite-size scaling behavior of the density and lifetime
for μ = 0.1. For the larger sizes we obtain β/ν⊥ = 0.78(1)
and z = 1.74(2), in good agreement with the best estimates
for DP in 2+ 1 dimensions. Simulation results for the two-
dimensional model are summarized in Table VI.
TABLE IV. One-dimensional SSLCP: results from quasistation-
ary simulations.
ν λc β/ν⊥ z mc ν⊥
−0.05 3.1489(1) 0.235(8) 1.63(5) 1.154(5)
−0.1 2.8878(1) 0.242(1) 1.612(12) 1.161(3) 1.08(3)
−0.2 2.0502(1) 0.253(6) 1.59(1) 1.170(6)
DP 3.29785 0.25208(5) 1.5807(1) 1.1736(1) 1.096854(4)
TABLE V. One-dimensional SSLCP: results from initial-decay
simulations.
μ L λc δ z
−0.05 50000 3.1489 0.1458(5) 1.45(2)
−0.1 50000 2.8878 0.1484(7) 1.45(3)
−0.2 20000 2.0503 0.1597(3) 1.53(1)
DP 0.1599 1.5807(1)
IV. FIELD-THEORETIC ANALYSIS
In this section we extend the field theory or continuum
representation of DP to the two-species case to determine
whether the presence of additional species changes the scaling
behavior. Since the theory of DP has been known for some
time, we give a bare outline of this analysis, referring the
reader to Refs. [20–24] for details. To begin, we modify the
lattice model so as to facilitate the definition of a continuum
description following the Doi-Peliti formalism [20,21], which
has been applied to DP in Refs. [22,24]. (The latter study
applies the Wilson renormalization group to the problem.)
In the Doi-Peliti formalism, the master equation governing
the evolution of the probability vector |P (t)〉 ≡ ∑C p(C,t)|C〉
(the sum is over all configurations) is written in the form
d|P 〉/dt = L|P 〉, where the evolution operatorL is composed
of creation and annihilation operators. Starting from this
“microscopic” description, one derives an effective action
S via a path-integral mapping. Then, taking the continuum
limit, one arrives at a field theory for the model. Of the many
lattice models that belong to the DP universality class, the
simplest to analyze in this manner is the Malthus-Verhulst
process (MVP). Here, each site i of a lattice hosts a number
ni > 0 of particles. The transitions at a given site are creation
(ni → n1 + 1) at rate λni and annihilation (ni → n1 − 1) at
rate μni + νni(ni − 1). In addition, particles hop between
nearest-neighbor sites at rate D.
For the MVP on a ring of 
 sites, one has the
set of basis configurations |n1, . . . ,n
〉. Letting ci
and c†i denote, respectively, annihilation and creation
FIG. 8. (Color online) The 2SCP in two dimensions: density of
active sites starting from a maximally active initial condition for
μ = 0.1 and λ values increasing from λ = 0.742 (bottom curve) to
from λ = 0.745 (top curve). The system size is L = 4000.
011121-6
SYMBIOTIC TWO-SPECIES CONTACT PROCESS PHYSICAL REVIEW E 86, 011121 (2012)
FIG. 9. (Color online) The 2SCP in two dimensions: density of
active sites starting from a maximally active initial condition for
μ = 0.1, 0.25, 0.5, 0.75, and 0.9 (from top to bottom) and λ = λc(μ)
(see Table VI). The slope of the dashed line is −0.45. The system
size is L = 4000.
operators associated with site i, we have, by definition,
ci |n1, . . . ,ni, . . . ,n
〉 = ni |n1, . . . ,ni − 1, . . . ,n
〉 and
c
†
i |n1, . . . ,ni, . . . ,n
〉 = |n1, . . . ,ni + 1, . . . ,n
〉. Then the
evolution operator for the MVP is
LMVP =
∑
i
[λ(c†i − 1)c†i ci + (1 − c†i )(μ + νci)ci]
+ D
2
∑
i
[(c†i − c†i+1)ci+1 + (c†i − c†i−1)ci−1]. (5)
Following the steps detailed in Ref. [22], one arrives at the
effective action for the MVP,
SMVP =
∫
dt
∫
dx[ ˆψ(∂t +w −D∇2)ψ + ν ˆψψ2 − λ ˆψ2ψ],
(6)
where w ≡ μ − λ, the continuum limit has been taken, and
terms higher than third order have been discarded, as they are
irrelevant to critical behavior. (We recall that ˆψ(x,t) is an aux-
iliary field that arises in themapping. The operator that governs
the evolution of the probability generating function is given by
the functional integral Ut =
∫ Dψ ∫ D ˆψ exp[−S(ψ, ˆψ)]; see
Refs. [21,22].)
The action of Eq. (6) is equivalent that of DP and serves
as the starting point for renormalization-group (RG) analyses
[8,9,24]. (One usually imposes the relation ν = λ via a
rescaling of the fields, but this is not needed here.) In the
RG analysis the bilinear term naturally defines the propagator,
while the cubic terms correspond to the vertices shown in
Fig. 11. These terms lead, via diagrammatic analysis, to
a nontrivial DP fixed point below dc = 4 dimensions. The
FIG. 10. (Color online) The 2SCP in twodimensions:QSmoment
ratio of particles vs λ for μ = 0.1 (the system sizes are L =
40, 80, 160, and 320 in order of steepness). The inset shows the
QS density of active sites (circles), density of doubly occupied sites
(squares), and lifetime of the QS state (triangles) for μ = 0.1.
one-loop diagrams that yield, to lowest order, the recursion
relations for parameters w, λ, and ν are shown in Fig. 12.
Now consider the two-species CP. To formulate a minimal
field theory, we consider a two-species MVP; call it MVP2.
Let mi and ni denote, respectively, the number of particles of
species A and B at site i and let ai and a†i , and bi and b
†
i , denote
the associated annihilation and creation operators. We require
the annihilation rate for species A to be a decreasing function
of ni and vice versa; a simple choice for the annihilation rate
of an A particle at site i is μ exp[−γ ni], where γ is a positive
constant, and similarly for B particles, with ni replaced by mi .
This corresponds to the evolution operator
LMVP2 =
∑
i
[λ(a†i − 1)a†i ai + (1 − a†i )(μe−γ bi†bi + νai)ai]
+D
2
∑
i
[(a†i − a†i+1)ai+1 + (a†i − a†i−1)ai−1]
+
∑
i
[λ(b†i − 1)b†i bi + (1− b†i )(μe−γ ai†ai + νbi)bi]
+D
2
∑
i
[(b†i − b†i+1)bi+1 + (b†i − b†i−1)bi−1]. (7)
To avoid ambiguity, we interpret the exponentials as being
in normal order, i.e., all creation operators to the left of
annihilation operators. Recalling that terms with four or
more fields are irrelevant, we may expand the exponentials,
retaining only the terms proportional to b†i bi and a
†
i ai . Using
: X : to denote the normal-ordered expression of X, it is
TABLE VI. Simulation: critical parameters for the two-dimensional 2SCP.
μ λc β/ν⊥ z δ mp mq
0.9 1.64515(5) 0.63(5) 1.95(5) >0.35 1.40(2) 1.52(2)
0.75 1.61640(5) 0.73(5) 1.78(6) 0.44(3) 1.32(3) 1.33(3)
0.5 1.47290(5) 0.74(3) 1.72(3) 0.46(2) 1.298(8) 1.322(8)
0.25 1.13730(5) 0.76(2) 1.73(2) 0.45(2) 1.30(2) 1.31(2)
0.1 0.743160(5) 0.78(1) 1.73(2) 0.46(1) 1.305(10) 1.315(12)
CP or DP 1.64874(4) 0.797(3) 1.7674(6) 0.4523(10) 1.3264(5)
011121-7
DE OLIVEIRA, DOS SANTOS, AND DICKMAN PHYSICAL REVIEW E 86, 011121 (2012)
FIG. 11. Two three-field vertices in the field theory of DP. Lines
exiting a vertex correspond to ˆψ , while those entering correspond
to ψ .
straightforward to show that
: e−γ b
†b := 1 − (1 − e−γ )b†b + I ≡ 1 − γ¯ b†b + I, (8)
where I consists of terms with four or more operators. (With
the truncation comes the possibility of a negative rate, but
this is of no consequence in the RG analysis.) Now, following
the usual procedure, we obtain the effective action for the
two-species MVP:
SMVP2 =
∫
dt
∫
dx[ ˆψ(∂t +w −D∇2)ψ + ν ˆψψ2 − λ ˆψ2ψ]
+
∫
dt
∫
dx[ϕˆ(∂t +w −D∇2)ϕ + νϕˆϕ2 − λϕˆ2ϕ]
−ν¯
∫
dt
∫
dx[ϕˆϕψ + ˆψψϕ], (9)
where ν¯ = γ¯ μ. Hereψ and ˆψ are fields associatedwith species
A; ϕ and ϕˆ are associated with species B. The first two lines
of expression (9) correspond to independent MVPs; the third
represents the symbiotic interaction between them. [While
such a minimal action could have been “postulated” directly,
we prefer to start with the microscopic expression of Eq. (7)
since it describes a valid stochastic process.]
There are two cubic terms in the action involving only
species A (i.e., the vertices shown in Fig. 11), two involving
only B (those of Fig. 11 drawn, say, with dashed lines),
and two vertices with a mixed pair of incoming lines as
well as a single outgoing line that may belong to either
species. One readily identifies the one-loop diagrams leading
to renormalization of the parameter ν¯. In contrast, no diagrams
(at any order) involving mixed-species vertices can affect
the recursion relations for the DP parameters w, ν, and λ.
The reason is that the presence of a mixed-species vertex
FIG. 12. One-loop diagrams in the field theory of DP, leading to
renormalization of w, μ, and λ, respectively.
anywhere in a diagram implies that the lines entering the
diagram are mixed, so that it can only contribute to the
recursion relation for ν¯. We conclude that the interaction
between species cannot alter the scaling behavior, which must
therefore remain that of DP. At one-loop order, there are
two fixed-point values for ν¯, namely, 2λ and zero, the latter
corresponding to independent processes.
V. CONCLUSION
We study symbiotic interactions in contact-process-like
models in one and two dimensions. For this purpose, we
propose a two-species model (2SCP), in which the death
rate is reduced (from unity to μ) on sites occupied by both
species. A related model (SSLCP), in which each species is
confined to its own sublattice, is also studied in one dimension
and found to exhibit similar behavior. Simulations reveal that
the phase transition between active and absorbing states is
continuous and that the critical creation rate λc is reduced
in the presence of symbiosis. This means that the loss of one
species will rapidly lead to extinction since the system is then a
basic contact process operating at λ < λc. Although this might
suggest identifying the density q of doubly occupied sites as
the order parameter, we find that the particle density ρ (which
includes a large contribution from singly occupied sites) scales
in the same manner as q.
Mean-field theory (in both the one- and two-site approxima-
tions) predicts a discontinuous phase transition in any number
of dimensions for μ sufficiently small. A discontinuous
transition between an active and an absorbing phase is not
expected in one-dimensional systems of the kind studied
here [15], nor do our simulations show any evidence of a
discontinuous transition in two dimensions. Nevertheless, we
cannot discard the possibility of such a transition for d > 2,
for small values of μ, or under rapid particle diffusion, which
generally favors mean-field-like behavior.
Overall, the critical behavior of the symbiotic models is
consistent with that of directed percolation. Corrections to
scaling are, however, more significant than in the basic CP, so
that a study restricted to smaller systems, or to only one kind
of simulation, could easily suggest non-DP behavior. These
corrections are stronger, and of longer duration, the smaller the
intensity of symbiosis. Thus, in the two-dimensional case, the
decay ofρ (in initial-decay studies) attains the expected power-
law regime (with aDP value for the decay exponent), except for
μ = 0.9, the weakest symbiosis studied. A similar tendency
is observed in the QS simulations of the one-dimensional
2SCP, for which the estimates for critical exponents and the
critical moment ratio mc differ most from DP values for
μ = 0.9.
In the initial-decay studies in one dimension, for smaller
intensities of symbiosis (i.e., μ = 0.9 and 0.75), we observe
anomalous growth of fluctuations in the order parameter.
The latter are characterized by mρ − 1 = var(ρ)/ρ2, which
is expected to grow ∼ t1/z, before saturating at its QS value.
The growth at long times corresponds to a z value significantly
smaller than that of DP. The exponent zq associated with the
growth of mq is substantially larger, though still slightly below
the DP value. In contrast with these anomalies, the spreading
exponents are found to take DP values in one dimension,
011121-8
SYMBIOTIC TWO-SPECIES CONTACT PROCESS PHYSICAL REVIEW E 86, 011121 (2012)
independent of the degree of symbiosis. Thus we are inclined
to regard the asymptotic scaling of the symbiotic models as
being that of DP and to interpret the deviations as arising from
finite-time and finite-size corrections. One might conjecture
that under strong symbiosis, the critical system is rapidly
attracted to the DP fixed point (although not as rapidly as
is the basic CP), whereas for weak symbiosis, it makes a long
excursion into a regime in which DP-like scaling is not evident
before finally returning to the vicinity of the DP fixed point.
The asymptotic scaling behavior is presumably associated
with large, sparsely populated but highly correlated regions
of doubly occupied sites, which, for reasons of symmetry,
behave analogously to DP space-time clusters. The presence
of isolated particles, which are relatively numerous and long
lived for weak symbiosis, could mask the asymptotic critical
behavior, on short scales. We defer further analysis of these
questions to future work.
Extending the field theory of DP to the two-species case,
we find that the irrelevance of four-field terms makes DP
extremely robust since the only possible three-field vertices are
already present in the single-species theory. Thismeans that the
interaction between species cannot alter the scaling behavior,
as already noted by Janssen in the case of multispecies DP pro-
cesses [12]. Our simulation results, as noted, support this con-
clusion. A more detailed field-theoretic analysis, including the
evolution of the lowest-order irrelevant terms,might shed some
light on the scaling anomalies observed in the simulations.
ACKNOWLEDGMENTS
We are grateful to Miguel A. Mun˜oz for helpful comments.
This work was supported by Conselho Nacional de Desen-
volvimento Cientı´fico e Tecnolo´gico and Fundac¸a˜o deAmparo
a` Pesquisa do Estado de Minas Gerais, Brazil.
[1] J. Marro and R. Dickman, Nonequilibrium Phase Transitions in
Lattice Models (CambridgeUniversity Press, Cambridge, 1999).
[2] M. A. Mun˜oz et al., in Procedings of the 6th Granada Seminar
on Computational Physics, edited by J. Marro and P. L. Garrido,
AIP Conf. Proc. No. 574 (AIP, New York, 2001).
[3] G. ´Odor, Universality in Nonequilibrium Lattice Sys-
tems: Theoretical Foundations (World Scientific, Singapore,
2007).
[4] M. Henkel, H. Hinrichsen, and S. Lu¨beck, Non-Equilibrium
Phase Transitions: Absorbing Phase Transitions (Springer,
Dordrecht, 2008).
[5] G. ´Odor, Rev. Mod. Phys. 76, 663 (2004).
[6] K. A. Takeuchi, M. Kuroda, H. Chate´, and M. Sano, Phys. Rev.
Lett. 99, 234503 (2007).
[7] L. Corte´, P. M. Chaikin, J. P. Gollub, and D. J. Pine, Nat. Phys.
4, 420 (2008).
[8] H. K. Janssen, Z. Phys. B 42, 151 (1981).
[9] P. Grassberger, Z. Phys. B 47, 365 (1982).
[10] I. Dornic, H. Chate´, J. Chave, and H. Hinrichsen, Phys. Rev.
Lett. 87, 045701 (2001).
[11] T. E. Harris, Ann. Probab. 2, 969 (1974).
[12] H. K. Janssen, J. Stat. Phys. 103, 801 (2001).
[13] H. Hinrichsen, Adv. Phys. 49, 815 (2000).
[14] M. M. de Oliveira and R. Dickman, Phys. Rev. E 84, 011125
(2011).
[15] H. Hinrichsen, arXiv:cond-mat/0006212.
[16] M. M. de Oliveira and R. Dickman, Phys. Rev. E 71, 016129
(2005); R. Dickman and M. M. de Oliveira, Physica A 357, 134
(2005).
[17] R. Dickman and J. Kamphorst Leal da Silva, Phys. Rev. E 58,
4266 (1998).
[18] P. Grassberger and A. de la Torre, Ann. Phys. (NY) 122, 373
(1979).
[19] R. da Silva, R. Dickman, and J. R. Drugowich de Felicio, Phys.
Rev. E 70, 067701 (2004).
[20] M. Doi, J. Phys. A 9, 1465 (1976); 9, 1479 (1976).
[21] L. Peliti, J. Phys. 46, 1469 (1985).
[22] R. Dickman and R. Vidigal, Braz. J. Phys. 33, 73 (2003).
[23] U. C. Ta¨uber, in Field-Theory Approaches to Nonequilibrium
Dynamics, edited byM.Henkel,M. Pleimling, and R. Sanctuary,
Lecture Notes in Physics, Vol. 716 (Springer, Heidelberg, 2007),
pp. 295–348.
[24] F. van Wijland, K. Oerding, and H. J. Hilhorst, Physica A 251,
179 (1998).
011121-9
30
Capı´tulo 3
Sobreviveˆncia do mais escasso no espac¸o
Esta introduc¸a˜o e´ baseada nos resultados do artigo [4], publicado em Journal of Statistical Mechanics.
E´ comum se afirmar que populac¸o˜es grandes de espe´cies interagentes tendem a ser menos susceptı´veis
a` extinc¸a˜o quando em competic¸a˜o com populac¸o˜es mais escassas. Seria interessante nos questionar se e´
possı´vel o contra´rio e, se a resposta for sim, sob quais condic¸o˜es.
Em um artigo recente [34] foi proposto um modelo em que ha´ a possibilidade de a espe´cie mais escassa
ser menos susceptı´vel a` extinc¸a˜o quando em competic¸a˜o com uma espe´cie mais numerosa, com carac-
terı´sticas ana´logas. Os autores utilizaram um modelo 0−dimensional e tal fenoˆmeno foi denominado de
sobreviveˆncia do mais escasso (SE). A condic¸a˜o para que o fenoˆmeno SE ocorra e´ que a espe´cie mais
escassa precisa ser mais competitiva na interac¸a˜o com a mais abundante. Mais competitiva quer dizer que
na maioria das vezes que um ser da espe´cie mais rara luta pela sobreviveˆncia com um ser da espe´cie mais
abundante, a primeira vence.
Ate´ aqui, na˜o ha´ nenhuma surpresa. Pore´m, o interessante e´ que ha´ um regime de equilı´brio onde a
descric¸a˜o feita em termos de equac¸o˜es diferenciais (descric¸a˜o contı´nua), que indica a prevaleˆncia de uma
dada espe´cie, fica em explı´cito contraste com a descric¸a˜o discreta feita atrave´s da equac¸a˜o mestra, que in-
dica uma maior chance de extinc¸a˜o da espe´cie prevalente. A descric¸a˜o probabilı´stica em termos da equac¸a˜o
mestra leva em conta a finitude das populac¸o˜es e a discreteza das interac¸o˜es entre as espe´cies. E o resul-
tado obtido nas duas descric¸o˜es, contı´nua e discreta, sa˜o conflitantes. Estamos diante de um tı´pico exemplo
onde uma descric¸a˜o feita em termos de equac¸o˜es diferenciais (a`s vezes denominada descric¸a˜o em termos
da teoria de campo me´dio) esta´ em pleno contraste com a descric¸a˜o mais detalhada e realista da equac¸a˜o
mestra. Tal descric¸a˜o em termos probabilı´sticos que leva em considerac¸a˜o a intrı´nseca aleatoriedade das
interac¸o˜es entre os agentes tem tido um papel muito relevante no desenvolvimento de va´rios ramos da
biologia teo´rica. E este fato representa e espinha dorsal de todo a minha pesquisa de doutorado.
O artigo que expo˜e o fenoˆmeno SE na˜o considera a possibilidade de as espe´cies se difundirem no espac¸o.
E´ conhecida a importaˆncia que o fenoˆmeno da difusa˜o pode ter nas dinaˆmicas populacionais de espe´cies
interagentes. Alguns fenoˆmenos conhecidos que decorrem desta extensa˜o espacial sa˜o a formac¸a˜o de
padro˜es espaciais e temporais, a formac¸a˜o de frentes de onda de invasa˜o de uma espe´cie sobre a outra, o
surgimento de um tamanho crı´tico mı´nimo para sustentar uma populac¸a˜o em um ambiente hostil, etc.
Em um artigo publicado em Journal of Statistical Mechanics [4], extendemos o modelo original a`
situac¸a˜o onde as espe´cies podem se difundir pelo espac¸o d−dimensional e verificamos que o fenoˆmeno
da sobreviveˆncia do mais escasso no espac¸o (SEE) se verifica, com algumas diferenc¸as. Utilizamos as
te´cnicas de mapeamento do processo estoca´stico subjacente em uma teoria de campos via procedimento
de Doi-Peliti [27]. Subsequentemente utilizamos os conceitos da teoria do grupo de renormalizac¸a˜o para
inferirmos como as intensidades das interac¸o˜es variam com a escala de observac¸a˜o. Construı´mos um dia-
grama que mostra as regio˜es no espac¸o de paraˆmetros onde ha´ coexisteˆncia entre as espe´cies. Constatamos
31
Capı´tulo 3. Sobreviveˆncia do mais escasso no espac¸o
que este diagrama e´ fundamentalmente diferente do diagrama proposto no artigo original [34]. Uma das
principais concluso˜es e´ que para qualquer vantagem competitiva que uma espe´cie tenha sobre a outra, na˜o
importa o qua˜o pequena ela seja, no longo prazo e para escalas de observac¸a˜o grandes∗, teremos que a
espe´cie menos abundante, se mais competitiva, sobrepuja a espe´cie mais numerosa. De certa forma o
fenoˆmeno SE ficou muito mais evidente e robusto na nova versa˜o SEE quando consideramos a possibili-
dade de difusa˜o.
Simulac¸o˜es de Monte Carlo confirmaram a existeˆncia do fenoˆmeno SEE e detalhes a respeito dele fo-
ram obtidos numericamente, como algumas caracterı´sticas relacionadas a`s velocidades de propagac¸a˜o das
frentes de onda das populac¸o˜es no espac¸o e as distribuic¸o˜es de probabilidade quase-estaciona´rias.
∗Suponhamos, por exemplo, na escala de dezenas ou centenas de quiloˆmetros.
32
Survival of the scarcer in space
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
J. Stat. Mech. (2013) P07004
(http://iopscience.iop.org/1742-5468/2013/07/P07004)
Download details:
IP Address: 150.164.255.201
The article was downloaded on 06/07/2013 at 20:33
Please note that terms and conditions apply.
View the table of contents for this issue, or go to the journal homepage for more
Home Search Collections Journals About Contact us My IOPscience
J.S
tat.M
ech.(2013)P
07004
ournal of Statistical Mechanics:J Theory and Experiment
Survival of the scarcer in space
Renato Vieira dos Santos1 and Ronald Dickman1,2
1 Departamento de F´ısica, Instituto de Cieˆncias Exatas, Universidade Federal
de Minas Gerais, CP 702, CEP 30161-970, Belo Horizonte, Minas Gerais,
Brazil
2 National Institute of Science and Technology for Complex Systems,
Universidade Federal de Minas Gerais, CP 702, CEP 30161-970, Belo
Horizonte, Minas Gerais, Brazil
E-mail: econofisico@gmail.com and dickman@fisica.ufmg.br
Received 13 April 2013
Accepted 16 June 2013
Published 4 July 2013
Online at stacks.iop.org/JSTAT/2013/P07004
doi:10.1088/1742-5468/2013/07/P07004
Abstract. The dynamics leading to extinction or coexistence of competing
species is of great interest in ecology and related fields. Recently a model of
intra- and interspecific competition between two species was proposed by Gabel
et al, in which the scarcer species (i.e., with smaller stationary population size)
can be more resistant to extinction when it holds a competitive advantage; the
latter study considered populations without spatial variation. Here we verify
this phenomenon in populations distributed in space. We extend the model
of Gabel et al to a d-dimensional lattice, and study its population dynamics
both analytically and numerically. Survival of the scarcer in space is verified for
situations in which the more competitive species is closer to the threshold for
extinction than is the less competitive species, when considered in isolation. The
conditions for survival of the scarcer species, as obtained applying renormalization
group analysis and Monte Carlo simulation, differ in detail from those found in
the spatially homogeneous case. Simulations highlight the speed of invasion waves
in determining the survival times of the competing species.
Keywords: renormalization group, stochastic particle dynamics (theory),
population dynamics (theory), stochastic processes
c© 2013 IOP Publishing Ltd and SISSA Medialab srl 1742-5468/13/P07004+19$33.00
J.S
tat.M
ech.(2013)P
07004
Survival of the scarcer in space
Contents
1. Introduction 2
2. Model 4
2.1. Effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2. Mean-field approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3. Renormalization group (RG) flow 6
3.1. Steady state close to critical points . . . . . . . . . . . . . . . . . . . . . . . 8
3.1.1. SPDE in the vicinity of point F. . . . . . . . . . . . . . . . . . . . . 8
3.1.2. SPDE in the vicinity of a stable fixed point. . . . . . . . . . . . . . . 9
3.1.3. Nature of phase transitions. . . . . . . . . . . . . . . . . . . . . . . . 10
4. Simulations 12
5. Conclusion 18
Acknowledgment 18
References 18
1. Introduction
Coexistence of species and maintenance of species diversity are key issues in ecology as
well as in conservation and restoration. A key idea in this context is the competitive
exclusion principle, which asserts that similar species competing for a limited resource
cannot coexist [1]. Individual-based stochastic models, both with and without spatial
structure, are useful for analyzing these questions [2]. There is some evidence [3] to
suggest that spatial structure should facilitate coexistence of similar competitors. This
is because when dispersal and interactions are localized, individuals tend to interact with
conspecific neighbors more frequently than would be suggested by the overall densities
at the landscape level; this is a common pattern in natural communities [4]. Computer
simulations suggest that such spatial structures could lead to the long-term coexistence
of similar competitors and hence to the maintenance of high levels of biodiversity [5].
This mechanism has been encapsulated in the segregation hypothesis [6], which states
that intraspecific spatial aggregation promotes stable coexistence by reducing interspecific
competition.
In the context of competition and extinction, the question of population size plays
a fundamental role. In the simplest birth-and-death models, for example, the extinction
probability of a viable, but initially small population is high, but decreases rapidly with
population size.3 Recently, Gabel et al [7] proposed a stochastic model of two-species
competition in which the species with smaller population size is, under certain conditions,
less susceptible to extinction than the more populous species. This phenomenon, dubbed
3 See chapter 2 of [2].
doi:10.1088/1742-5468/2013/07/P07004 2
J.S
tat.M
ech.(2013)P
07004
Survival of the scarcer in space
survival of the scarcer (SS) in [7], is somewhat surprising since conventional wisdom
on population dynamics suggests that a smaller population is more susceptible to
extinction due to demographic fluctuations. The authors of [7] use a variant of the
Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) approximation to obtain the tails of the
quasi-stationary (QS) probability distribution [8, 9]. (Here and below, the quasi-stationary
regime characterizes the behavior at long times, conditioned on survival.) They show that
asymmetry in interspecific competition can induce survival of the scarcer species.
While the analysis of [7] holds for well mixed (spatially uniform) populations, in
view of the above-mentioned segregation hypothesis it is natural to inquire whether the
same conclusions apply for populations distributed in space. In this paper we study a
stochastic model similar to that of [7], but with organisms located on a d-dimensional
lattice, and investigate the conditions required for survival of the scarcer in space (SSS).
On reproduction, a daughter organism appears at the same site as the mother; competition
only occurs between organisms at the same site. In addition to these reactions, organisms
also diffuse on the lattice. We study the model using two complementary approaches:
field-theoretic analysis and direct numerical (Monte Carlo) simulation.
Since the field-theoretic study requires specialized techniques, we summarize the
method for readers less familiar with this approach. Starting from the master equation, we
construct a representation involving creation and annihilation operators. This allows us
to obtain a functional integral formulation, that is, a mapping of the stochastic process to
a field theory. This now standard procedure is known as the Doi–Peliti mapping [10]–[12].
We consider the population dynamics in a critical situation, i.e., competing populations
close to extinction. The dynamic renormalization group (DRG) is used to study the
nonequilibrium critical dynamics of the populations, specifically, to determine how the
model parameters transform under rescaling of length and time. The analysis furnishes a
set of reduced stochastic evolution equations that describe the population dynamics for
parameters close to those of a fixed point of the DRG. Analysis of these equations lets us
map out regions of coexistence and of single-species survival in the space of reproduction
and competition rates. In particular, we find regions in which the species predicted by
mean-field theory to have the larger population in fact goes extinct.
We use Monte Carlo simulations to probe the QS population densities and lifetimes
of the two species. We find SSS in a small but significant region of parameter space, in
which the less populous species is more competitive. The simulations yield insights into
the spatiotemporal pattern of population growth and decay, the speed of species invasion,
and the QS probability distribution.
The remainder of this paper is organized as follows. In section 2 we describe the
model and write the effective action associated with the stochastic process. A brief
analysis of mean-field theory, valid in dimensions greater than the critical dimension is
performed. In section 3 we use some known results on multispecies directed percolation to
obtain the renormalization group flow in parameter space. This analysis, combined with
numerical simulations of the associated stochastic partial differential equations, allows us
to determine the regions in parameter space in which SSS occurs. Results of Monte Carlo
simulations are presented in section 4. Section 5 closes the paper with our conclusions.
doi:10.1088/1742-5468/2013/07/P07004 3
J.S
tat.M
ech.(2013)P
07004
Survival of the scarcer in space
Figure 1. Phase diagram of a well mixed system in the g–δ plane, as furnished
by the analysis of [7]. Dashed blue curve: gmf = (1 + δ)/(1 + ). Full red curve:
gsw = (1 +mδ)/(1 +m).
2. Model
The model proposed in [7] may be described, with some minor modifications in the rates
as specified below, by the following set of reactions:
A
α
⇀A + A B
β
⇀B + B
A + A
α′
⇀ ∅(A) B + B β′⇀ ∅(B)
A + B
ζ
⇀B A + B
ξ
⇀A.
(1)
The reactions in the first line correspond to reproduction of species A and B. Those in
the second line describe intraspecific competition resulting in the death of one individual
(A/B) or of both (∅), while the third line represents interspecific competition. α (β), α′
(β′) and ζ (ξ) are the rates of the reactions associated with species A (B).
The relation between the rates in equation (1) and those of the original model [7] is
(in the case of mutual annihilation): α↔ 1, β ↔ g, α′ ↔ 1/K, β′ ↔ 1/K, ζ ↔ /K, and
ξ ↔ δ/K. δ < 1 implies an asymmetry in competition between species that favors B at
the expense of A. In this case species B is more competitive than A; the opposite occurs
if δ > 1.
The authors of [7] constructed a phase diagram in the g–δ plane (shown schematically
in figure 1). The phase diagram includes results of both mean-field theory (MFT) and
analysis of the master equation for a stochastic population model in a well mixed system,
i.e., without spatial structure. In MFT, the populations of the two species are equal when
gmf(δ) = (1 + δ)/(1 + ). Analysis of the master equation in the QS regime yields equal
extinction probabilities when gsw(δ) = (1+mδ)/(1+m) with m ≡ [(ln 2)−1 − 1]−1. These
conditions are plotted in figure 1; they intersect at the point (g, δ) = (1, 1). The phase
diagram includes two ‘normal’ regions, in which the more populous species is less likely
to go extinct, and two ‘anomalous’ regions, in which the scarcer species is more likely to
survive. Thus in region I (where δ < 1 and B is more competitive in the sense defined
doi:10.1088/1742-5468/2013/07/P07004 4
J.S
tat.M
ech.(2013)P
07004
Survival of the scarcer in space
above), species A is more numerous in the quasi-stationary state and simultaneously more
susceptible to extinction, while in region II (where δ > 1 and A is more competitive),
species B is more numerous, yet more likely to go extinct first. Regions I and II are
anomalous since they correspond to an inversion of the usual dictum that susceptibility
to extinction grows with diminishing population size. Our goal in this work is to determine
whether a stochastic model of two-species competition with spatial structure, in which
organisms diffuse in d-dimensional space, exhibits a similar phenomenon. In section 2.1
we develop a continuum description for this spatial stochastic process.
2.1. Effective action
We consider a stochastic implementation of the reactions of equation (1) on a d-
dimensional lattice, adding diffusion (nearest-neighbor hopping) of individuals of species
A and B at rates DA and DB respectively. Following the standard Doi–Peliti procedure
mentioned in section 1, ignoring irrelevant terms (in the renormalization group sense)
and with the so-called Doi shift [12] already performed, we obtain the following effective
action:
S(φ¯, φ, ψ¯, ψ) =
∫
ddx
∫
dt
{
φ¯[∂t +DA(σA −∇2)]φ− αφ¯2φ+ jα′φ¯φ2 + α′φ¯2φ2 + ζψφφ¯
}
+
∫
ddx
∫
dt
{
ψ¯[∂t +DB(σB −∇2)]ψ − βψ¯2ψ + jβ′ψ¯ψ2 + β′ψ¯2ψ2
+ ξφψψ¯
}
, (2)
where σA ≡ −α/DA, σB ≡ −β/DB and j ≡ 1 (2) for individual death (mutual
annihilation). Action S has four fields. The expected values of the fields φ and ψ (evaluated
by performing functional integrals over the four fields, using the weight exp[−S]), represent
the mean population densities of species A and B, respectively. The fields with overbars
have no immediate physical interpretation, but are related to intrinsic fluctuations of the
stochastic dynamics. In particular, terms quadratic in φ¯ and ψ¯ correspond to noise in
the associated stochastic evolution equations [13]. The terms ∝ φφ¯2 and ψψ¯2 correspond,
respectively, to intrinsic fluctuations of species A and B. In the expansion of the action
in powers of the fields, the lowest order term involving the product φ¯ψ¯ would be φψφ¯ψ¯.
Since it is of fourth order in the fields, this term is irrelevant, that is, the coefficient of this
term flows to zero under repeated renormalization group transformations; it has therefore
been excluded from the action. Fluctuations of one species nevertheless affect the other
species via the coupling terms ∝ ζ and ξ in equation (2).
Using the definitions of σA and σB and the change of variables φ¯→ θAφ¯, φ→ θ−1A φ,
ψ¯ → θBψ¯, and ψ → θ−1B ψ, where θA ≡
√
jα′/α, θB ≡
√
jβ′/β, we can write the effective
action in the form:
S(φ¯, φ, ψ¯, ψ) =
∫
ddx
∫
dt
{
φ¯[∂t +DA(σA −∇2)]φ
+ ψ¯[∂t +DB(σB −∇2)]ψ
+ gAφφ¯(φ− φ¯) + gBψψ¯(ψ − ψ¯) + hBφψψ¯ + hAφψφ¯
}
, (3)
with
gA ≡
√
jαα′ hA ≡ θ−1A ζ gB ≡
√
jββ′ hB ≡ θ−1B ξ. (4)
doi:10.1088/1742-5468/2013/07/P07004 5
J.S
tat.M
ech.(2013)P
07004
Survival of the scarcer in space
Dimensional analysis yields
[hA] = [hB] = p
2−d/2 = [gA] = [gB],
where [X] denotes the dimensions of X, and p has dimensions of momentum [12]. The
upper critical dimension is therefore dc = 4, as for single-species processes such as the
contact process and directed percolation.
2.2. Mean-field approximation
For d ≥ dc, mean-field analysis, which ignores fluctuations in φ and ψ, yields the
correct critical behavior. Imposing the conditions δS/δφ = 0, δS/δφ¯ = 0, δS/δψ = 0 and
δS/δψ¯ = 0, we obtain the mean-field equations
∂φ
∂t
= DA∇2φ+ αφ− gAφ2 − hBφψ
∂ψ
∂t
= DB∇2ψ + βψ − gBψ2 − hAφψ.
(5)
Let L be a typical length scale and define X = gAφ/α, Y = gBψ/β, s = DAt/L
2, and
define a rescaled coordinate x′ = x/L. Then equation (5) can be written in dimensionless
form as
∂X
∂s
= ∇2X + γ (X −X2 − aXY ) ≡ ∇2X + f(X, Y )
∂Y
∂s
= D∇2Y + γ (cY − cY 2 − bXY ) ≡ D∇2Y + g(X, Y ) (6)
with γ ≡ αL2/DA, a ≡ βhB/(αgB), b ≡ βhA/(αgB), c ≡ gAβ2/(gBα2), D ≡ DBβgA/
(DAαgB), f(X, Y ) ≡ γ (X −X2 − aXY ) and g(X, Y ) ≡ γ (cY − cY 2 − bXY ).
Coexistence is possible in the stationary state if the conditions c > b and a < 1 hold,
implying hA < βgA/α and hB < αgB/β. These conditions depend monotonically on
parameters α, β, gA and gB, analogous to the coexistence conditions found in [7]. The
same is not true in spatial stochastic theory, as we shall see in section 3.
3. Renormalization group (RG) flow
In this section we apply a renormalization group analysis to determine regions of
coexistence and of single-species survival in the space of reproduction and competition
rates. Some years ago, Janssen analyzed a class of reactions of the kind defined in
equation (1) [14]. This work considered multispecies reactions of the form
Xi ↔ 2Xi Xi → ∅ Xi +Xj → kXi + lXj, (7)
where i and j are species indices and k, l are either zero or unity; this process is
called multicolored directed percolation (MDP), with different colors referring to different
species. The RG analysis of [14] shows that knowledge of the directed percolation fixed
points is sufficient to determine the fixed points of the full MDP. As shown in that
work, the parameter combinations gA/DB ≡ uA and gB/DA ≡ uB, related to intraspecific
competition, flow under renormalization group transformations to the stable DP fixed
point u∗A = u
∗
B = u
∗ = 2/3 with  = 4− d > 0.4 Therefore, regarding the renormalization
4 Not to be confused with parameter  used in equation (1).
doi:10.1088/1742-5468/2013/07/P07004 6
J.S
tat.M
ech.(2013)P
07004
Survival of the scarcer in space
Figure 2. RG flow of the interspecific competition parameters vB and vA. Only
the first quadrant is relevant due to the requirement of positive rates. There is an
unstable fixed point O at (vA, vB) = (0, 0) (black dot); a hyperbolic fixed point F
at (vA, vB) = (u
∗, u∗) (red dot), and two stable fixed points G and H, located at
(vA, vB) = (0, 2u
∗) and (vA, vB) = (2u∗, 0) respectively (blue dots). The separatrix
vA = vB is the boundary between the basins of attraction of G and H.
of intraspecific competition parameters, inclusion of other species behaving as in DP does
not alter the fixed point.
Janssen’s analysis shows that there are four fixed points for interspecific competition
parameters hA/DA ≡ vA and hB/DB ≡ vB, which from reactions (1) are related to ξ and
ζ. The first two are (v∗A, v
∗
B) = (0, 0), which is unstable, and
(v∗A, v
∗
B) =
(
DA +DB
DA
2
3
,
DA +DB
DB
2
3

)
(8)
which is a hyperbolic fixed point if DA = DB.
5 For DA 6= DB, it was conjectured [14]
that the stability of the fixed points remains the same despite small changes in the
renormalization group flow diagram topology. The other two fixed points for DA = DB
are stable; they are given by (v∗A, v
∗
B) = (0, 2u
∗) and (v∗A, v
∗
B) = (2u
∗, 0). To summarize, for
DA = DB ≡ D0, the interspecies competition parameters flow to one of the following four
d-dependent fixed points (denoted as O,F,G, and H below),
(vA, vB) =
(
θ−1A ζ
D0
,
θ−1B ξ
D0
)
→ {O : (0, 0), F : (u∗, u∗),
G : (2u∗, 0), H : (0, 2u∗)} . (9)
Figure 2 shows the renormalization group flow diagram. The unstable fixed point O
corresponds to complete decoupling between the two species, and the hyperbolic point F to
5 In the nomenclature of [14], if species are of the same flavor.
doi:10.1088/1742-5468/2013/07/P07004 7
J.S
tat.M
ech.(2013)P
07004
Survival of the scarcer in space
symmetric coupling. In the symmetric subspace, any nonzero initial value of v ≡ vA = vB,
flows to F. For vB > vA (δ > 1), (vA, vB) flows to G, so that species A is effectively unable
to compete with B. Similarly, if vA > vB (δ < 1), the flow attains point H. In section 3.1
we discuss the stationary states associated with these fixed points.
3.1. Steady state close to critical points
Population dynamics in the critical regime is governed by a pair of coupled stochastic
partial differential equations (SPDE), which are readily deduced from the action of
equation (3) [12]:
∂φ
∂t
= D0∇2φ+ αφ− gAφ2 − hBφψ + η1 (10)
and
∂ψ
∂t
= D0∇2ψ + βψ − gBψ2 − hAψφ+ η2 (11)
where the noise terms η1(x, t) and η2(x, t) satisfy 〈η1(x, t)〉 = 〈η2(x, t)〉 = 0, and
〈η1(x, t)η1(x, t)〉 = 2gAφ(x, t)δd(x− x′)δ(t− t′), (12)
〈η2(x, t)η2(x, t)〉 = 2gBψ(x, t)δd(x− x′)δ(t− t′). (13)
Note that these multiplicative noise terms depend on the square root of their respective
fields, and were obtained directly from the action, without any additional hypotheses.
3.1.1. SPDE in the vicinity of point F. In the symmetric subspace, δ = 1, the RG flow is
to the fixed point F, and parameters gA/D0, gB/D0, hA/D0, and hB/D0 take the associated
values. Therefore the SPDEs in (10) and (11) can be written as [15]:
∂φ
∂t
= D0∇2φ+ φ−D0u∗φ2 −D0u∗φψ + η1, (14)
∂ψ
∂t
= D0∇2ψ + gψ −D0u∗ψ2 −D0u∗ψφ+ η2 (15)
where we set α = 1 and β = g. With the rescaling φ → φ/(D0u∗), ψ → ψ/(D0u∗) and
x→ (1/D0)1/2x, we can write equations (14) and (15) in the form:
∂φ
∂t
= ∇2φ+ φ− φ2 − φψ + η1, (16)
∂ψ
∂t
= ∇2ψ + gψ − ψ2 − ψφ+ η2 (17)
with rescaled noise
〈η1(x, t)η1(x, t)〉 = 2 (D0)2−d/2 (u∗)2φ(x, t)δd(x− x′)δ(t− t′) (18)
〈η2(x, t)η2(x, t)〉 = 2 (D0)2−d/2 (u∗)2ψ(x, t)δd(x− x′)δ(t− t′). (19)
For d < 4, the rescaled noise intensity σ grows with diffusion rate D0; for d = 1 we have
σ2 = 2D
3/2
0 u
∗2 with u∗2 ≈ 2. Figures 3(a) and (b) show results of numerical simulations of
equations (16) and (17) in d = 1 with g = 1. They show, for a one-dimensional lattice of
doi:10.1088/1742-5468/2013/07/P07004 8
J.S
tat.M
ech.(2013)P
07004
Survival of the scarcer in space
Figure 3. Numerical simulations symmetric SPDE. (a) Symmetric case with
σ = 0.0 and g = 1. Initial populations are φ(x, 0) = 0.4 and ψ(x, 0) = 0.6.
(b) Symmetric case with σ = 0.03 and g = 1. Initial populations are φ(x, 0) = 0.4
and ψ(x, 0) = 0.6.
L = 128 sites in the interval (−1, 1), the population densities averaged over 1000 Monte
Carlo realizations for two different values of the noise intensity. The time interval T is
T = 400, partitioned into N = 40 000 steps so that ∆t ≡ T/N = 400/40 000 = 0.01. Initial
conditions are φ(x, 0) = 0.4 and ψ(x, 0) = 0.6. For σ = 0, population densities are almost
time-independent, as shown in figure 3(a). This is not the case for larger noise values.
In this case, the two populations tend to the same value (see figure 3(b)). Although
figures 3(a) and (b) represent averages over many realizations, we have verified coexistence
in individual runs.
These simulations were performed using standard integration techniques for the
Langevin equation with the XMDS2 software [16]. Since we are interested in showing only
the initial temporal trends of the two population densities in the vicinity of fixed points,
the use of standard techniques of integrating the Langevin equations is sufficient to reveal
the qualitative nature of the solutions. Near a phase transition to an absorbing state, one
of the population densities tends to zero and the standard numerical integration scheme
fails. This standard algorithm is not suitable for extracting the more accurate results
associated with critical exponents or asymptotic decays. In this case we would have to use
more sophisticated algorithms, such as those proposed in [17]–[19].
3.1.2. SPDE in the vicinity of a stable fixed point. Now consider the case δ 6= 0. With
δ > 1, the parameters flow to point G and equations (10) and (11) become
∂φ
∂t
= D0∇2φ+ φ− u∗φ2 + η1, (20)
∂ψ
∂t
= D0∇2ψ + gψ − u∗ψ2 − 2u∗ψφ+ η2, (21)
where we put α = 1 and β = g. With a rescaling similar to that used above, we have
∂φ
∂t
= ∇2φ+ φ− φ2 + η1, (22)
∂ψ
∂t
= ∇2ψ + gψ − ψ2 − 2φψ + η2 (23)
doi:10.1088/1742-5468/2013/07/P07004 9
J.S
tat.M
ech.(2013)P
07004
Survival of the scarcer in space
Figure 4. Asymmetric case (δ = 1.5) with σ = 0.03 and g = 2.2. Initial
populations for A and B species are φ(x, 0) = 0.1 and ψ(x, 0) = 0.9, respectively.
with η1 and η2 satisfying (18) and (19), respectively. For δ < 1 the situation is completely
analogous to the case δ > 1, and the resulting equations are as above, with the exchange
of φ and ψ and changing the position of the factor g accordingly.
If we neglect diffusion and noise terms, we have a system of ordinary differential
equations having a fixed point associated with the stable coexistence state given by
(φ∗, ψ∗) = (1, g − 2).6 Therefore, the condition for coexistence is g > 2. If g < 2, only
species A survives for δ > 1. Figure 4 shows the result of a simulation with g = 2.2 in which
the equations are integrated including both diffusion and noise. Numerical experiments
indicate that the effect of these terms is to increase the value of g ' 2 for coexistence.
Higher noise values imply higher thresholds g for coexistence.
For δ < 1 similar reasoning applies. The results are summarized in the phase diagram
of figure 5. There are four stationary phases: a pure-A phase for δ > 1 and g < 2, a
symmetric pure-B phase for δ < 1 and g < 1/2, and two (disjoint) coexistence phases.
The latter arise when the less competitive species proliferates sufficiently faster than the
more competitive one.
3.1.3. Nature of phase transitions. From the equations that omit the diffusion and noise
terms, one may infer the nature of the phase transitions in the diagram of figure 5. From
table 1, which shows the fixed points of the equations for different values of δ, we see that
in general the values in the final column do not match for δ 6= 1. This fact indicates that
the vertical blue line in the diagram (5), i.e., the line δ = 1, is a line of discontinuous phase
transitions. Similarly, examining the expressions in the final column of table 1, we infer
the nature of phase transitions along the horizontal lines at g = 1/2 and 2. In regions
of coexistence, the densities of the minority species grow continuously from zero; thus
these transitions are continuous. One should of course recognize that the predictions for
the phase diagram are essentially qualitative; quantitative predictions for nonuniversal
properties such as phase boundaries are not possible once irrelevant terms have been
discarded. Regarding the nature of the transitions, we expect the continuous ones, as
6 For the case δ < 1, the fixed points are (φ∗, ψ∗) = (1− 2g, g), and the coexistence condition is 0 < g < 1/2.
doi:10.1088/1742-5468/2013/07/P07004 10
J.S
tat.M
ech.(2013)P
07004
Survival of the scarcer in space
Figure 5. Stationary phase diagram in the g–δ plane for the stochastic model with
spatial structure. The reference points have coordinates: b = (1, 0), p = (0, 1/2),
r = (3, 2). For δ > 1, when g < 2 only species A is present, while for g > 2 there is
coexistence. Similarly, for δ < 1, when g > 1/2 only species B is present, and for
g < 1/2 there is coexistence. The diagonal line connecting points p and r is the
equal-population criterion furnished by mean-field theory, gmf(δ) = (1+δ)/(1+),
with  = 1. In regions I and II, the more populous species (A or B, respectively),
as predicted by mean-field theory, in fact goes extinct. The thick, dashed, blue
and black curves represent gmf and gsw for the value of  that maximizes the area
between them, i.e.,  = 1/
√
m ≈ 0.66.
Table 1. Simplified equations and their fixed points for different values of δ.
δ Equation Fixed Point
= 1 φ˙ = φ− φ2 − φψ and ψ˙ = gψ − ψ2 − φψ (φ∗, ψ∗) = (0, g)
> 1 φ˙ = φ− φ2 and ψ˙ = gψ − ψ2 − 2φψ (φ∗, ψ∗) = (1, 2− g)
< 1 φ˙ = gφ− φ2 and ψ˙ = ψ − ψ2 − 2φψ (φ∗, ψ∗) = (g, 1− 2g)
furnished by this mean-field-like analysis, do in fact belong to the DP universality class
[14]. The line of discontinuous transitions contains a portion (between levels p and r) that
is between absorbing subspaces (only on species present in each phase), and other portions
that separate (from the viewpoint of the species undergoing extinction) an active and an
absorbing phase. Discontinuous transitions of this nature are not expected to occur in one
dimension [20].
According to mean-field theory, a point in region II corresponds to ρB > ρA. If we
increase the competitiveness of species A, increasing δ while maintaining g fixed, there
will be a point beyond which species B no longer has the greater population density.
Improvement of the competitiveness of species A will make it the more populous species
if organisms are well mixed. In the case of a spatially structured population, by contrast,
any competitive advantage of species A, no matter how small, is sufficient to dominate the
B population (if the latter does not proliferate very rapidly) when the population densities
doi:10.1088/1742-5468/2013/07/P07004 11
J.S
tat.M
ech.(2013)P
07004
Survival of the scarcer in space
are very low, as is the case near criticality. Thus in a situation in which mean-field theory
predicts species B to be the majority, we can have species A exclusively. This can be seen
as a strong version of the survival of the scarcer phenomenon.
To compare the predictions of the mean-field and spatially structured descriptions,
a pair of dashed lines are plotted in figure 5, representing equations for gmf and gsw
as defined previously, for parameters such that the area between them is maximum,
thereby maximizing the region in which SS occurs. (The maximum area is obtained using
 = 1/
√
m with m = [(ln 2)−1 − 1]−1, in equation (1)). Given the much larger area of the
region exhibiting SS in the spatial model, compared with that in the mean-field analysis
(i.e., the area between the two dashed lines for 1 < δ < 3,) we see that the SS phenomenon
can be greatly intensified when spatial structure is included. Analogous reasoning applies
for region I (green). (For the parameter values used, however, this reasoning does not
apply for δ > 3, since in this case the area between the dashed lines can be arbitrarily
large. Also we no longer have a region analogous to region II with species A only.) In
this way we see that the phase diagram for the spatial stochastic model is rather different
from that found in [7]. Moreover, under certain conditions, the survival of the scarcer
phenomenon is strengthened.
4. Simulations
Since the preceding analysis involves approximations whose reliability is difficult to assess,
we perform simulations of a simple lattice model to verify SSS. This approach permits us
to access the QS regime of the spatial model, in finite systems. We consider the following
spatial stochastic process, defined on a lattice of Ld sites. Each site i is characterized by
nonnegative occupation numbers ai and bi for the two species. The organisms (hereafter
‘particles’) of the two species evolve according to the following rates.
• Particles of either species hop to a neighboring site at rate D.
• Particles of species A(B) reproduce at rate λA(λB).
• At sites with two or more A particles, mutual annihilation occurs at rate αAai(ai−1),
and similarly for B particles, at rate αBbi(bi − 1).
• At sites having both A and B particles, the competitive reaction B→ 0 occurs at rate
ζAaibi and the reaction A→ 0 occurs at rate ζBaibi.
The model is implemented in the following manner. In each time step, of duration ∆t,
each process is realized, at all sites, in the sequence: hopping, reproduction, annihilation,
competition.
In the hopping substep, each site is visited, and the probability of an A particle
hopping to the right (under periodic boundaries) is set to ph = Dai∆t/(2d). A random
number z, uniform on [0, 1) is generated, and if z < ph the site is marked to transfer
a particle to the right. Once all sites have been visited, the transfers are realized. The
same procedure is applied, in parallel, for B particles hopping to the right. Subsequently,
hopping in the other 2d− 1 directions is realized in the same manner.
In the reproduction substep, the A particle reproduction probability at each site i is
taken as pc = λAai∆t. A random number z is generated and a new A particle created at
this site if z < pc. An analogous procedure is applied for reproduction of B particles.
doi:10.1088/1742-5468/2013/07/P07004 12
J.S
tat.M
ech.(2013)P
07004
Survival of the scarcer in space
In the annihilation substep, a probability pa = αAai(ai − 1)∆t is defined at each site,
and mutual annihilation (ai → ai − 2) occurs if a random number is < pa. Again, an
analogous procedure is applied for annihilation of B particles.
Finally, at sites harboring both A and B particles, probabilities pA = ζAaibi∆t and
pB = ζBaibi∆t are defined. The process B→ 0 occurs if a random number z < pA, while
the complementary process A→ 0 occurs if pA ≤ z < pA + pB. Note that at most one of
the processes (A→ 0 and B→ 0) can occur at a given site in this substep.
The time step ∆t is chosen to render the reaction probabilities relatively small. To do
this, before each step we scan the lattice and determine the maximum, over all sites, of
ai, bi, and aibi. With this information we can determine the maximum reaction rate over
all sites and all reactions. Then ∆t is taken such that the maximum reaction probability
(again, over sites and reactions) be 1/5. In this way, the probability of multiple reactions
(e.g., two A particles hopping from i to i+ 1, etc) is at most 1/25, and can be neglected
to a good approximation, particularly in the regime in which occupation numbers are
typically small.
We simulate the process on a ring (d = 1) using quasi-stationary (QS) simulations,
intended to sample the QS probability distribution [21, 22]. In these studies the system
is initialized with one A and one B particle at each site. When either species goes extinct
(an absorbing subspace for the process), the simulation is reinitialized with one of the
active configurations (having nonzero populations for both species) saved during the run.
Following a brief transient, the particle densities fluctuate around steady values. We search
for parameter values such that species A is more numerous, despite being much less
competitive than species B (i.e., ζB  ζA).
Given the large parameter space, certain rates are kept fixed in the study. We set
D = αA = αB = 1/4. To understand the competitive dynamics we first need to determine
the conditions for single-species survival. We use QS simulations as well as spreading
simulations [23] to estimate the critical value of λ for survival of a single species, given
α = 1/4. In spreading simulations the initial configuration is that of a single site with
one particle, and all other sites empty. We search for the value of λ associated with
a power-law behavior of the survival probability P (t) and the mean population size
n(t). These studies yield λc = 1.267(1) as the critical point for survival of a single
species, i.e., without interspecies competition. (Here and in the following, figures in
parentheses denote statistical uncertainties in the final digit.) The phase transition is
clearly continuous; details on scaling behavior will be reported elsewhere.
In the two-species studies we set λA = 1.6 and ζA = 0.005, so that species A is well
above criticality but weakly competitive. We then vary λB and ζB, monitoring the QS
population densities ρA and ρB of the two species, as well as their QS lifetimes, τA and
τB. The latter are estimated by counting the number of times, in a long QS simulation,
that one or the other species goes extinct. Survival of the scarcer is then characterized by
the conditions ρA > ρB and τA < τB. For the parameter values studied, we observe SSS
for λB ≈ λc and ζB  ζA. Figure 6 illustrates the variation of the population densities,
and of the ratio τA/τB, as ζB is varied for fixed λB = 1.25, on a ring of 200 sites. In this
case, ρA > ρB throughout the range of interest; the scarcer species, B, survives longer
than A for ζB ≥ 0.25. Figure 7 shows a typical evolution, for ζB = 2 and other parameters
as in figure 6. (The spatiotemporal pattern observed for other parameter sets exhibiting
SSS is qualitatively similar.) The system is divided into A-occupied regions, B-occupied
doi:10.1088/1742-5468/2013/07/P07004 13
J.S
tat.M
ech.(2013)P
07004
Survival of the scarcer in space
Figure 6. Quasi-stationary population densities ρA (blue) and ρB (red), and
lifetime ratio τA/τB (black) versus ζB, for parameters as specified in the text.
Figure 7. A sample evolution on a ring of 200 sites, of duration 4000 time units
(with time increasing downward). Blue: sites with ai > 0 and bi = 0; red: sites
with ai = 0 and bi > 0; black: sites with both species present.
regions, and voids. The voids arising in B-occupied regions are larger, as this species is
nearer criticality. Species A, which is well above criticality, rapidly invades empty regions,
but is in turn subject to invasion by the more competitive species B. Sites bearing both
species are quite rare, occurring only at the frontiers between regions occupied by a single
species.
doi:10.1088/1742-5468/2013/07/P07004 14
J.S
tat.M
ech.(2013)P
07004
Survival of the scarcer in space
Figure 8. Simulations in one dimension: SSS is observed in the region bounded
by the two curves; see text for parameters. Inset: the same data plotted using
δ = ζA/ζB instead of ζB on the horizontal axis.
Figure 8 shows the result of a systematic search in the ζB–λB plane on a ring of L = 200
sites. These data represent averages of 200 realizations, each lasting 105 time units. SSS
is observed in the region bounded by the two curves; the lower curve corresponds to
τA = τB while on the upper we have ρA = ρB. There is an interval of λB values, including
λc, on which SSS is observed for any value of ζB greater than a certain minimum. For
larger values of λB, SSS occurs only within a narrow range of ζB. In the latter region we
have α > ζB > ζA, so that intraspecies competition is stronger than competition between
species. Although we use a rather small system to facilitate the search, SSS is not restricted
to this system size. For L = 800, for example, the range of λB values admitting SSS is more
restricted (from about 1.24 to 1.27) but at the same time the effect can be more dramatic.
For λB = 1.27 and ζB = 0.2, for example, we find ρA = 0.74, ρB = 0.49, and τB ≈ 100τA.
Studies using λA = 1.35 and ζA = 0.01 yield a qualitatively similar diagram, leading us
to conjecture that the form of the region exhibiting SSS in one dimension is generically
that shown in figure 8. To summarize, we verify SSS in a limited but significant region of
parameter space.
In the inset of figure 8 the data are plotted in the δ–λB plane, for comparison with
the theoretical predictions shown in figure 5. (Recall that parameter g corresponds to
the ratio λB/λA, which is smaller than unity here. The parameter δ = ζA/ζB, i.e., the
ratio of the interspecies competition rates.) The region exhibiting SSS is rather different
from the predictions of both MFT and stochastic field theory. In particular the lower
doi:10.1088/1742-5468/2013/07/P07004 15
J.S
tat.M
ech.(2013)P
07004
Survival of the scarcer in space
Figure 9. Mean density profiles ρA(x, t) (maximum at right) and ρB(x, t)
(maximum at left) starting from neighboring single-species domains. Parameters:
λA = 1.6, λB = 1.29, ζA = 0.005, ζB = 0.16.
boundary in the δ-λB plane is not horizontal and it is unclear whether the SSS region
extends to δ = 1. (As λB is increased, SSS is observed in an ever more limited range of
ζB values, making numerical work quite time-consuming.) Here it is important to recall
that irrelevant terms (in the renormalization group sense) are discarded in the theoretical
analysis, so that predictions for the phase diagram are only qualitative.
The evolution shown in figure 7 suggests that survival of a given species depends
on the speed of invasion into territory occupied by the competing species; we study
this issue via the following procedure. We first use simulations to prepare collections
of configurations drawn from the QS distribution for each species in isolation. (We saved
1000 such configurations, on a ring of L = 200 sites, for each λ value of interest.) To
determine the speed of invasion we prepare initial configurations for the two-species system
by placing a randomly chosen, saved configuration with species A only at sites 1, . . . , L,
beside a similar configuration, with species B only, occupying sites L + 1, . . . , 2L. In the
subsequent dynamics, the two species interact at the boundary, leading to invasion by one
species or the other in different realizations. Figure 9 shows the evolution of the A and B
density profiles, averaged over many realizations (3000 or more). Given the average density
profiles ρA(x, t) and ρB(x, t) we identify the interface position Xi(t) of species i via the
condition ρi[Xi(t), t] = ρ¯i/2, with ρ¯i the QS density of species i in isolation. Plotting the
interface positions versus time, we find that they attain steady velocities after a certain
initial transient; the velocities are plotted versus ζB in figure 10. (For the system size used
here, steady velocities are attained after about 500 time units.) The interface velocity vB
of species B increases slowly with ζB, but that of species A (vA) falls rapidly, becoming
doi:10.1088/1742-5468/2013/07/P07004 16
J.S
tat.M
ech.(2013)P
07004
Survival of the scarcer in space
Figure 10. Steady state interface velocities for species A (blue) and B (red) versus
ζB; other parameters as in figure 9.
Figure 11. Quasi-stationary probability distribution for parameters λA = 1.6,
λB = 1.26, ζA = 0.005, and ζB = 0.3, system size L = 200. Red corresponds to
maximum and violet to zero probability. The maxima along the x and y axes
correspond to the single-species QS distributions.
negative for ζB greater than about 0.13. For the parameters considered here, the mean
lifetime of the scarcer species (B) is longer for ζB ≥ 0.113; very near this value, vA becomes
smaller than vB. This suggests that, as one would expect, the preferential extinction of the
more populous species is associated with a smaller rate of spreading, so that A domains
eventually die out due to invasion by B.
To close this discussion of simulation results, we present in figure 11 a portrait of the
QS probability distribution in the SSS region. The basic features identified in the spatially
uniform case (i.e., a well defined maximum corresponding to coexistence with NA > NB, as
well as single-species QS distributions), persist in the presence of spatial structure. We may
doi:10.1088/1742-5468/2013/07/P07004 17
J.S
tat.M
ech.(2013)P
07004
Survival of the scarcer in space
speculate that the higher extinction probability for species A is associated with the more
diffuse barrier in the small-NA region (near the y-axis) as compared with the corresponding
barrier near the x axis. More detailed simulations, including other parameter sets, larger
systems, and simulations on the square lattice, are planned for future work.
5. Conclusion
We study a spatial stochastic model of two-species competition, inspired by the spatially
uniform system analyzed in [7]. Based on Janssen’s results for multicolored directed
percolation [14], we find that the flow of parameters under the renormalization group
is such that a small competitive advantage may be amplified to the point of excluding the
more numerous, but less competitive species, unless the latter reproduces rapidly. Thus
the less competitive population can go extinct, regardless of its size as given by mean-field
theory. These effects tend to be stronger, the smaller the dimensionality d of the system.
Our result shows that the survival of the scarcer phenomenon observed in [7] persists, in
some cases in a stronger form, in the presence of spatial structure. These differences are
evident when comparing figures 1 and 5.
Monte Carlo simulations of a one-dimensional lattice model verify the existence of
SSS when the more competitive species has a reproduction rate near the critical value,
while that of the less competitive one is above criticality. The latter species, although
more numerous in the quasi-stationary regime, loses out due to invasion by the more
competitive species.
Our results suggest that, in an ecological setting, a species with smaller population
density can in fact out-compete one with a higher density, even when the populations
are not well mixed, so that the spatiotemporal pattern is one of shifting domains. Under
conditions that favor survival of the scarcer, when both species are present in the same
niche, the more competitive species (B) tends to survive longer, even if its density is
smaller than that of the less competitive species (A). In isolation, however, species A
is less susceptible to extinction than is species B, again considered in isolation. This
suggests that, as long as both species are present globally, a cyclical dynamics of the form
A → B → 0 → A, . . ., may be observed in localized regions. That is, a region occupied
by species A is susceptible to invasion by B, which in turn is susceptible to extinction,
permitting the return of species A, and so on. Sequences of this kind are indeed seen
in figure 7. Given the simplicity of the model and the limitations of the present theoretical
and numerical analysis, the above conclusions should be seen as preliminary at best. They
may nevertheless suggest directions for further research in ecological modeling.
Acknowledgment
This work is supported by CNPq, Brazil.
References
[1] Hardin G, 1960 Science 131 1292
[2] Renshaw E, 1991 Modelling Biological Populations in Space and Time (Cambridge: Cambridge University
Press)
[3] Murrell D J, Purves D W and Law R, 2001 Trends Ecol. Evol. 16 529
doi:10.1088/1742-5468/2013/07/P07004 18
J.S
tat.M
ech.(2013)P
07004
Survival of the scarcer in space
[4] Condit R et al, 2000 Science 288 1414
[5] Silander J A Jr and Pacala S W, 1985 Oecologia 66 256
[6] Pacala S W and Deutschman D H, 1995 Oikos 74 357
[7] Gabel A, Meerson B and Redner S, 2013 Phys. Rev. E 87 010101
[8] Assaf M and Meerson B, 2006 Phys. Rev. E 74 041115
[9] Assaf M and Meerson B, 2010 Phys. Rev. E 81 021116
[10] Doi M, 1976 J. Phys. A: Math. Gen. 9 1479
[11] Peliti L, 1985 J. Physique 46 1469
[12] Tauber U C, Howard M and Vollmayr-Lee B P, 2005 J. Phys. A: Math. Gen. 38 79
[13] Cardy J, 1996 Scaling and Renormalization in Statistical Physics (Cambridge: Cambridge University Press)
[14] Janssen H, 2001 J. Stat. Phys. 103 801
[15] Barabasi A L and Stanley E, 1995 Fractal Concepts in Surface Growth (Cambridge: Cambridge University
Press)
[16] Dennis G R, Hope J J and Johnsson M T, 2013 Comput. Phys. Commun. 184 201
[17] Dickman R, 1994 Phys. Rev. E 50 4404
[18] Moro E, 2004 Phys. Rev. E 70 045102
[19] Dornic I, Chate´ H and Mun˜oz M A, 2005 Phys. Rev. Lett. 94 100601
[20] Hinrichsen H, 2000 Adv. Phys. 49 815
[21] De Oliveira M M and Dickman R, 2005 Phys. Rev. E 71 016129
[22] Dickman R and De Oliveira M M, 2005 Physica A 357 134
[23] Grassberger P and De La Torre A, 1979 Ann. Phys., NY 122 373
doi:10.1088/1742-5468/2013/07/P07004 19
Capı´tulo 4
A importaˆncia de ser discreto no sexo
Esta introduc¸a˜o e´ baseada nos resultados do artigo [5].
Uma das mais fascinantes questo˜es da biologia evolutiva envolve as razo˜es da manutenc¸a˜o da reproduc¸a˜o
sexuada entre os seres vivos. Esse processo esta´ associado a um elevado dispeˆndio energe´tico (disputas ter-
ritoriais, acasalamento e meiose) e expo˜e os indivı´duos a uma se´rie de riscos, como predac¸a˜o, parasitismo
e ferimentos causados em disputas por feˆmeas. Apesar disso, esse mecanismo e´ adotado pela maioria
das espe´cies multicelulares e apenas poucas espe´cies conhecidas de vertebrados adotam normalmente a
reproduc¸a˜o assexuada [35, 36, 37].
A onipresenc¸a da reproduc¸a˜o sexuada e´ especialmente intrigante quando consideramos que este modo
de reproduc¸a˜o esta´ associado a diversos custos [38][39]. Em primeiro lugar, existem os va´rios custos
associados ao acasalamento. Em muitas espe´cies, e´ preciso tempo e energia para garantir um companheiro.
Por exemplo, para garantir a polinizac¸a˜o, muitas plantas investem recursos substanciais na exposic¸a˜o floral
e nas recompensas de ne´ctar. Ale´m disso, o ato de reproduc¸a˜o sexuada e´ muitas vezes mais lento do que
o ato de reproduc¸a˜o assexuada, como pode ser visto em muitos micro´bios. Durante o acasalamento, os
indivı´duos normalmente sa˜o menos capazes de reunir recursos e fugir de predadores. O acasalamento
tambe´m introduz o risco de doenc¸as sexualmente transmissı´veis e elementos gene´ticos parasitas.
Nas espe´cies com sexos separados ou com tipos de acasalamento distintos, podem surgir conflitos sexu-
ais dispendiosos - Por exemplo, o fluido seminal da Drosophila conte´m toxinas que reduzem a aptida˜o das
feˆmeas acasaladas. Tambe´m existem, na reproduc¸a˜o sexuada, os custos de produc¸a˜o dos va´rios descenden-
tes, tais como o custo duplo do sexo: na reproduc¸a˜o sexuada, a unidade de reproduc¸a˜o e´ o casal, enquanto
que na reproduc¸a˜o assexuada, e´ o indivı´duo. A na˜o ser que o casal que se reproduz sexualmente possa pro-
duzir o dobro de descendentes do indivı´duo assexuado, os indivı´duos sexuados tera˜o necessariamente uma
produc¸a˜o reprodutiva per capita menor. Em um extremo, se os casais sexuados e indivı´duos assexuados
produzem o mesmo nu´mero me´dio de descendentes, devido ao fato de um parceiro sexuado na˜o contribuir
com recursos para a prole, a produc¸a˜o por indivı´duo reprodutivo nas espe´cies assexuadas e´ o dobro das
espe´cies sexuadas, portanto, o custo duplo do sexo. Por u´ltimo, e´ arriscado produzir descendentes mistu-
rando aleatoriamente genes com os de outro indivı´duo. Por todas as contas, o sexo deveria ser um beco
sem saı´da evolutivo, uma relı´quia que deveria ser observada apenas raramente.
A teoria da evoluc¸a˜o tem, em sua maior parte, mostrado que a resposta para o paradoxo do sexo e´ mais
elusiva do que inicialmente se pensava. A maioria dos bio´logos esta´ conforta´vel com a ide´ia de que o sexo
evoluiu para fornecer variabilidade, mas modelos matema´ticos provaram que esse conforto e´ injustificado:
o sexo na˜o precisa aumentar a variabilidade, a variabilidade na˜o precisa ser bene´fica e a evoluc¸a˜o na˜o
precisa ser a favor do sexo, mesmo quando ela faz aumentar a variabilidade e variabilidade e´ bene´fica [40].
Resolver o paradoxo do sexo exige que abracemos mais as complexidades do mundo real, e que utilizemos
modelos evolutivos de populac¸o˜es finitas, que sa˜o distribuı´das no espac¸o, e que esta˜o sujeitas a` selec¸a˜o
53
Capı´tulo 4. A importaˆncia de ser discreto no sexo
gerada por va´rias forc¸as ecolo´gicas, incluindo espe´cies co-evoluindo como predadores, competidores e
parasitas.
Neste artigo [5] apresento uma possı´vel explicac¸a˜o para esse aparente “paradoxo do sexo”. Proponho que
uma das razo˜es possı´veis para a predominaˆncia da reproduc¸a˜o sexuada na natureza se encontra no cara´ter
discreto das interac¸o˜es entre os indivı´duos (sexuados) e entre os indivı´duos e seus parasitas/predadores.
Tal cara´ter induz flutuac¸o˜es estatı´sticas endo´genas∗ na dinaˆmica populacional que, por sua vez, induzem a
emergeˆncia de fenoˆmenos a primeira vista inesperados. Devido a discreteza das interac¸o˜es, que esta´ inti-
mamente associada a` finitude das populac¸o˜es, os seres sexuados tendem a agir coletivamente e um agrupa-
mento entre os agentes emerge em dimenso˜es espaciais menores que d = 3.† Este fato decorre diretamente
da propriedade recorrente dos passeios aleato´rios em baixas dimenso˜es.‡ Consequeˆncia imediata desta
propriedade e´ que a densidade populacional localmente elevada favorece os encontros entre os indivı´duos
sexuados, fazendo sua populac¸a˜o florescer. O contra´rio ocorre para populac¸o˜es assexuadas: densidade
elevada localmente favorece encontros entre os indivı´duos e seus parasitas/predadores. Encontros entre os
indivı´duos sexuados e seus parasitas/predadores tambe´m ocorrem, mas seus efeitos sa˜o amenizados por
uma espe´cie de blindagem que os indivı´duos das camadas mais externas do aglomerado exercem sobre os
indivı´duos mais internos. Desta forma, a existeˆncia de sexo na natureza seria uma consequeˆncia puramente
fı´sica da natureza discreta das interac¸o˜es e pode na˜o ter nada a ver com gene´tica.
∗Endo´geno: diz-se de uma causa interna.
†Esta dependeˆncia na dimensa˜o espacial nos leva a uma conjectura: As espe´cies assexuadas tendem a existir nos oceanos, na
atmosfera, ou em algum outro meio “efetivo” tridimensional.
‡Para d < 3, dois indivı´duos perfazendo passeios aleato´rios devem necessariamente se encontrar para tempos suficientemente longos.
54
The Importance of Being Discrete in
Sex
Renato Vieira dos Santos∗
UFLA - Universidade Federal de Lavras
DEX - Departamento de Ciências Exatas
CEP: 37200-000, Lavras, Minas Gerais, Brazil.
∗econofisico@gmail.com
Abstract
The puzzle associated with the cost of sex, an old problem of evolutionary
biology, is discussed here from the point of view of nonequilibrium statistical
mechanics. The results suggest, in a simplified model, that the prevalence of
sexual species in nature can be a natural and necessary consequence of the
discrete character of the nonlinear interactions between couples and their
pathogens/parasites. Mapped into a field theory, the stochastic processes per-
formed by the species are described by continuous fields in space and time.
The way that the model’s parameters scale with subsequent iterations of the
renormalization group gives us information about the stationary emergent
properties of the complex interacting systems modeled. We see that the com-
bination of one aspect of the Red Queen theory with the stochastic processes
theory, including spatiotemporal interactions, provides interesting insights
into this old Darwinian dilemma.
| Paradox of sex | Red Queen theory | Stochastic processes | Renormalization group
1 Introduction
Sex is an evolutionary puzzle. In several ways, sexual reproduction is less efficient
when compared with the asexual method [1]. All offspring produced by asexual
individuals will be able to reproduce, whereas sexual beings need to spend en-
ergy on creating males and females that do not reproduce separately. Hence the
resources spent on producing sons are a cost of sexual reproduction and asexual
species economize on males. John Maynard Smith [2] summarized this argument
as follows:
“Suppose a population consists of a mixture of sexual and partheno-
genetic females, the former producing equal numbers of male and
(sexual) female offspring, and the latter only parthenogenetic females
like themselves. If the two kinds of female lay equal numbers of
eggs, and if survival probabilities are equal, then the parthenogenetic
type will have a twofold selective advantage, and will increase in fre-
quency very rapidly. Sexual reproduction means that a female wastes
half her energy producing males.”
He also noted that a sexual individual uses only half of its genetic material
on its descendents, while an asexual individual uses all his sexless genes. That
is, in the evolutionary race where passing on genes to the next generation is one
of the greatest goals, sexual organisms starts with a disadvantage of almost 50%,
which is known as the cost of meiosis [1]. There is also the risk of infection
by sexually transmitted diseases, the fact that sexual reproduction is often slower
than asexual reproduction, and that during mating, individuals are typically less
able to gather resources and evade predators. In addition to these disadvantages,
and perhaps more important, is the cost of having to find a mate [3]. If insects
are excluded, approximately one-third of animal species are hermaphrodites [4].
Hermaphroditism is even more widespread in plants. The difficulty of finding
mates is widely implicated in the evolution of hermaphroditism, so its widespread
occurrence suggests that sexual organisms incur significant costs to locate mates
[5]. Sex, therefore, seems to be a luxury that should not exist. Consequently,
many works about its evolution look for it’s compensatory benefits.
Since sexual reproduction exists, biologists try to find out what great benefit it
brings to living beings. Maynard Smith argued that sex could only have evolved
if this mysterious benefit at least outweighed the great cost of meiosis. But what,
after all, could this benefit be? To answer this question, an audacious theory about
the origin and perpetuation of sex was proposed in [6]. According to this work,
the parasites are everywhere and will always seek, by their nature, to explore their
hosts. As the generation time of parasites is many times smaller than that of hosts,
1
and their evolution rates therefore many times higher, the only way out for the
hosts is to produce offspring with greater genetic variability through sexual repro-
duction. Therefore, competition with parasites that develop very fast genetically,
favors sexual reproduction, which enables a more efficient genetic evolution [7, 8].
The world in which this model is inserted became known as the Red Queen’s
world, a name given in [9] in reference to a passage in the fable Alice in the
mirrors [10]. In this passage, Alice flees the army (of cards) of the Red Queen,
but can not distance herself from her pursuers. The Red Queen then says: “Now,
here, you see, it takes all the running you can do, to keep in the same place”. Alice
would be caught only if she stopped running. Things have to change to remain the
same.
According to [6], an arms race has been underway between hosts and parasites
since life appeared on Earth. The parasites are always breaking the defensive
barriers imposed by the host’s genotype, while the host, with the help of sex,
continually creates new defenses. In the absence of sex, the hosts would remain
essentially the same, while the parasites would accumulate adaptations that would
enable them to break all the defensive systems of the former. Sooner or later,
the hosts would be virtually devoured from the inside out. To escape the parasite
army besieging them, the only remaining option is to just keep running. The co-
evolutive cycle of parasites and hosts reflects this eternal pursuit. For criticism to
the Red Queen theory, see [11, 12].
The aim of this paper is to investigate the prevalence of sexual reproduction
observed in nature through simple models of reaction-diffusion inspired by the
Red Queen theory, but not fully equivalent to it. The role of the parasite may
be replaced by any pathogen which diffuses through space and fatally harms the
species. There is also no need for any aspect related to genetics [13].
Over the years, evolutionary theory has shown that the answer to the para-
dox of sex is more elusive than we initially thought. While most biologists are
comfortable with the idea that sex exists to provide genetic variability, several
mathematical models have shown that this comfort is unjustified. According to
[14], “...the theory of population genetics, as complete as it may be in itself, fails
to deal with many problems of primary importance for an understanding of evo-
lution.” After all, it is risky to produce offspring by randomly mixing genes with
those of another individual. In the words of [15]:
“... sex need not increase variability, variability need not be bene-
ficial and evolution need not favour sex, even when it does increase
variability and variability is beneficial.”
There is also a well-documented pattern of high frequency of sex in undis-
turbed, biologically complex habitats [16, 17, 18, 19] where disease and other
2
“natural enemies” are prevalent [16, 17, 20]. The recognition of the predomi-
nance of sex in stable communities has devalued models hypothesizing that sex
is common because it provides a selective advantage when abiotic conditions are
unpredictable [17, 21, 22].
Moreover, the need to take into account aspects of complexity that are rou-
tinely neglected is widely recognized [15]. One reason that an answer to the para-
dox of sex has been so elusive is that many mathematical models have focused on
populations that are infinite in size, unstructured, and isolated from other species.
Satisfactory explanations for the paradox of sex should consider finite populations
of agents that interact in an environment where structure and complexity are able
to emerge.
To incorporate a few of these elements, we take into account the discrete na-
ture of the species interactions with itself and with its pathogens/parasites, in a
d−dimensional space. Discreteness of the interactions is a consequence of the
finiteness of the populations, interactions with parasites characterize the non-
insulation explicitly, and a d−dimensional space enables structure and complexity
to emerge.
We know that if we want information about the emergent aggregate macro-
scopic behavior of complex systems, we’ll need to consider the corpuscular char-
acter of interacting species [23]. We will achieve this goal by employing dynamic
renormalization group (DRG) theory to obtain the renormalization group (RG)
flow in the parameter space [24, 25]. Starting from the microscopic formulation
of the model described by reactions, this RG flow will allow us to understand how
the model parameters scale in space and time. In turn, this information will be
helpful in determining the final equilibrium state of the aggregates of interacting
species. Such aggregates imply the advantage of sex on the population- or group-
level [26, 27, 28, 29]. This approach is consistent with the perspective of exten-
sive time and space scales taken by macroecology and biogeography [30, 31, 32].
Large-scale spatial correlations are likely to be important for understanding the
evolutionary influence of predator and prey or host and pathogen ecology [33].
It’s worth noting that it is not our intention to refute the wide existing knowl-
edge of population genetics about the paradox of sex. It is intended only to call
attention to an intrinsic feature of sexual reproduction that’s been neglected so
far. The longevity of the problem indicates that the solution requires a variety of
causal effects, and what we propose here could be just one of these effects.
1.1 Models
In this subsection we present the two models used. They are simplified models
that attempt to capture only the essential aspects of population dynamics. The
first refers to the competition between an asexual species and a pathogen that can
3
harm it, eventually inducing death. The second is the analogous model for the
sexual species. The incorporation of the pathogen in interactions was inspired by
the Red Queen theory regarding the host’s parasites. In principle, however, any
other death-inducing agent can be imagined. The models are as follows:
Asexual species model
The model for asexual species is described by the following reactions, occurring
in a d−dimensional lattice:
A
λ
⇀ 2A A + B
µ
⇀ B (1)
The first reaction on the left describes the reproduction of species A which
occurs at rate λ per time unit, at a given site of the lattice. The second reaction
describes the attack that species A suffers from its pathogen B. In this attack,
species A will always be annihilated at a rate of µ per time unit per population
size unit. Note that in a model that takes the spatial character of the interactions in
a d dimensional lattice, pathogen/parasites only diffuse.∗ They are not created or
annihilated. This captures their essence of being everywhere and always seeking,
by their nature, to exploit their hosts, as mentioned above.
Sexual species model
The model for sexual species is described by the following reactions:
2A
λ
⇀ 3A A + B
µ
⇀ B (2)
The first reaction describes species A′s reproduction that occurs at rate λ per
time unit per population size unit, always at a given site of the lattice. Because two
agents are required to reproduce a third, this reaction captures the cost of finding
a mate. Everything else is as in the previous model.
This is not the first time that a model of this type is proposed for population
dynamics incorporating the Allee effect† [34, 35, 36] on the lattice. For recent
references, see [37, 38]. In the process known as the quadratic contact process
(QCP) [39], we also have similar reactions. QCP is sometimes called the process
of sexual reproduction [40].
It is very important to keep in mind that we are considering the critical situ-
ation where the concentration of species in the lattice is initially very small and
∗This diffusion process (for the species B, for example) could be represented by B +∅ → ∅+ B,
where ∅ is an empty site of the lattice.
†For all sexual populations, there is a density threshold below which the probability of finding
a mate is too low to ensure sufficient reproduction for the population to remain viable.
4
the dynamics is dominated by diffusion [41]. The average number of agents per
site is initially much lower than 1 and sexual reproduction is therefore penalized.
Only in this critical regime does it make sense to use the renormalization group
techniques.
2 Results
In the following subsection (2.1) we consider A or B as the average density of
agents in the sites of the lattice.
2.1 Mean field theory
Asexual model
Using the law of mass action, we obtain the differential equations for asexual
species: A˙ = DA∇2 A + λA − µAB and B˙ = DB∇2B with DA and DB being
diffusion coefficients. The diffusion processes only tend to homogenize popu-
lations in space and nabla operators will be neglected from now on in this sub-
section. The B population is a constant on average denoted by 〈NB〉 and therefore
A˙ = (λ−µ〈NB〉)A ≡ mA,which defines m = λ−µ〈NB〉. We see that if µ〈NB〉 < λ,
m > 0 and the B population tends exponentially to infinity. Otherwise, m < 0 and
the B population becomes extinct. If m = 0, the A population remains constant.
Sexual model
In this case, the equation for population dynamics already disregards diffusion
terms and settings κ ≡ µ〈NB〉 is A˙ = λA2 − κA ≡ −dV/dA with V ≡ −λA3/3 +
κA2/2. V is an effective potential that allows a pictorial view of the dynamics, as
illustrated in Figure (1). The point P on the potential maximum has coordinates
(Amax,Vmax) = (κ/λ, κ3/6λ2). For any initial population A(0) < κ/λ, the popula-
tion tends to be extinct. This fact is illustrated in figure (1) by the tendency of the
red ball to moves down the curve to the origin, and characterizes the aforemen-
tioned Allee effect. If A(0) > κ/λ, population tends to infinity, a fact represented
by the tendency of the green ball to get lost in the bottomless potential hole.
In the next subsections we will see how the κ, λ and µ parameters change with
successive renormalization group iterations, or, in other words, how the discrete
nonlinear species interactions in space-time induce variations in the numerical
parameter values. These changes can transform very significantly the potential
barrier to be overcome (given by κ3/6λ2) by the population.
5
Figure 1: Effective potential V (A). The black point P has coordinates
(Amax,Vmax) = (κ/λ, κ3/6λ2).
2.2 Statistical Field Theory
Asexual model
We are interested in the way the parameters λ and m vary when we increase the
observation scales of time and space. Using the results of statistical field the-
ory [42, 43, 44], we get the following flow equations for these parameters in the
asexual model (see section 5):
dµ
dl
=  µ +
µ2
2piD¯
(3a)
dm
dl
= 2m − 〈NB〉µ
2
2piD¯
, (3b)
where  = 2−d (d is the dimension of space), l = ln (s) (s is the escale parameter,
see section 5), and D¯ ≡ (DA + DB)/2. Competition parameter µ increases indef-
initely with the renormalization group iterations, favoring the extinction of the
asexual species. This fact is interpreted in section 3 using the re-entrant property
of diffusive systems in low dimensions.
Equations (3) are identical to those obtained in [23, 45] for the reactions B
µ
⇀
∅, A + B λ⇀ 2B + A, with bare mass mAB ≡ µ − λ〈NA〉, with 〈NA〉 representing
the average number of A. This model is known as the AB model [46]. It has been
originally proposed in [23, 47, 46, 48, 45] to discuss the origin of life in terms of
auto-catalysis, and it has been applied in some research areas such as ecology and
economy [49, 50, 51]. The spatial version of this model shows that self-replication
can be locally maintained with B growing exponentially, even when average A
6
concentration would not be sufficient to sustain growth in a homogeneous vessel.
This fact is a consequence of the tendency of µ to grow with the scale s, as shown
by the equations (6), and from the definition of mAB . Exactly the opposite will
occur in the case of the model with asexual population, since in this case m =
λ − µ〈NB〉 and therefore µ is subtracted rather than added.
Figures (2a) and (2b) show the RG flow diagrams associated with equations
(6) for µ ≥ 0. On the left we have the case of  < 0 (or d > 2). The black
dot is the fixed point given by (µ∗,m∗) = (2piD¯, D¯pi2〈NB〉). The diagonal line
represents an eigenvector indicating two distinct behaviors of the diagram near the
fixed point. The horizontal dotted line represents m = 0. Above the straight line
and for m > 0, the RG flow tends to take m to infinity. In this case the asexual
species population explodes. This happens for a sufficiently small µ. The opposite
occurs below the line (i.e., for sufficiently large µ), with the RG flow inducing m
to negative values, inducing the population to extinction, even with the mean field
theory indicating explosion. We may call this phenomenon Discreteness Inducing
Extinction (DIE).
More interesting is the figure on the right, where the DIE phenomenon is cer-
tain across the parameter space (for µ > 0) for d ≤ 2 (or  ≥ 0). On the surface,
asexual species always die.
(a) RG flow for  < 0. (b) RG flow for  ≥ 0.
Figure 2: RG flow asexual model
Sexual model
Let’s compare the effects caused by the discrete character of the interactions in
asexual and sexual reproduction. For the model of sexual reproduction, there is
7
one more parameter (λ) flowing due to the iterations of the renormalization group.
This is due to the nonlinearity of the process 2A
λ
⇀ 3A. The flow equations are:
dµ
dl
=  µ +
µ2
2piD¯
(4a)
dκ
dl
= 2κ − 〈NB〉µ
2
2piD¯
(4b)
dλ
dl
= λ +
λ2
piD¯
(4c)
where  = 2 − d, l = ln (s), and D¯ ≡ (DA + DB)/2. The new λ RG flow does not
influence κ and therefore RG flows involving κ and µ are as shown in figures (2a)
and (2b) by replacing m with κ in the vertical axes. We can now examine how the
potential barrier (given by Vmax = κ3/6λ2) in figure (1) varies when the parameters
are renormalized. We see that on the surface (or in smaller dimensions:  ≥ 0),
this potential barrier disappears quickly, favoring the sexual species. This happens
because of how quickly λ approaches infinity (with decreasing κ), making Vmax →
0. The barrier that prevented the sexual population’s development is increasingly
transposable if there is enough space and time for the interactions to occur. And
this fact arises from the interaction’s discreteness. This is the importance of being
discrete in sex. Furthermore, according to our simplified models, the only chance
for an asexual species to exist in nature, is in a three dimensional space. This
finding leads to the conjecture that most asexual species existing today, live in the
oceans, in the air, or in other “effective” three-dimensional media.
3 Discussion
3.1 General discussion
A possible objection to the sexual model is that various particles can accumulate
in one lattice site, leading to a divergence in the population due to the reaction
2A
λ→ 3A. In principle, the amount of particles per site can be infinite, but for our
purposes, it does not matter. One possible argument goes as follows:
If we establish the same initial conditions‡ for the two models, with both
species on the verge of extinction, the sexuated species will win unequivocally,
after a transient. And this occurs twice: once because of the certainty of extinc-
tion of the asexual species (at least for d = 2) and once because of the trend of
‡Random initial positions of individuals in the lattice, without the overlapping of two or more
individuals at the same site.
8
sexuated species to diverge. It does not matter if ultimately the amount of indi-
viduals per site becomes greater than one. From this point on, our minimal model
is no longer valid and constraints must be added, but the battle has already been
won by the sexuated species.
Is it possible to understand physically why the description based on differ-
ential equations (or mean field theory) becomes invalid for long timeframes and
large scales in lower dimensions? To answer this question, let’s first consider a
different situation, for the sake of simplicity. Consider a one-dimensional lattice,
where nearly all lattice sites are populated by particles of species A, such that the
only reaction occurring is the annihilation reaction A + A
α→ ∅. In principle, this
reaction can take place everywhere. The initial dynamics is therefore accurately
described by the solution of the kinetic rate equation ∂ρ∂t = DA∇2ρ − αρ2, where
ρ(x, t) is the concentration of A individuals in the lattice and DA is the diffusion
constant. Therefore, initially, ρ ∝ t−1. At later times, the lattice becomes more
and more diluted and the reaction rate is limited by the first passage time of a
random walk in d = 1, scaling exactly with t−1/2. Hence, the long time behavior
of ρ is renormalized to ρ ∝ t−1/t−1/2 = t−1/2. This is a simple example of how the
reactions limited by diffusion might provide different results from the equivalent
models described in terms of differential equations.
The probability of species B (the parasites) finding species A is 1 for d < 3
for asexual species at the limit of the reaction limited by diffusion because of the
re-entrant property of random walks in low dimensions. Physically, this means
that the diffusing particle will thoroughly sweep its neighborhood. It’s therefore
highly probable that it will react with another particle in its vicinity. Hence, it is
reasonable to expect that after a short period of time, the system will be in a state
where there are almost no individuals of species A. This phenomenon no longer
occurs in d > 2 and asexual species may exist.
In the case of sexual reproduction, two nonlinear phenomena are in compe-
tition: 2A
λ→ 3A and A + B µ→ ∅. As we have seen, however, the former wins
from the latter and the tendency to form clusters of sexed individuals predomi-
nates. This is due to the “shielding” effects that the boundaries of the clusters
have on the individuals within it. It is therefore reasonable to expect that after
a short period of time the system will be in a state where there are a lot of iso-
lated particles. This leads to the interesting question whether “ancestral” sexed
reproductions are responsible for habitat heterogeneity (through the formation of
clusters of individuals) rather than the heterogeneity of the environment induc-
ing mating [52, 53, 54, 55]. We are referring to the chicken or the egg causality
dilemma. Taking this idea to the extreme, we might ask: is there sex because of
demographic noise?
9
3.2 Some remarks
Some specific comments related to our results and to the literature.
Importance of the space
Several authors have indicated the importance of considering spatial distribu-
tion to explain the paradox of sex [56, 57, 58, 59, 52, 60, 61]. A population’s spa-
tial structure is well known to have significant evolutionary consequences [62].
For prey-predator or host-pathogen interactions, theory has convincingly shown
that spatial structure can lead to the evolution of a reduced predator-pathogen at-
tack rate [63, 64, 65]. This involves mechanisms that are analogous to the shield-
ing mentioned before.
The evolutionary influence of a population’s spatial structure is particularly
compelling because it bears on the fundamental evolutionary question of the emer-
gence of cooperation. Theory suggests that spatial structure is a key requirement
for the emergence of cooperation [66, 67, 68, 69, 70, 71]. Notably, spatial struc-
ture appears to resolve the tragedy of the commons [72, 73, 74]. Spatial structure
also seems to imply biodiversity and species coexistence [75, 76, 77, 78, 79].
Importance of Discreteness
Many fields of science have come to the realization that complex phenomena
can be explained by supposing the existence of discrete underlying levels that can
be described using integers. Atomic theory is perhaps the best example. In bi-
ology, a consistent theory of evolution was developed by introducing the concept
of genes as the quantum of inheritance information [80]. Various phenomena of
modern physics, such as the photoelectric effect, the radiation spectrum of stars,
the universal temperature dependence of the specific heat of solids, can be ex-
plained assuming an underlying discrete level of some physical quantity.
Demographic stochasticity, which that in this paper emerges from the dis-
creteness of the interactions, is known to be important in population dynamics
[81]. The inclusion of stochasticity into non-linear mathematical models affects
the mean dynamics [62, 82, 83]. We can cite as example the ability of demo-
graphic stochasticity to excite macroscopic-scale coherent oscillations, known as
quasi-cycles [84]. An extension to the spatial case where spatiotemporal patterns
are induced by demographic noise has also been observed [85]. Generation [86]
and exacerbation [87] of the Allee effect are also consequences of the discrete
character of the interactions associated with demographic noise, which can even
induce survival when mean field theory indicates extinction [23]. The importance
of this intrinsic noise in the microscopic dynamics of cellular systems has also
10
been studied intensively in recent years [88], leading to a flurry of papers in this
area [89]. These facts indicate a considerable effervescence in research associated
with stochastic phenomena related to the inevitable discrete character of the inter-
actions [90].
Sex and density
Our results suggest that in the long term the sexual species agglomerate in
regions of high relative density. Some experimental studies have associated high
population density to an advantage for sexual reproduction. In [91], the frequency
of sexual reproduction in a population is found to increase with the intensity of
intraspecific competition in a lichen species. Several mathematical models indi-
cate positive dependence between density and sexual reproduction [92]. Many of
them are related to the Tangled Bank hypothesis [93], an idea based on previous
studies about the role of spatial heterogeneity in the maintenance of sex [94, 95].
Geographic Parthenogenesis
There is some observational evidence of the phenomenon which we call shield-
ing [96]. One of the characteristics of so-called geographic parthenogenesis (GP)
is adaptation at the margin. Geographical parthenogenesis is a term that describes
the phenomenon that related sexual and asexual organisms have different distri-
bution areas, and adaptation at the margin of asexual species is one of their char-
acteristics. The sexual species tends to concentrate in the central regions, being
surrounded by the asexual species [96]. A mathematical model [97] suggests that
asexuals could more easily adapt to environmental conditions in boundary areas.
Another characterization is that GP is the phenomenon where sexual popula-
tions are restricted to a core area while parthenogenetic lineages occupy a wider
geographic range [96]. Such segregation is not always complete. Sometimes there
is coexistence between the two reproduction modes as is common in freshwater
ostracods [98, 99]. Understanding the coexistence of sexuals and asexuals on a
local scale is not simple because of the competitive exclusion principle [100, 101].
However, there are theoretical indications that the discreteness of interactions be-
tween individuals in populations can also induce coexistence [79].
4 Conclusion
Understanding how complex ecosystems work often relies on simplified models
that disregard many details of the actual system while retaining the essential infor-
mation [102, 103]. In this paper we propose a simplified model that can shed some
11
light on the question of the predominance and maintenance of sexual reproduction
in nature despite all its costs. Sex is a reproductive ritual that is inherently more
complex than its rival asexual method. And this inherent complexity gives rise to
some counterintuitive features. The complexity aspect discussed here refers only
to the nonlinear interactions between species and their pathogens/parasites, and to
the cost of finding a mate in the case of sexual selection. Mathematically, this cost
implies a nonlinearity (coming from the reaction 2A→ 3A) which is absent in the
asexual reproduction model. And from this nonlinearity in the interactions, purely
physical conditions emerge that favor sexual reproduction. We need not consider
anything about genetics for example.
Another important actor in this context of complexity is the discrete charac-
ter of the interactions. This actor is solely responsible for the DIE phenomenon
in asexual species, where the extinction is possible for d < 2 and certain for
d ≥ 2, even if the mean field theory indicates otherwise. The intrinsic stochas-
ticity induced by this discreteness is also responsible for effectively raising the λ
parameter, as seen through the RG flow for the sexual model. This fact allows for
the development of sexual populations, despite its considerable costs for finding a
mate, even in situations not covered by the mean field theory. A phenomenon able
to induce this increase is the aggregation in clusters of interacting agents. A well-
known property of diffusion is the re-entrancy of the visited sites in low space
dimensions. In particular, for d = 1 and d = 2, the probability that the diffusing
particle will ever return (t → ∞) to the starting point is equal to 1. Physically, it
means that the diffusing particle thoroughly sweep its neighborhood and thus it is
highly probable that it will react with another particle in its vicinity. Hence, it is
reasonable to expect that after short period of time the system will be in a state
where there is a lot of isolated particles.§ The clustering of sexual agents favors a
localized increase in the λ reproduction rate, allowing their permanence and de-
velopment. These aggregates, emergent phenomena in our model, helps to explain
the group-level advantage of sex and their associated experimental observations
[104, 105]. We must not forget that an individual cannot reproduce arbitrarily fast.
This imposes an upper limit for λ. Briefly, I propose that:
One aspect of the explanation for the maintenance of sexual reproduction in
nature is in the scale of populations, far above the molecular scale of the gene,
and manifests itself as an emergent property of the discrete interactions in the
intermediate scale of the individuals.
§A similar phenomenon also occurs among species and their pathogens/parasites, making the
death rate also rises, explaining the extinction of asexual species in d ≤ 2. However, this increase
is insufficient to overcome the increased of the sexual species for d ≤ 2.
12
5 Materials and Methods
Asexual model
In the Doi-Peliti theory, the master equation which describes the evolution of the
probability vector |P(t)〉 ≡ ∑C p(C, t) |C〉,with the sum being performed over all
configurations, is written in the form d |P〉/dt = H |P〉, where the Hamiltonian
operator H is composed of creation and destruction operators. Starting from this
“microscopic” description, one derives an effective action S via a path-integral
representation [24]. Then, taking the continuum limit, one arrives at a field theory
for the model.
For the asexual model, Doi-Peliti action, already with the following Doi shifts
φ˜→ 1 + φ¯, ψ˜ → 1 + ψ¯, ψ → ψ + 〈NB〉, and φ→ φ, is
S[φ¯, φ, ψ¯,ψ] =
∫
dd x
∫
dt
[
φ¯(∂t − m − DA∇2)φ
+ ψ¯(∂t − DB∇2)ψ − λφ¯2φ + µφ¯φψ
+ µ〈NB〉φ¯ψ¯φ + µφ¯ψ¯φψ
]
(5)
where m ≡ λ − µ〈NB〉 is the bare mass. φ and ψ are fields associated with the
populational densities of A and B respectively, while φ¯ and ψ¯ are related to their
statistical fluctuations.
Let us assume that the parameters µ and λ are adjusted so that the species
are on the verge of extinction for perturbation theory to be valid [106]. Feynman
diagrams associated with the action (5) are shown in Figure (2).
Figure 3: Feynman diagrams
13
5.1 Dynamical renormalization group
We use the field theory techniques to find the renormalization group flow in the
parameter space. The system will be analyzed using the standard renormalization
group technique, imposing the change of scale x → sx, t → szt, φ→ s−d−ηφ,ψ →
s−d−ηψ (similarly to φ¯, ψ¯), and Λ → Λ/s, where s is the renormalization group
scale factor, η is a critical exponent, and Λ is a momentum cuttoff. Performing the
standard perturbation theory procedures [107], using the diagrams combinations
I I and I I I (propagator renormalization, see Figure (3) left) and I I and IV (vertice
renormalization, see Figure (3) right), we find the following flow equations for the
model parameters in the limit of Λ→ ∞ [23]:
dµ
dl
=  µ +
µ2
2piD¯
(6a)
dm
dl
= 2m − 〈NB〉µ
2
2piD¯
, (6b)
where  = 2 − d, l = ln (s), and D¯ ≡ (DA + DB)/2.
Figure 4: Propagator renormalization diagram I I + I I I (left) and vertice renor-
malization diagram I I + IV (right).
Sexual model
The Doi-Peliti effective action for the sexual reproduction model is:
S[φ¯, φ, ψ¯,ψ] =
∫
dd x
∫
dt
[
φ¯(∂t + κ − DA∇2)φ
+ ψ¯(∂t − DB∇2)ψ − λφ¯φ2 + µφ¯φψ
+ µ〈NB〉φ¯ψ¯φ + µφ¯ψ¯φψ2λφ¯2φ2
− λφ¯3φ2
]
(7)
where κ ≡ µ〈NB〉. Field interpretations are as before.
An important feature of this model is the diagram V in figure (5). We also
should replace the diagram I in figure (2) with the diagram V I in figure (5).
14
Figure 5: Some Feynman diagrams in sexual model.
Figure 6: Vertice renormalization (Feynman diagram V + V I) for sexual model.
Now, the crucial point is that the parameter λ can be explicitly renormalized
using diagrams V and V I (see Figure (6)). Performing the basic steps mentioned
before, we have the following RG flow equations for the sexual species model:
dµ
dl
=  µ +
µ2
2piD¯
(8a)
dκ
dl
= 2κ − 〈NB〉µ
2
2piD¯
(8b)
dλ
dl
= λ +
λ2
piD¯
(8c)
where  = 2−d, l = ln (s), and D¯ ≡ (DA+DB)/2. The diagram in Figure (6) which
renormalizes λ is equal to the diagram in Figure (4) at right, which renormalizes
µ,with the difference of a factor 2 in the former. Associating this similarity to the
fact that the propagators are also very similar,¶ the results are almost identical and
we obtain an expression for the RG flow for λ very similar to the expression of µ.
¶The propagator for the field φ in equation (7) is given by Gφφ[k,ω] = (Dk2 + κ − iτω)−1.
Replacing κ by −m we obtain the propagator for the field φ in equation (5). The propagators for
the fields ψ,Gψψ[k,ω], are identical for both models.
15
6 Acknowledgments
RVS is grateful to Linaena Méricy da Silva for helpful comments. This work
was supported by the Conselho Nacional de Desenvolvimento Científico e Tec-
nológico, Brazil.
16
Bibliography
[1] Lehtonen J, Jennions MD, Kokko H (2012) The many costs of sex. Trends
in ecology & evolution 27: 172–178.
[2] Smith J, Dawkins R (1993) The Theory of Evolution. Canto Series. Cam-
bridge University Press.
[3] Daly M (1978) The cost of mating. The American Naturalist 112: 771–
774.
[4] Jarne P, Auld J (2006) Animals mix it up too: the distribution of self-
fertilization among hermaphroditic animals. Evolution 60: 1816–1824.
[5] Eppley S, Jesson L (2008) Moving to mate: the evolution of separate and
combined sexes in multicellular organisms. Journal of evolutionary biology
21: 727–736.
[6] Hamilton W, Zuk M (1982) Heritable true fitness and bright birds: a role
for parasites? Science .
[7] Lively CM (2010) A review of red queen models for the persistence of
obligate sexual reproduction. Journal of Heredity 101: S13–S20.
[8] Liow LH, Van Valen L, Stenseth NC (2011) Red queen: from populations
to taxa and communities. Trends in ecology & evolution 26: 349–358.
[9] van Valen L (1973) A new evolutionary law. Evolutionary Theory 1: 1–30.
[10] Carroll L (2000) Alice’s Adventures in Wonderland and Through the Look-
ing Glass. Perma-Bound Books.
[11] Otto SP, Nuismer SL (2004) Species interactions and the evolution of sex.
Science 304: 1018–1020.
[12] Elzinga J, Chevasco V, Mappes J, Grapputo A (2012) Low parasitism rates
in parthenogenetic bagworm moths do not support the parasitoid hypothesis
for sex. Journal of evolutionary biology 25: 2547–2558.
17
[13] Agrawal AF (2006) Evolution of sex: why do organisms shuffle their geno-
types? Current Biology 16: R696–R704.
[14] Lewontin RC (1961) Evolution and the theory of games. J Theor Biol 1:
382-403.
[15] Otto SP, Lenormand T (2002) Resolving the paradox of sex and recombi-
nation. Nature Reviews Genetics 3: 252–261.
[16] Levin DA (1975) Pest pressure and recombination systems in plants. Amer-
ican Naturalist : 437–451.
[17] Glesener RR, Tilman D (1978) Sexuality and the components of environ-
mental uncertainty: clues from geographic parthenogenesis in terrestrial
animals. American Naturalist : 659–673.
[18] BARRETT SC, ECKERT CG (1990) 14 variation and evolution of mating
systems in seed plants. Biological approaches and evolutionary trends in
plants : 229.
[19] Hamilton WD, Axelrod R, Tanese R (1990) Sexual reproduction as an
adaptation to resist parasites (a review). Proceedings of the National
Academy of Sciences 87: 3566–3573.
[20] Bell G (1982) The masterpiece of nature: the evolution and genetics of
sexuality. CUP Archive.
[21] Smith JM, Maynard-Smith J (1978) The evolution of sex. Cambridge Univ
Press.
[22] Price MV, Waser NM (1982) Population structure, frequency-dependent
selection, and the maintenance of sexual reproduction. Evolution : 35–43.
[23] Shnerb N, Louzoun Y, Bettelheim E, Solomon S (2000) The importance
of being discrete: Life always wins on the surface. Proceedings of the
National Academy of Sciences 97: 10322.
[24] Tauber UC, Howard M, Vollmayr-Lee BP (2005) Applications of field-
theoretic renormalization group methods to reaction-diffusion problems. J
Phys A-Math Gen 38: 79.
[25] Santos RV, Dickman R (2013) Survival of the scarcer in space. Journal of
Statistical Mechanics: Theory and Experiment 2013: P07004.
18
[26] Rice WR, Chippindale AK (2001) Sexual recombination and the power of
natural selection. Science 294: 555–559.
[27] Colegrave N (2002) Sex releases the speed limit on evolution. Nature 420:
664–666.
[28] Poon A, Chao L (2004) Drift increases the advantage of sex in rna bacte-
riophage φ6. Genetics 166: 19–24.
[29] Goddard MR, Godfray HCJ, Burt A (2005) Sex increases the efficacy of
natural selection in experimental yeast populations. Nature 434: 636–640.
[30] Beck J, Ballesteros-Mejia L, Buchmann CM, Dengler J, Fritz SA, et al.
(2012) What’s on the horizon for macroecology? Ecography 35: 673–683.
[31] Loreau M (2010) From Populations to Ecosystems: Theoretical Founda-
tions for a New Ecological Synthesis (MPB-46). Monographs in Popula-
tion Biology. Princeton University Press.
[32] Brown J (1995) Macroecology. Biology/Ecology. University of Chicago
Press.
[33] Messinger SM, Ostling A (2013) Predator attack rate evolution in space:
The role of ecology mediated by complex emergent spatial structure and
self-shading. Theoretical Population Biology 89: 55 - 63.
[34] Allee W, Bowen ES (1932) Studies in animal aggregations: mass protection
against colloidal silver among goldfishes. Journal of Experimental Zoology
61: 185–207.
[35] Allee W, Park O, Emerson A, Park T, Schmidt K (1949) Principles of ani-
mal ecology. WB Saunders London.
[36] Courchamp F, Berec L, Gascoigne J (2008) Allee effects in ecology and
conservation. Oxford University Press, Oxford , New York.
[37] Windus A, Jensen H (2007) Allee effects and extinction in a lattice model.
Theoretical population biology 72: 459–467.
[38] Gastner MT, Oborny B, Ryabov AB, Blasius B (2011) Changes in the gra-
dient percolation transition caused by an allee effect. Physical Review Let-
ters 106: 128103.
[39] Durrett R (1999) Stochastic spatial models. Siam Review 41: 677–718.
19
[40] Preece T, Mao Y (2009) Sustainability of dioecious and hermaphrodite
populations on a lattice. Journal of theoretical biology 261: 336–340.
[41] Mattis DC, Glasser ML (1998) The uses of quantum field theory in
diffusion-limited reactions. Reviews of Modern Physics 70: 979.
[42] Parisi G (1988) Statistical Field Theory. Fron-
tiers in Physics. Addison-Wesley. URL
http://books.google.com.br/books?id=OF8sAAAAYAAJ.
[43] Mussardo G (2010) Statistical Field Theory: An Introduction to Exactly
Solved Models in Statistical Physics. Oxford Graduate Texts. OUP Oxford.
URL http://books.google.com.br/books?id=JnLnXmBmCzsC.
[44] Täuber UC (2007) Field-theory approaches to nonequilibrium dynamics.
In: Ageing and the Glass Transition, Springer. pp. 295–348.
[45] Shnerb N, Bettelheim E, Louzoun Y, Agam O, Solomon S (2001) Adapta-
tion of autocatalytic fluctuations to diffusive noise. Phys Rev E 63: 021103.
[46] Louzoun Y, Shnerb N, Solomon S (2007) Microscopic noise, adaptation
and survival in hostile environments. The European Physical Journal B-
Condensed Matter and Complex Systems 56: 141–148.
[47] Agranovich A, Louzoun Y, Shnerb N, Moalem S (2006) Catalyst-induced
growth with limited catalyst lifespan and competition. Journal of theoreti-
cal biology 241: 307–320.
[48] Louzoun Y, Solomon S, Atlan H, Cohen I (2003) Proliferation and compe-
tition in discrete biological systems. Bulletin of mathematical biology 65:
375–396.
[49] Challet D, Solomon S, Yaari G (2009) The universal shape of economic
recession and recovery after a shock. Economics: The Open-Access, Open-
Assessment E-Journal 3.
[50] Solomon S, Goldenberg J, Mazursky D (2003) World-size global markets
lead to economic instability. Artificial life 9: 357–370.
[51] Yaari G, Nowak A, Rakocy K, Solomon S (2008) Microscopic study reveals
the singular origins of growth. The European Physical Journal B 62: 505–
513.
[52] Becks L, Agrawal AF (2010) Higher rates of sex evolve in spatially hetero-
geneous environments. Nature 468: 89–92.
20
[53] Candolin U, Salesto T, Evers M (2007) Changed environmental conditions
weaken sexual selection in sticklebacks. Journal of Evolutionary Biology
20: 233–239.
[54] Candolin U, Heuschele J (2008) Is sexual selection beneficial during adap-
tation to environmental change? Trends in Ecology & Evolution 23: 446–
452.
[55] Myhre LC, Forsgren E, Amundsen T (2013) Effects of habitat complex-
ity on mating behavior and mating success in a marine fish. Behavioral
Ecology 24: 553–563.
[56] Peck JR, Jonathan Y, Barreau G (1999) The maintenance of sexual repro-
duction in a structured population. Proceedings of the Royal Society of
London Series B: Biological Sciences 266: 1857–1863.
[57] Barraclough TG, Herniou E (2003) Why do species exist? insights from
sexuals and asexuals. Zoology 106: 275–282.
[58] Halkett F, Kindlmann P, Plantegenest M, Sunnucks P, Simon J (2006) Tem-
poral differentiation and spatial coexistence of sexual and facultative asex-
ual lineages of an aphid species at mating sites. Journal of evolutionary
biology 19: 809–815.
[59] Hartfield M, Keightley PD (2012) Current hypotheses for the evolution of
sex and recombination. Integrative Zoology 7: 192–209.
[60] Ament I, Scheu S, Drossel B (2008) Influence of spatial structure on the
maintenance of sexual reproduction. Journal of theoretical biology 254:
520–528.
[61] Kokko H, Heubel KU, Rankin DJ (2008) How populations persist when
asexuality requires sex: the spatial dynamics of coping with sperm para-
sites. Proceedings of the Royal Society B: Biological Sciences 275: 817–
825.
[62] Houchmandzadeh B, Vallade M (2003) Clustering in neutral ecology. Phys-
ical Review E 68: 061912.
[63] Pels B, de Roos AM, Sabelis MW (2002) Evolutionary dynamics of prey
exploitation in a metapopulation of predators. The American Naturalist
159: 172–189.
[64] Van Baalen M, Sabelis MW (1995) The milker-killer dilemma in spatially
structured predator-prey interactions. Oikos : 391–400.
21
[65] Rauch E, Sayama H, Bar-Yam Y (2002) Relationship between measures of
fitness and time scale in evolution. Physical Review Letters 88: 228101.
[66] Doebeli M, Knowlton N (1998) The evolution of interspecific mutualisms.
Proceedings of the National Academy of Sciences 95: 8676–8680.
[67] Killingback T, Doebeli M, Knowlton N (1999) Variable investment, the
continuous prisoner’s dilemma, and the origin of cooperation. Proceedings
of the Royal Society of London Series B: Biological Sciences 266: 1723–
1728.
[68] Nowak MA, May RM (1992) Evolutionary games and spatial chaos. Nature
359: 826–829.
[69] Lion S, Gandon S (2009) Habitat saturation and the spatial evolutionary
ecology of altruism. Journal of evolutionary biology 22: 1487–1502.
[70] Lehmann L, Keller L, Sumpter DJ (2007) The evolution of helping and
harming on graphs: the return of the inclusive fitness effect. Journal of
evolutionary biology 20: 2284–2295.
[71] Hauert C, Doebeli M (2004) Spatial structure often inhibits the evolution
of cooperation in the snowdrift game. Nature 428: 643–646.
[72] Hardin G (1968) The tragedy of the commons. New York .
[73] J Rankin D, López-Sepulcre A (2005) Can adaptation lead to extinction?
Oikos 111: 616–619.
[74] Bargum K, Sundström L (2007) Multiple breeders, breeder shifts and in-
clusive fitness returns in an ant. Proceedings of the Royal Society B: Bio-
logical Sciences 274: 1547–1551.
[75] Ai D, Gravel D, Chu C, Wang G (2013) Spatial structures of the environ-
ment and of dispersal impact species distribution in competitive metacom-
munities. PloS one 8: e68927.
[76] Hugueny B, Cornell HV, Harrison S (2007) Metacommunity models predict
the local-regional species richness relationship in a natural system. Ecology
88: 1696–1706.
[77] Leibold MA, Holyoak M, Mouquet N, Amarasekare P, Chase J, et al.
(2004) The metacommunity concept: a framework for multi-scale com-
munity ecology. Ecology letters 7: 601–613.
22
[78] Logue JB, Mouquet N, Peter H, Hillebrand H (2011) Empirical approaches
to metacommunities: a review and comparison with theory. Trends in ecol-
ogy & evolution 26: 482–491.
[79] Santos RV (2013) Discreteness inducing coexistence. Physica A: Statistical
Mechanics and its Applications 392: 5888 - 5897.
[80] Fisher RA (1999) The genetical theory of natural selection: a complete
variorum edition. Oxford University Press.
[81] Black A, McKane A (2012) Stochastic formulation of ecological models
and their applications. Trends in ecology & evolution .
[82] Houchmandzadeh B (2008) Neutral clustering in a simple experimental
ecological community. Physical review letters 101: 078103.
[83] Houchmandzadeh B (2009) Theory of neutral clustering for growing pop-
ulations. Physical Review E 80: 051920.
[84] McKane A, Newman T (2005) Predator-prey cycles from resonant amplifi-
cation of demographic stochasticity. Physical review letters 94: 218102.
[85] Butler T, Goldenfeld N (2011) Fluctuation-driven turing patterns. Physical
Review E 84: 011112.
[86] Lande R (1998) Demographic stochasticity and allee effect on a scale with
isotropic noise. Oikos : 353–358.
[87] Dennis B (2002) Allee effects in stochastic populations. Oikos 96: 389–
401.
[88] Paulsson J (2004) Summing up the noise in gene networks. Nature 427:
415–418.
[89] Thattai M, Van Oudenaarden A (2001) Intrinsic noise in gene regulatory
networks. Proceedings of the National Academy of Sciences 98: 8614–
8619.
[90] Houchmandzadeh B (2013) The remarkable discreteness of being. arXiv
preprint arXiv:13064535 .
[91] Hestmark G (1992) Sex, size, competition and escape—strategies of repro-
duction and dispersal in lasallia pustulata (umbilicariaceae, ascomycetes).
Oecologia 92: 305–312.
23
[92] Song Y, Drossel B, Scheu S (2011) Tangled bank dismissed too early. Oikos
120: 1601–1607.
[93] Bell G (1982) The Masterpiece of Nature: The Evolution and Genetics
of Sexuality. Croom Helm applied biology series. Croom Helm. URL
http://books.google.com.br/books?id=q5g9AAAAIAAJ.
[94] Williams G (1975) Sex and Evolution. Monographs in population biology.
Princeton University Press.
[95] Smith JM (1976) A short-term advantage for sex and recombination
through sib-competition. Journal of theoretical biology 63: 245–258.
[96] Schmit O, Adolfsson S, Vandekerkhove J, Rueda J, Bode S, et al. (2013)
The distribution of sexual reproduction of the geographic parthenogen
eucypris virens (crustacea: Ostracoda) matches environmental gradients in
a temporary lake. Canadian Journal of Zoology 91: 660–671.
[97] Peck JR, Yearsley JM, Waxman D (1998) Explaining the geographic distri-
butions of sexual and asexual populations. Nature 391: 889–892.
[98] Chaplin JA, Havel JE, Hebert PD (1994) Sex and ostracods. Trends in
ecology & evolution 9: 435–439.
[99] Horne D, Baltanás A, Paris G (1998) Geographical distribution of repro-
ductive modes in living non-marine ostracods. Sex and parthenogenesis:
evolutionary ecology of reproductive modes in non-marine ostracods Back-
huys, Leiden, the Netherlands : 77–99.
[100] Hardin G (1960) The competitive exclusion principle. Science 131: 1292.
[101] Hutchinson G (1961) The paradox of the plankton. American Naturalist :
137–145.
[102] Levin SA (1992) The problem of pattern and scale in ecology: the robert h.
macarthur award lecture. Ecology 73: 1943–1967.
[103] Bascompte J, Solé R (1998) Modeling spatiotemporal dynamics in ecology.
Environmental intelligence unit. Springer.
[104] Greig D, Borts RH, Louis EJ (1998) The effect of sex on adaptation to high
temperature in heterozygous and homozygous yeast. Proceedings of the
Royal Society of London Series B: Biological Sciences 265: 1017–1023.
24
[105] Becks L, Agrawal AF (2012) The evolution of sex is favoured during adap-
tation to new environments. PLoS biology 10: e1001317.
[106] van Wijland F, Oerding K, Hilhorst HJ (1998) Wilson renormalization of a
reaction–diffusion process. Physica A: Statistical and Theoretical Physics
251: 179–201.
[107] Zinn-Justin J (2002) Quantum Field Theory and Critical Phenomena (In-
ternational Series of Monographs on Physics). Clarendon Press, 4 edition.
25
82
Capı´tulo 5
Uma possı´vel explicac¸a˜o para a frequeˆncia
varia´vel das ce´lulas tronco do caˆncer nos
tumores
Esta introduc¸a˜o e´ baseada nos resultados do artigo [1], publicado em PLoS ONE.
O ressurgimento de tumores tempos depois de eles terem sido combatidos pelos tratamentos convenci-
onais (cirurgia, quimioterapia e radioterapia) e´ um dilema que aflige ha´ muito tempo me´dicos e pesqui-
sadores que lidam com caˆncer. Estudos recentes [41, 42] conseguiram mostrar a ac¸a˜o de ce´lulas-tronco
tumorais∗ na retomada desse crescimento, oferecendo um novo alvo para a luta contra a doenc¸a.
Uma controve´rsia existe no que se refere a` frequeˆncia com que estas ce´lulas aparecem nos mais diversos
tumores [43, 44, 45, 46, 47, 48, 49, 50]. No inı´cio, cogitou-se que esta frequeˆncia era muito baixa, indo de
0,0001% a 0,1% do total de ce´lulas [51]. Posteriormente foi verificada a possibilidade de esta frequeˆncia
ser bem maior, com experimentos indicando uma frac¸a˜o de ate´ 41% [52]. A relevaˆncia da determinac¸a˜o da
quantidade de ce´lulas tronco do caˆncer (CTCs) nos tumores se faz perceber da necessidade de se focar os
tratamentos clı´nicos na eliminac¸a˜o destas ce´lulas. Se for percebido que as CTCs representam grande parte
dos tumores, tais tratamentos devera˜o atuar sobre todo o tumor como sempre foi feito. Caso contra´rio,
te´cnicas mais especı´ficas para tratar ce´lulas especı´ficas devera˜o ser desenvolvidas. A importaˆncia deste
aspecto fica evidente em [53], onde os pesquisadores da a´rea afirmam que o trabalho mais importante nos
u´ltimos anos anteriores a 2011 foi [54], o pioneiro na identificac¸a˜o de grandes populac¸o˜es de CTCs nos
tumores so´lidos.
Neste primeiro artigo, de uma sequeˆncia de dois nesta tese, propomos um modelo estoca´stico simples,
sem estrutura espacial, que tem como hipo´teses fundamentais os mecanismos conhecidos de divisa˜o celular
[55], bem como de plasticidade celular [56, 57, 58]. Este modelo apresenta a propriedade de transic¸a˜o in-
duzida por ruı´do† em sua distribuic¸a˜o estaciona´ria de probabilidade Pst . Este ruı´do tem origem nos eventos
complexos aos quais as ce´lulas esta˜o inevitavelmente expostas no microambiente celular, e a bimodalidade
de Pst e´ interpretada de acordo com as observac¸o˜es experimentais. Desta forma, se as perturbac¸o˜es es-
tatı´sticas com origem no microambiente na˜o sa˜o suficientemente intensas, a populac¸a˜o de CTCs adquire
uma distribuic¸a˜o unimodal. Caso contra´rio, a distribuic¸a˜o e´ bimodal e, como tal, pode justificar as dis-
crepaˆncias observadas nos experimentos, a`s vezes indicando frac¸o˜es pequenas, a`s vezes frac¸o˜es grandes.
Considerac¸o˜es a respeito da heterogeneidade dos tumores bem como implicac¸o˜es clı´nicas dos resultados
sa˜o feitas. Em particular, conclui-se que, dentro das hipo´teses do modelo, os tratamentos convencionais
∗Ce´lulas com caracterı´sticas de ce´lulas tronco, ou seja, grande capacidade de renovac¸a˜o e diferenciac¸a˜o, que seriam responsa´veis
pelo desenvolvimento e manutenc¸a˜o dos tumores.
†Modelado como um processo Gaussiano conhecido como ruı´do branco.
83
Capı´tulo 5. Uma possı´vel explicac¸a˜o para a frequeˆncia varia´vel das ce´lulas tronco do caˆncer nos tumores
contra o caˆncer podem na˜o apenas ser ineficazes, como podem estimular o desenvolvimento das CTCs.
Esta conclusa˜o depende da interpretac¸a˜o atribuı´da a tais formas de tratamento como fontes de variabilidade
estatı´stica no microambiente celular. Se estes resultados estiverem corretos, pode ser que as recidivas
muitas vezes fatais da doenc¸a, a`s vezes na forma de meta´stazes, possam ser explicadas pelo estı´mulo
fornecido a`s ce´lulas tronco do caˆncer no esta´gio inicial dos tratamentos convencionais.
84
A Possible Explanation for the Variable Frequencies of
Cancer Stem Cells in Tumors
Renato Vieira dos Santos1*, Linaena Me´ricy da Silva2*
1Departamento de Fı´sica, Instituto de Cieˆncias Exatas, Universidade Federal de Minas Gerais, Belo Horizonte, Minas Gerais, Brasil, 2 Laborato´rio de Patologia Comparada,
Instituto de Cieˆncias Biolo´gicas, Universidade Federal de Minas Gerais, Belo Horizonte, Minas Gerais, Brasil
Abstract
A controversy surrounds the frequency of cancer stem cells (CSCs) in solid tumors. Initial studies indicated that these cells
had a frequency ranging from 0:0001 to 0:1% of the total cells. Recent studies have shown that this does not always seem to
be the case. Some of these studies have indicated a frequency of 40%. In this paper we propose a stochastic model that is
able to capture this potential variability in the frequency of CSCs among the various type of tumors. Considerations
regarding the heterogeneity of the tumor cells and its consequences are included. Possible effects on conventional
treatments in clinical practice are also described. The model results suggest that traditional attempts to combat cancer cells
with rapid cycling can be very stimulating for the cancer stem cell populations.
Citation: dos Santos RV, da Silva LM (2013) A Possible Explanation for the Variable Frequencies of Cancer Stem Cells in Tumors. PLoS ONE 8(8): e69131.
doi:10.1371/journal.pone.0069131
Editor: Je´re´mie Bourdon, Universite´ de Nantes, France
Received March 9, 2013; Accepted June 4, 2013; Published August 7, 2013
Copyright:  2013 Santos, da Silva. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported by the Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico, Brazil. The funders had no role in study design, data
collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: econofisico@gmail.com; linaena.mericy@gmail.com
Introduction
In recent years there has been increasing evidence for the Cancer
Stem Cell (CSC) hypothesis [1–4], according to which tumor
formation is a result of genetic and epigenetic changes in a subset
of stem-like cells, also known as tumor-forming or tumor-initiating cells
[5]. Cancer stem cells (CSCs) were first identified in leukemia and
more recently in several solid tumors such as brain, breast, cervix
and prostate tumors [4]. It has been suggested that these are the
cells responsible for initiating and maintaining tumor growth [6].
In this paper, we study a model for tumor growth assuming the
existence of cancer stem cells, or tumor initiating cells [6–8].
The conceptual starting point relevant to the CSC theory is
constructed from the known tumor heterogeneity. We now know
that cells in a tumor aren’t all identical copies of each other, but
that they display a striking array of characteristics [9–13]. The
CSC theory recognizes this fact and develops its consequences.
And one of the most immediate consequences for clinical practice
is that conventional treatments can attack the wrong cell type. The
appeal of the CSC idea can be described through the following
analogy: just as killing the queen bee will lead to the demise of the
hive, destroying cancer stem cells, should, in theory, stop the
tumor from renewing itself. Unfortunately, things are never that
simple. In the hive, workers react quickly to the death of queen by
replacing her with a new one. And there is some evidence [8,14]
suggesting that the same may occur in a tumor due to a
phenomenon known as cell plasticity, which allows differentiated
tumor cells to turn into cancer stem cells, should the situation call
for this. One goal of the present study is to evaluate the possible
effects of this plasticity. Analogies with super organisms such as bee
colonies are taken much more seriously in [15].
Stem cells in general (the same applies to CSCs) tend to be
found on specific areas of a tissue where one particular
microenvironment, called niche [16,17], promotes the maintenance
of their vital functions. Such a niche is specialized in providing
factors that prevent differentiation and thus maintain the stemness
of CSCs and, ultimately, the tumor’s survival. Stem cells and niche
cells interact with each other through adhesion molecules and
paracrine factors. This complex network of interactions exchanges
molecular signals and maintains the unique characteristics of stem
cells, namely, pluripotency and self-renewal.
In this paper, we are interested in investigating a controversy
related to the frequency in which CSCs appear in various tumors
[18–25]. In the initial version of the CSC theory, it was believed
that these cells were a tiny fraction of the total, ranging from
0.0001 to 0.1 % [26]. However, more recent studies have shown a
strong dependence of the number of CSCs present in the tumor
with the experimental xenograft model used. In explicit contrast to
what was previously thought, in [27] a proportion of CSCs of
approximately 25% was observed. Other studies have confirmed
this observation [26,28,29] with the possibility of a proportion of
up to 41% [30]. In [31] the authors provide evidence that this
discrepancy may be due to the possibility of phenotypic switching
between different tumor cells. Phenotypic switching is interpreted
as the possibility of a more differentiated cancer cell being able to,
under the appropriate conditions, dedifferentiate into cancer stem
cell. This is the cellular plasticity mentioned above.
In [32] it is suggested that inconsistencies in the numbers of
cancer stem cells reported in the literature can also be explained as
a consequence of the different definitions used by different
researchers. Different assays will give different numbers of cells,
which can be orders of magnitude away from each other. Articles
[31] and [32] provide different explanations for the discrepancy in
PLOS ONE | www.plosone.org 1 August 2013 | Volume 8 | Issue 8 | e69131
the frequency of CSCs. Our arguments are consistent with the
results of [31].
Considering that the complexity of the cellular microenviron-
ment can be modeled by the insertion of a Gaussian noise into the
equation that describes the population dynamics, we show that a
noise-induced transition occurs. That corresponds to the emer-
gence of a bimodal stationary probability distribution. This
happens when the noise intensity s exceeds a critical limit value
scr:
In this paper we show that cell plasticity [14,33,34], combined
with a complex network of interactions modeled as noise, can
induce discrepant (too small or too large) stationary CSC
populations. Effects related to tumor heterogeneity and clinical
treatments will be discussed at the end, occasion in which the
model parameters possess the appropriate biological interpreta-
tions.
Methods
Model Assumptions
In the model used in this paper, cancer stem cells can perform
three types of divisions, according to [35]:
N symmetric self-renewal: cell division in which both
daughter cells have the characteristics of the mother stem cell,
resulting in an expanding population of stem cells;
N symmetric differentiation: a stem cell divides into two
progenitor cells;
N asymmetric self-renewal a cancer stem cell (denoted by C)
is generated and a progenitor cell (mature cancer cell, denoted
by P) is also produced;
We have developed a simple mathematical model for the
stochastic dynamics of CSCs in which the three division types
possess intrinsic replication rates, which are assumed to be time-
independent. We assume, therefore, that besides the three
described types of division, there is also the possibility of a
transformation in which a progenitor cell can acquire character-
istics of stem cells where, for all practical purposes, we may regard
it as having become a dedifferentiated CSC. This hypothesis has
experimental support [36]. These dedifferentiated cells do not
become cancer stem cells, but rather develop CSC like behavior
by re-activating a subset of genes highly expressed in normal
hematopoietic stem cells [14]. The biological mechanisms
underlying this transformation are described in [31], for example.
As mentioned previously, we refer to this process as cell plasticity.
Finally, we assume that cells are well mixed, so that we can ignore
spatial effects.
The model proposed is a natural extension of what is proposed
in [37]. We also incorporates the possibility of competition
between CSCs and between the progenitor cells in order to limit
the exponential growth of the linear model in [37]. This is
described in the next subsection.
The basic model
We assume that the dynamics of cancer stem cells (C) and
progenitor cells (P) are governed by the following reactions:
C '
k1
k’2=V2
CzC
P '
k3
k’4=V4
PzP
CI
k5
CzP
CI
k6
PzP
PI
k7
1
PI
k8
C ð1Þ
The first and second reactions, in the forward sense, models cell
proliferation, which occurs at a rate of k1 and k3, respectively. The
constants k’2 and k’4 are associated with the reverse process and
describe the intensity of competition between the CSCs and
progenitors cells, respectively, and prevents their unlimited
exponential growth. Many studies, experimental and theoretical,
justify this approach [38–47]. As long as no mechanical nor
nutritional restrictions apply, the tumor cells go on replicating with
a constant duplication time. After a while, however, several
constraints force the development of a necrotic core, and growth
slows down towards some asymptotic level of saturation. V2 and
V4 are constants related to the carrying capacity of the model. The
third reaction involving k5 originates from the asymmetric
transformation of CSCs in CSC daughter and progenitor cell
types. The reaction involving the k6 rate is related to a
symmetrical division of the stem cell, which gives rise to two
progenitor cells. The penultimate reaction is associated with the
progenitor cell’s death at rate k7: Finally, k8 is the rate of
dedifferentiation. All rates have dimension (time){1: The specific
time unit (months, quarters, years, etc.) will depend on the type
and aggressiveness of the tumor.
Using the law of mass action, we can write
dC
dt
~k1C{k2C
2{k6Czk8P
dP
dt
~k3P{k4P
2z(k5z2k6)C{(k7zk8)P
8><
>: ð2Þ
with k2:k2’=V2, k4:k4’=V4: Setting VC:k1=k2, VP:k3=k4,
k9:k5z2k6 and k10:k7zk8 and making the substitutions
C~VCx, P~VC
ffiffiffiffiffiffiffiffiffiffiffiffi
k9=k2
p
y and t~t=k6, equation (2) can be
written as (see Appendix S1)
dx
dt
~Ax(1{x){xzBy:f (x,y)
dy
dt
~Ey(1{Fy)zBx{Gy:g(x,y)
8><
>: ð3Þ
with
Cancer Stem Cells Enjoy Noise
PLOS ONE | www.plosone.org 2 August 2013 | Volume 8 | Issue 8 | e69131
A:
k1
k6
B:
ffiffiffiffiffiffiffiffiffi
k8k9
p
k6
E:
k3
k6
F:
VC
VP
ffiffiffiffiffi
k9
k8
s
G:
k10
k6
8>>>>>>>>><
>>>>>>>>>:
ð4Þ
As Lf =Ly~Lg=Lx~B, equation (3) represents a gradient system
[48] with potential V (x,y) given by (see Appendix S1)
V (x,y)~
1
6
(3{3Az2Ax)x2{Bxyz
1
6
(3G{3Ez2EFy)y2: ð5Þ
As a consequence [49]:
1. The eigenvalues of the linearization of equation (3) evaluated at
equilibrium point are real.
2. If (x0; y0) is an isolated minimum of V then (x0; y0) is an
asymptotically stable solution of (3).
3. If (x(t); y(t)) is a solution of (3) that is not an equilibrium point
then V (x(t),y(t)) is a strictly decreasing function and is
perpendicular to the level curves of V(x,y):
4. There are no periodic solutions of (3).
Sufficiently small F (VP&VC ) implies large differences in C and
P equilibrium populations. For parameters A~B~G~1, E~3
and F~0:01, (x0; y0)~(8:4; 70:6): If we set F~0:0001 keeping
the other parameters fixed, we have (x0; y0)~(82; 6710):
Adiabatic elimination
The proposed model in (1) is in fact a general model of stem
cells and does not carry any specific characteristic of cancer stem
cells. All properties considered, such as plasticity and changes in
the microenvironment conditions (to be included later), are also
found in normal, stem cell tissue systems. The features associated
with cancer stem cells are related to the large carrying capacity of
progenitor cells when compared with the carrying capacity of
CSCs. This fact is represented numerically by the choice of model
parameters made below and is important because it allows a
simplification using the adiabatic approximation.
We can write (2) as (see Appendix S1)
x’~A’x(1{x){xzB’y
y’~E’y(1{y)zF ’x{G’y

ð6Þ
with x’:
dx
dt’
, y’:
dy
dt’
, t’:t=k6 and
A’:
k1
k6
B’:
k2k3k8
k1k4k6
E’:
k3
k6
F ’:
k1k4k9
k2k3k6
G’:
k10
k6
8>>>>>>>><
>>>>>>>>:
ð7Þ
Figure (1) shows the numerical solutions of equations (6, Top) (the
rescaled equation) and (2, Bottom) for the parameter values shown
in table 1 (which correspond to A’~8, B’~5|10{4, E’~10,
F ’~0:6 and G’~1, and b is a general parameter with dimension
time{1 required for dimensional consistency in the following
analysis):
Considering the global rate b (we use b:1 throughout the text)
and assuming k5~r5b, k6~r6b, we make the usual assumption
Figure 1. Numerical solutions of differential equations. Top:
Numerical solution for reescaled equation (6). Horizontal axis is time t:
x(t) and y(t) represent the rescaled population of cancer stem cells and
progenitor cells, respectively. Bottom: Numerical solution for equation
(2). C(t) and P(t) represent he population of cancer stem cells and
progenitor cells, respectively. P? and C? represent the limits of C(t)
and P(t) when t??, respectively. Parameters values: k1~1{k5{k6,
k2~4|10
{13, k3~1, k4~10
{13, k5~0:1, k6~0:1, k7~0:1 and
k8~0:00001: P?~9:6|10
12 and C?~1:8|10
12: C?=P?~0:1875:
doi:10.1371/journal.pone.0069131.g001
Cancer Stem Cells Enjoy Noise
PLOS ONE | www.plosone.org 3 August 2013 | Volume 8 | Issue 8 | e69131
r1zr5zr6~1 [50] and write k1~r1b~(1{r5{r6)b~
b{k5{k6, where r1, r2, and r3 are probabilities. The values for
r5 and r6 are consistent with those estimated in [50]. For these
parameter values, VC:
k1
k2
~2|1012 and VP:
k3
k4
~1|1013 (see
Appendix S1). These are rescaled parameters in x and y variables,
respectively. Stationary values for P(t) and C(t) are
P?~9:6|10
12 cells and C?~1:8|10
12 cells, respectively.
Adjusting the k2 and k4 parameters, we can easily obtain more
suitable values for the CSC and progenitor cell equilibrium
populations, according to possible new experimental results.
Employing standard adiabatic elimination methods, we can
write equation (6) as
x’~A’ x(1{x){
x
A’
z
B’
A’
y
 
Ey’~y(1{y)zEF ’x{G’y
8<
: ð8Þ
where:1=E’: If we consider%1 (this is equivalent to considering
the progenitor cell division rate sufficiently large) we can perform
adiabatic approximation [51,52] in (8) and, setting y’~0, we
obtain the following equation for x, expanding in Taylor series up
to first order in :
x’~x{mxzax(1{x) ð9Þ
where x:B’(1{G’)~
k8k2(k3{k10)
k1k4k6
, m:1{EB’F ’~1{
k8k9
k3k6
and a:A’~
k1
k6
: Note that x can be positive or negative
depending on the magnitude of k3 and k10:
If we set a small enough value for with respect to G’, B’ and F ’,
we can further simplify and write x~B’ and m~1: We observe
that the plasticity phenomenon (associated with k8) is crucial for
the existence of the constant term x: For this reason, from now on
we will consider the parameter x as representing the plasticity
phenomenon in the reduced equation (9).
The deterministic equation
For comparison with the stochastic study of the next section, we
will briefly review the deterministic analysis of the problem. An
analytic solution of Eq. (9) is possible. For the initial condition
x(0)~N0, one has
x(t)~
1
2a
d{
ffiffiffi
k
p
Tan
1
2
t
ffiffiffi
k
p
zArcTan
d{2N0affiffiffi
k
p
   
ð10Þ
with d:a{m and k:{d2{4ax: The physically relevant stable
fixed point is
x~
a{mz
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a2{2amzm2z4ax
p
2a
: ð11Þ
The x scaled population size dynamics can be thought of as
analogous to the motion of a particle in a potential V0(x), seeking
its minimum point, with V0(x):{
ð
f(x)dx with
f(x)~xzdx{ax2 from (9). Thus, V0 is given by the cubic
polinomial,
V0(x)~
x3a
3
{
dx2
2
{xx:
We see from (11) that by increasing either x or d, the minimum
x of V0 moves to the right in the potential, thus favoring CSCs
population. Such behavior, of course, is expected, since an
increase of x means an increase in frequency in which the induced
plasticity mechanism occurs, and an increase of d is an increase of
the symmetric renewal rate of cancer stem cells, both of which
increase the population.
Results
Noise in the CSCs niche
Environmental noise. In tumor tissue, the growth rate and
other parameters are influenced by many environmental factors,
e.g., degree of vascularization of tissues, supply of oxygen and
nutrients, immunological state of the host, chemical agents, gene
expression, protein synthesis, mechanical stress, temperature,
radiation, etc [50,53–55]. Given the many perturbations affecting
the CSC niche, we expect parameters such as growth rate to be
random, rather than fixed, to give a more reliable description. We
propose a simplification in the interaction mechanisms between
cancer stem cells and their niche by adding an external Gaussian
white noise in an attempt to capture the essential aspects of this
complexity in a mathematically tractable way.
It is worth noting that in conjunction with nonlinear interac-
tions, noise can induce many interesting phenomena, such as
stochastic resonance [56], noise-induced phase transitions [57],
noise-induced pattern formation, and noise-induced transport
[51,58].
Including external noise. To model the effect of external
noise, focusing initially on the CSCs proliferation rate (by making
a?azj(t), j(t) is the noise with the statistical properties
described below), we modify the deterministic equation (9) as
follows:
x’~x{mxz(azj)x 1{xð Þ~x{mxzax 1{xð Þzx 1{xð Þj, ð12Þ
where j:j(t’) is a Gaussian white noise with statistical properties
Sj(t’)T~0 and Sj(t’)j(t’’)T~s2d(t’{t’’), s2 is the variance of
j(t’): Furthermore, x is considered a constant related to the
plasticity phenomenon and a,m have interpretations similar to
those of equation (9), where a now represents the average
symmetric division rate. The noise term in equation (12) represents
fluctuations in parameter a, due to the complexity of the
microenvironment, as discussed above. We include noise in this
Table 1. Parameter Values.
Parameters k1 k2 k3 k4 k5 k6 k7 k8 b
Values b-k5-k6 4610
213 1 10213 0.1 0.1 0.1 1025 1
doi:10.1371/journal.pone.0069131.t001
Cancer Stem Cells Enjoy Noise
PLOS ONE | www.plosone.org 4 August 2013 | Volume 8 | Issue 8 | e69131
term because it is more important in the CSCs population
dynamics, since it is this parameter that regulates symmetric
reproduction (C?CzC). Later on we will add yet another noise
in the plasticity constant.
We can write the Langevin equation (12) as a stochastic
differential equation (considerations concerning the interpretation
of the multiplicative term, i.e., if Itoˆ or Stratonovich or other, will
be made below) in the form of
dxt~f(xt,t)dtzsD(xt,t)dBt
: x{mxtzaxt 1{xtð Þ½ dtzsxt 1{xtð ÞdBt,
ð13Þ
where we define the drift f(xt,t) and diffusion D(xt,t) functions
and where dBt is the Wiener process increment [52,59,60]. The
stationary probability distribution Pst(x) of the stochastic process
defined by (13) is given by [52]
Pst(x)~N exp {2
V (x)
s2
 
ð14Þ
where N is a normalization constant and V (x) is the stochastic
effective potential defined by
V (x)~{
ð
f(x)
D(x)2
dxzh
s2
2
ln½D(x): ð15Þ
Here h~1 refers to the Stratonovich interpretation of (13) and
h~2 to the Itoˆ version. Substituting the drift and diffusion
functions, we get
V (x)~
xmzx{2xx
x{x2
z(a{mz2x) ln
x{1
x
 
z
h
2
s2 ln½x(1{x) ð16Þ
and
Pst(x)~Ne
{
2
xmzx{2xx
x{x2
z(a{mz2x) ln
x{1
x
 
z
1
2
hs2 ln½x(1{x)
 
s2 : ð17Þ
The maximum xm of Pst, which corresponds to the minimum of
V (x), can be obtained from the following equation [61]:
f(xm){h
s2
2
D(xm)
d½D(xm)
dxm
~0: ð18Þ
We see that for s~0, xm corresponds to the value given by x

in eq. (11). From the drift and diffusion functions, we get:
{x3hs2{
1
2
x2 2a{3hs2
	 

z
1
2
x 2a{2m{hs2
	 

zx~0: ð19Þ
The condition for (19) possessing three real roots (corresponding
to the two extremes of Pst) is [62]:
1
16
2mzhs2{2a
	 
2
4a2z4h(a{4m)s2zh2s4
	 

z 2a{3hs2
	 

2a2z3h(a{3m)s2
	 

x{27h2s4x2w0: ð20Þ
For example, for the parameters values h~2, a~8, m~1 and
x~0:00045, the critical value scr above which a transition is
induced in Pst is scr~2:68:
Figures (2) show, in Stratonovich interpretation (h~1), (the
results do not change qualitatively if we use Itoˆ. For a discussion
quite enlightening about the controversial dilemma Itoˆ/Stratono-
vich, see [63]) the effect of increasing the noise intensity in the
stochastic effective potential V (x) (Top) and in the stationary
probability distribution Pst(x) (Middle). Below is the x{s plane.
The shaded region corresponds to high values of s where Pst is
bimodal. Note that the presence of plasticity (represented by xw0)
implies the survival of cells populations regardless of noise
intensity. Inclusion of external noise can induces the appearance
of a bimodal stationary probability distribution, which leads to a
result quite different from the deterministic case: while the
population in the deterministic case will necessarily reach the
value x, in the stochastic case the population is unlikely to reach
x if s is above its critical value scr: It is much more likely to
possess a nonzero (if xw0), very small population (left peak of Pst)
or a very large one (right peak of Pst). This peak positioned to the
right is associated with a population near the maximum value
x&1 in the rescaled variable x~C=VC : It stands for the
possibility that the population of cancer stem cells possess a value
close to C~VCx
&2|1012: This represents a significant fraction
of the population of progenitor cells P, a fraction that depends
mainly on the equilibrium value x of the deterministic equation
given by (11), never exceeding this threshold. When we insert noise
in the plasticity x this is no longer the case.
The inhibition of the host’s immune system, which can result in
a decrease of the microenvironmental complexity, is equivalent in
our model to a decrease of s: Therefore, a xenograft performed in
immunosuppressed mice may, over time, present significantly
large CSC populations. This may have been the case for the
experiments conducted in [27]. On the other hand, the left peak in
Pst may represent a tiny fraction of the CSCs population, as
commonly reported in the pioneering experiments mentioned in
the introduction, in which less immunosuppressed mice were used.
If x~0 and swscr, it is much more likely that the population
becomes extinct as shown in figure (3).
Figures (4) and (5) show five trajectories of the relevant
stochastic process, constructed using the Euler algorithm [64],
with initial condition x(0)~0:1, for svscr and swscr, respec-
tively. The black curve represents the solution for s~0: We see in
Figure (5) that for high values of s, some trajectories can exhibit
spontaneous regression of the CSCs. This seems plausible in light
of the supporting evidence from many clinical reports [65].
Figure (6) shows the effect of a on V (x) (Top) and Pst (Middle).
Sufficiently small values of a refer to unimodal distributions with
left asymmetry (blue curve/dot). Intermediate values correspond
to bimodal distributions (shaded area in the a{s plane, red
curve/dot). Sufficiently high levels of a correspond to unimodal
distributions with right asymmetry (black curve/dot).
We conclude in this section that the cell plasticity phenomenon
is necessary for the existence of a cancer stem cell population as a
small fraction of total tumor cells. Of course, microenvironmental
conditions consistent with high noise levels are also necessary.
Cancer Stem Cells Enjoy Noise
PLOS ONE | www.plosone.org 5 August 2013 | Volume 8 | Issue 8 | e69131
Cancer Stem Cells Enjoy Noise
PLOS ONE | www.plosone.org 6 August 2013 | Volume 8 | Issue 8 | e69131
Colorful background noise
We can ask ourselves what effects the variability induced by
noise in P cells produce in the C population. In equation (9),
reminiscences of the presence of P cells are manifested by the
presence of x: We can imagine this term as representing a source
of background noisy for C cells. The question that immediately
arises is: what are the effects of a noise on the proliferation rate a
combined with other noise related to the plasticity in constant x?
To answer this question, let’s add the noise j(t) and g(t) as
x?xzj(t) and a?azg(t) and write the equations
_x~x{mxzax(1{x)zj(t)zg(t)x(1{x)
:h(x)zg1(x)j(t)zg2(x)g(t)
ð21Þ
Figure 2. Effects of noise intensity on V (x) and Pst(x): Effect of s on V (x) (on the top) and Pst(x) (in the middle) for parameters k1~1{k5{k6,
k2~4|10
{13, k3~1, k4~10
{13, k5~0:1, k6~0:1, k7~0:1 and k8~0:00001: Horizontal axis represents population size x: Blue, dashed curve:
s~1:5: Red, dotted: s~2:68: Black, thick: s~5:9: Below we also show the x{s plane with s in the horizontal axis.
doi:10.1371/journal.pone.0069131.g002
Figure 3. Effects of noise intensity on V (x) and Pst(x): Effect of s on V (x) (on the top) and Pst(x) (at bottom) for x~0: Horizontal axis
represents population size x: Blue, dashed curve: s~2: Red, dotted: s~3: Black, thick: s~3:74: Other parameters are as in figure (2). For sufficiently
high values of s, the CSCs population is extinguished.
doi:10.1371/journal.pone.0069131.g003
Cancer Stem Cells Enjoy Noise
PLOS ONE | www.plosone.org 7 August 2013 | Volume 8 | Issue 8 | e69131
_j~{
1
t
jz
1
t
f(t) ð22Þ
where h(x):x{mxzax(1{x), g1(x):1 and g2(x):x(1{x)
and g(t) and f(t) are white noises with the following properties
Sf(t)T~Sg(t)T~0 ð23Þ
Sf(t)f(t’)T~2sd(t{t’) ð24Þ
Sg(t)g(t’)T~2Cd(t{t’) ð25Þ
Sf(t)g(t’)T~Sg(t)f(t’)T~2l
ffiffiffiffiffiffi
sC
p
d(t{t’), ð26Þ
where s and C are the noise intensity of f(t) and g(t) respectively,
and l is the correlation between noises. Equation (22) represents
the Ornstein-Uhlenbeck process that displays exponential corre-
lation function described in equation (27) below with correlation
time t: This stochastic process is called ‘‘colored noise’’.
The two dimensional Markovian process defined by equations
(21)–(26) is stochastically equivalent to the one-dimensional non-
Markovian process described by (21), (24) and (25), with Gaussian
colored noise j(t) [52]:
Sj(t)T~0, Sj(t)j(t’)T~
s
t
exp {
1
t
Dt{t’D
 
: ð27Þ
We are considering the possibility of a colored noise in x (for
correlation time tw0). Thus we intend to capture the effects of
noise in the plasticity more realistically.
Following [66], the stationary probability distribution is given
by
Pst(x)~N
C(t,x)ffiffiffiffiffiffiffiffiffi
B(x)
p exp ðx h(x’)C(t,x’)
B(x’)
dx’
 
ð28Þ
where N is a normalization constant and B(x) and C(t,x) are
given by
B(x)~C½g1(x)2z2l
ffiffiffiffiffiffi
Cs
p
g1(x)g2(x)zs½g2(x)2
and
C(t,x)~1{t h’(x){
g’1(x)
g1(x)
h(x)
 
:
In figure (7) we show the stationary probability distribution with
l~0:9, t~0, s~1|10{9, C~5 (blue), C~10 (red, dotted) and
C~15 (black, dashed). Now we see that even for very small s (the
background noise intensity due to x), extinction of CSCs is possible
for sufficiently high C (the noise due to a), which does not occur
when x is deterministic. For t=0 this statement becomes more
evident, as shown in figure (8) where we used the same parameter
values of previous figure with C~10 except that t~0:1 for blue
thick curve and t~0 for red dotted curve. The conclusion is that
the induction of fluctuations in the population of progenitor cells
(represented by the background noise due to x) can promote CSC
extinction.
Some remarks on the interpretation of s, C and l:
Before we continue the discussion about the effects of
background noise, we will make some considerations about the
interpretation that we assign to the parameters s, C and l:
About s : Given equation (9), we can interpret the system
formed by CSCs as an isolated system that exchanges ‘‘particles’’
(P cells) with the external environment and ‘‘feels’’ the
disturbances of the medium through the parameter x, the window
of communication with the outside. The intensity of these external
disturbances is represented by parameter s, and j(t) can therefore
be interpreted as an external noise, external to the system formed
by CSCs. When the body of the tumor is subjected to the effects of
clinical treatments such as radiotherapy, chemotherapy or
thermotherapy [67], the increase in the intensity of this parameter
can be considerable.
Figure 4. Some possible trajectories for the population
dynamics with weak noise. The rugged curves show four
realizations of stochastic process (13) with s~1:0: The black curve
shows the deterministic case, s~0:
doi:10.1371/journal.pone.0069131.g004
Figure 5. Some possible trajectories for the population
dynamics with strong noise. The rugged curves show four
realizations of stochastic process (13) with s~6:0: The black curve
shows the deterministic case, s~0: Some cases demonstrate the
possibility of spontaneous remission.
doi:10.1371/journal.pone.0069131.g005
Cancer Stem Cells Enjoy Noise
PLOS ONE | www.plosone.org 8 August 2013 | Volume 8 | Issue 8 | e69131
About C : The direct contact of CSCs with their immediate
microenvironment (their niche) is what enables exchange of
nutrients and complex biochemical interactions that allow for cell
life. Variability in this context represented by C, can be interpreted
as an internal noise (internal noise here is not related in any way to
the internal demographic noise as modeled by master equations).
This internal noise affects the cell proliferation rate a:
About l : A very important aspect about cancer, as mentioned
in the introduction, is that tumors contain heterogeneous
populations of cells, which may contribute differently in extent
and mechanism to the progression of malignancy [68]. Tumor
heterogeneity is possibly one of the most significant factors that
most treatment methods fail to address sufficiently. While a
particular drug may exhibit initial success, the eventual relapse
into tumor growth is due in many cases to subpopulations of
cancer cells that are either not affected by the drug mechanism,
possess or acquire a greater drug resistance, or have a localized
condition in their microenvironment that enables them to evade or
withstand the treatment. These various subpopulations may
include cancer stem cells, mutated clonal variants, and tumor-
associated stromal cells, in addition to cells experiencing a spatially
different condition such as hypoxia within a diffusion-limited
tumor region.
This important aspect is related to different forms in which the
various sub-populations respond to various types of internal and
external stimuli. Thus, we argue that the correlation coefficient l
Figure 6. Effect of a on V (x) (on the top) and Pst(x) (in the
middle) and the a{s plane at bottom (a on the vertical axis
and s on the horizontal axis). The parameters are: s~1:5 in all
figures. Blue-dashed: a~2:2: Red-dotted: a~2:6 and Black-thick: a~5:
doi:10.1371/journal.pone.0069131.g006
Figure 7. Stationary probability distribution for different
values of C: Pst(x) with parameters l~0:9, t~0, s~1|10
{9, C~5
(blue), C~10 (red dotted) and C~15 (black, dashed). Horizontal axis
represents population size x: Fluctuations in the progenitor population
P can stimulate CSCs extinction.
doi:10.1371/journal.pone.0069131.g007
Figure 8. Dependence of Pst with t: Pst(x) with parameters l~0:9,
t~0 (red curve), t~0:1 (blue curve), s~1|10{9, C~10: Horizontal
axis represents population size x: High values of t facilitates CSCs
extinction.
doi:10.1371/journal.pone.0069131.g008
Cancer Stem Cells Enjoy Noise
PLOS ONE | www.plosone.org 9 August 2013 | Volume 8 | Issue 8 | e69131
between the noise acts as a measure of this heterogeneity between
the two populations we are considering. Since each noise is related
primarily to a specific cell type, we have that parameter l
‘‘measured’’ different responses of these cells to these stimuli. If the
different subpopulations behave more or less in the same manner
when subjected to various stimuli (low heterogeneity), l tends to
approach 1. If the behaviors are independent, l&0: If the
responses to the stimuli tend to be opposite (great heterogeneity), l
tends to approach 21.
Figure (9) (Top) shows the possible effect of changes in l in
stationary probability distribution for the parameters values shown
in the description. The results for t=0 are analogous. Below is the
l{s diagram. In the yellow region the stationary probability
distribution is bimodal. We see that negative values of l favor the
survival of cancer stem cells. This result is no surprise, since it is
known that the heterogeneity of the tumor provides the
phenotypic variation required for natural selection to act to
increase the robustness (a property that allows a system to mantain
its function despite internal and external perturbations) of the
tumor [10].
Possible effect of conventional treatments
The proposed model in this paper is idealized and highly
simplified. In addition, it does not rely on biological data for some
Figure 9. Effect of l on Pst(x) (top) with parameters s~10, l~0:5 (Blue, thick line), s~10, l~{0:5 (Red, dotted line), s~20, l~{0:5
(Black, dashed line), t~0, C~10{1, m~1, a~8, x~0:00045: Horizontal axis represents population size x: Bottom: l{s plane with s in the
horizontal axis.
doi:10.1371/journal.pone.0069131.g009
Cancer Stem Cells Enjoy Noise
PLOS ONE | www.plosone.org 10 August 2013 | Volume 8 | Issue 8 | e69131
values of the k parameters. Therefore, the conclusions we can get
from it in this section are merely theoretical speculations. Having
said this, let’s try to estimate the effects that conventional
treatments may have on the CSC population.
In the proposed model we imagine that such treatments work
directly on progenitor cells, since such treatments are designed to
act mainly in cells that reproduce faster [69]. Thus, the effect on
CSCs is indirect via background noise in a manner that is
analogous to what was discussed above. Now we have the
possibility of noise intensity s being much larger. Treatments act
to eliminate progenitor cells and the tendency, therefore, is for
parameter x to approach zero. Since this is the parameter that
connects the ‘‘underlying world’’ of cancer stem cells to the world
of progenitor cells, we could imagine that the contact between the
worlds is lost. This is no problem, however, because now we think
of the background noise as an additive noise that arises as a result
of external perturbations to the CSCs. Thus, we can consider
equation (21) with x~0 and think about the noise j(t) as is
commonly understood when you introduce an additive noise in
the equations ‘‘phenomenologically’’ or ‘‘by hand’’.
For large values of s, the parameter of greater relevance is t:
Figure (10) shows the effect on the stationary probability
distribution: Positive values, even small ones, help cancer stem
cells considerably not going extinct. The most important, however,
is another fact, which is explicitly shown in this figure: The main
consequence of exploring the possibility of an intense additive
noise is that the population of cancer stem cells may be
considerably greater than the maximum population of the
deterministic model C?~VCx
: This means that the effects of
conventional treatments that act primarily in the fast cycling cells,
here represented by progenitor cells, can be extremely exciting for
CSC proliferation. Cancer stem cells enjoy noise.
Discussion
The importance of cellular plasticity in the conclusions we have
drawn so far, is evident. In [32] the authors point out potential
conceptual difficulties associated with the phenotypic switching
hypothesis. They argue that if cancer cells can turn into cancer
stem cells, then the very notion of CSC becomes blurred, since in
this way the cancer cells could dedifferentiate at any time and
acquire the potential immortality of CSCs. In the authors words,
‘‘the distinction between phenotypic switching and the original
conventional model, run the risk of becoming purely semantic.’’
From a clinical perspective, this means that the existence or not of
the CSCs is irrelevant, since we must try to kill all tumor cells and
not just focus on tumor initiating cells. However, the fact that we
have to kill the greatest possible amount of tumor cells does not
mean that we have to try to do it in the same way for all of them.
In [70], a near-twofold reduction in the density of brain tumors in
mice was observed when authors combined standard anticancer
drugs with the selective killing of CSCs, if compared with standard
agents alone. With regard to the phenotypic switching property,
selectively killing a population of CSCs can make room for
progenitor cells to dedifferentiate and occupy this vacant niche
space. Trying to limit ‘‘stemness’’ instead, by changing conditions
of the niche that supports the life of CSCs, may be a more
promising therapeutic strategy. This idea is in line with what is
thought to be necessary for major mass extinctions [71].
Until now the properties of cancer stem cells were tested only in
transplantation assays and their very existence have been
questioned several times [6–8]. In [72], the authors use a lineage
tracing technique that allows permanent, in vivo fluorescent marking
of stem cells and their progeny, trying to put an end to the
controversy of the existence of cancer stem cells in solid tumors.
They unraveled the in vivo mode of tumor growth in its native
environment and found that the majority of labeled tumor cells in
benign skin tumors have only limited proliferative potential,
whereas a fraction has the capacity to persist in the long term,
giving rise to progeny that occupies a significant part of the tumor.
Progression to cancer in benign skin tumors was associated with
expansion of the CSC population and a decrease in the production
of non-stem cells. This suggests that tumor evolution enriches the
CSC population. Designing therapies that prevent increases in
stemness may be a means to restrict tumor progression into cancer.
Conclusion
We propose a model to describe the population dynamics of
cancer cells, using the theory of cancer stem cells (CSCs). Our
analysis allows us to address a controversy related to the frequency
of such cells in tumors. Initially it was thought that these cells were
relatively rare, comprising at most *1% of the cancer cell
population. More recent experiments, however, suggest that the
CSC population need not be small. Taking into account the
cellular plasticity property, which permits more mature cells to
dedifferentiate into cells with characteristics of stem cells, we show
that the discrepancy observed in the frequency of these cells is
entirely consistent with the original hypothesis of the existence of
cancer stem cells, as long as favorable conditions related to the
complexity of the microenvironment are met. We assume that
these conditions can be described by the inclusion of noise in the
rate of tumor growth or in the rate at which the plasticity
phenomenon occurs.
In the model where we take into account only the noise in the
rate of CSC proliferation, we conclude that there is the possibility
of the stationary probability distribution being bimodal. In the
model that also incorporates noise in parameter x associated to the
cellular plasticity phenomenon, the possibility of extinction arises
and the fraction of CSCs in the tumor can assume quite high
values, exceeding the threshold C?: The ‘‘color’’ of this noise
stimulates the CSC population. The correlation coefficient
between noises is interpreted as a measure of heterogeneity
between progenitor cells and cancer stem cells, since different cells
respond to stimuli in different ways. This heterogeneity also excites
the CSC population.
Figure 10. Effect of t on Pst(x) with parameters t~0:0 (Blue,
thick line), t~0:05 (Red, dotted line), t~0:1 (Black, dashed line),
C~10, s~10, l~{0:5 m~1, a~8, x~0:0: Horizontal axis represents
population size x:
doi:10.1371/journal.pone.0069131.g010
Cancer Stem Cells Enjoy Noise
PLOS ONE | www.plosone.org 11 August 2013 | Volume 8 | Issue 8 | e69131
In future work we plan to extend the model to include spatial
distribution. We will also investigate the possibility of a model
based on a master equation to investigate the effects of
demographic stochasticity.
Supporting Information
Appendix S1 Appendix to a possible explanation for the
variable frequencies of cancer stem cells in tumors.
(PDF)
Acknowledgments
RVS is grateful to Ronald Dickman for his helpful comments. We thank
the referees of PLOS ONE for the rigorous and competent reviewing.
Author Contributions
Conceived and designed the experiments: RVS LMS. Analyzed the data:
RVS. Wrote the paper: RVS LMS.
References
1. Reya T, Morrison SJ, Clarke MF, Weissman IL (2001) Stem cells, cancer, and
cancer stem cells. Nature 414: 105–111.
2. Clarke MF, Fuller M (2006) Stem cells and cancer: Two faces of eve. Cell 124:
1111–1115.
3. Vermeulen L, Sprick MR, Kemper K, Stassi G, Medema JP (2008) Cancer stem
cells – old concepts, new insights. Cell Death and Differentiation aop.
4. Dalerba P, Cho RW, Clarke MF (2007) Cancer stem cells: models and concepts.
Annual review of medicine 58: 267–284.
5. Bomken S, Fisˇer K, Heidenreich O, Vormoor J (2010) Understanding the cancer
stem cell. British journal of cancer 103: 439–445.
6. Lewis M (2008) Faith, heresy and the cancer stem cell hypothesis. Future
oncology (London, England) 4: 585.
7. Hill RP (2006) Identifying cancer stem cells in solid tumors: case not proven.
Cancer Research 66: 1891–1895; discussion 1890.
8. Welte Y, Adjaye J, Lehrach HR, Regenbrecht CR (2010) Cancer stem cells in
solid tumors: elusive or illusive? Cell Commun Signal 8: 6.
9. Denison TA, Bae YH (2012) Tumor heterogeneity and its implication for drug
delivery. Journal of Controlled Release.
10. Tian T, Olson S, Whitacre J, Harding A (2011) The origins of cancer robustness
and evolvability. Integr Biol 3: 17–30.
11. Shackleton M, Quintana E, Fearon E, Morrison S (2009) Heterogeneity in
cancer: cancer stem cells versus clonal evolution. Cell 138: 822–829.
12. Marusyk A, Polyak K (2010) Tumor heterogeneity: causes and consequences.
Biochimica et Biophysica Acta (BBA)-Reviews on Cancer 1805: 105–117.
13. Marusyk A, Almendro V, Polyak K (2012) Intra-tumour heterogeneity: a looking
glass for cancer? Nature Reviews Cancer.
14. Rapp UR, Ceteci F, Schreck R (2008) Oncogene-induced plasticity and cancer
stem cells. Cell Cycle 7: 45.
15. Grunewald T, Herbst S, Heinze J, Burdach S (2011) Understanding tumor
heterogeneity as functional compartments-superorganisms revisited. Journal of
translational medicine 9: 79.
16. Lander A, Kimble J, Clevers H, Fuchs E, Montarras D, et al. (2012) What does
the concept of the stem cell niche really mean today? BMC biology 10: 19.
17. Iwasaki H, Suda T (2009) Cancer stem cells and their niche. Cancer science 100:
1166–1172.
18. Ishizawa K, Rasheed Z, Karisch R, Wang Q, Kowalski J, et al. (2010) Tumor-
initiating cells are rare in many human tumors. Cell stem cell 7: 279–282.
19. Stewart J, Shaw P, Gedye C, Bernardini M, Neel B, et al. (2011) Phenotypic
heterogeneity and instability of human ovarian tumor-initiating cells. Proceed-
ings of the National Academy of Sciences 108: 6468.
20. Vargaftig J, Taussig D, Griessinger E, Anjos-Afonso F, Lister T, et al. (2011)
Frequency of leukemic initiating cells does not depend on the xenotransplan-
tation model used. Leukemia.
21. Sarry J, Murphy K, Perry R, Sanchez P, Secreto A, et al. (2011) Human acute
myelogenous leukemia stem cells are rare and heterogeneous when assayed in
nod/scid/il2ra˜c-deficient mice. The Journal of Clinical Investigation 121: 384.
22. Zhong Y, Guan K, Zhou C, Ma W, Wang D, et al. (2010) Cancer stem cells
sustaining the growth of mouse melanoma are not rare. Cancer letters 292: 17–
23.
23. Baker M (2008) Melanoma in mice casts doubt on scarcity of cancer stem cells.
Nature 456: 553.
24. Johnston M, Maini P, Jonathan Chapman S, Edwards C, Bodmer W (2010) On
the proportion of cancer stem cells in a tumour. Journal of theoretical biology
266: 708–711.
25. Baker M (2008) Cancer stem cells, becoming common. Nature Reports Stem
Cells.
26. Schatton T, Murphy G, Frank N, Yamaura K, Waaga-Gasser A, et al. (2008)
Identification of cells initiating human melanomas. Nature 451: 345–349.
27. Quintana E, Shackleton M, Sabel M, Fullen D, Johnson T, et al. (2008) Efficient
tumour formation by single human melanoma cells. Nature 456: 593–598.
28. Kelly P, Dakic A, Adams J, Nutt S, Strasser A (2007) Tumor growth need not be
driven by rare cancer stem cells. Science 317: 337.
29. Williams R, Den Besten W, Sherr C (2007) Cytokine-dependent imatinib
resistance in mouse bcr-abl+, arf-null lymphoblastic leukemia. Genes &
development 21: 2283.
30. Boiko A, Razorenova O, van de Rijn M, Swetter S, Johnson D, et al. (2010)
Human melanomainitiating cells express neural crest nerve growth factor
receptor cd271. Nature 466: 133–137.
31. Gupta P, Chaffer C, Weinberg R (2009) Cancer stem cells: mirage or reality?
Nature medicine 15: 1010–1012.
32. Zapperi S, La Porta CAM (2012) Do cancer cells undergo phenotypic switching?
the case for imperfect cancer stem cells markers. Scientific reports.
33. Chaffer C, Brueckmann I, Scheel C, Kaestli A, Wiggins P, et al. (2011) Normal
and neoplastic nonstem cells can spontaneously convert to a stem-like state.
Proceedings of the National Academy of Sciences 108: 7950.
34. Strauss R, Hamerlik P, Lieber A, Bartek J (2012) Regulation of stem cell
plasticity: Mechanisms and relevance to tissue biology and cancer. Molecular
Therapy.
35. Morrison S, Kimble J (2006) Asymmetric and symmetric stem-cell divisions in
development and cancer. Nature 441: 1068–1074.
36. Leder K, Holland E, Michor F (2010) The therapeutic implications of plasticity
of the cancer stem cell phenotype. PloS one 5: e14366.
37. Turner C, Stinchcombe AR, Kohandel M, Singh S, Sivaloganathan S (2009)
Characterization of brain cancer stem cells: a mathematical approach. Cell
Prolif 42: 529–40.
38. Laird AK (1964) Dynamics of tumour growth. British journal of cancer 18: 490.
39. Choe SC, Zhao G, Zhao Z, Rosenblatt JD, Cho HM, et al. (2011) Model for in
vivo progression of tumors based on co-evolving cell population and vasculature.
Scientific reports 1.
40. Gliozzi AS, Guiot C, Delsanto PP (2009) A new computational tool for the
phenomenological analysis of multipassage tumor growth curves. PloS one 4:
e5358.
41. Herman AB, Savage VM, West GB (2011) A quantitative theory of solid tumor
growth, metabolic rate and vascularization. PloS one 6: e22973.
42. Vaidya VG, Alexandro FJ Jr (1982) Evaluation of some mathematical models for
tumor growth. International Journal of Bio-Medical Computing 13: 19–35.
43. Weedon-Fekjær H, Lindqvist BH, Vatten LJ, Aalen OO, Tretli S, et al. (2008)
Breast cancer tumor growth estimated through mammography screening data.
Breast Cancer Res 10: R41.
44. Guiot C, Delsanto PP, Carpinteri A, Pugno N, Mansury Y, et al. (2006) The
dynamic evolution of the power exponent in a universal growth model of tumors.
Journal of theoretical biology 240: 459–463.
45. Guiot C, Degiorgis PG, Delsanto PP, Gabriele P, Deisboeck TS (2003) Does
tumor growth follow a universal law? Journal of theoretical biology 225: 147–
151.
46. Castorina P, Zappala` D (2006) Tumor gompertzian growth by cellular energetic
balance. Physica A: Statistical Mechanics and its Applications 365: 473–480.
47. Von Bertalanffy L (1957) Quantitative laws in metabolism and growth. The
quarterly review of biology 32: 217–231.
48. Perko L (2000) Differential Equations and Dynamical Systems. Texts in Applied
Mathematics. Springer.
49. Hirsch M, Smale S, Devaney R (2004) Differential Equations, Dynamical
Systems, and an Introduction to Chaos. Pure and Applied Mathematics.
Academic Press.
50. Tomasetti C, Levy D (2010) Role of symmetric and asymmetric division of stem
cells in developing drug resistance. Proceedings of the National Academy of
Sciences 107: 16766–16771.
51. Berglund N, Gentz B (2006) Noise-induced phenomena in slow-fast dynamical
systems: a samplepaths approach. Probability and its applications. Springer.
52. Gardiner C (2009) Stochastic methods: a handbook for the natural and social
sciences. Springer series in synergetics. Springer.
53. Burness M, Sipkins D (2010) The stem cell niche in health and malignancy. In:
Seminars in cancer biology. Elsevier, volume 20, 107–115.
54. Whiteside T (2008) The tumor microenvironment and its role in promoting
tumor growth. Oncogene 27: 5904–5912.
55. Maffini M, Soto A, Calabro J, Ucci A, Sonnenschein C (2004) The stroma as a
crucial target in rat mammary gland carcinogenesis. Journal of cell science 117:
1495–1502.
56. Gammaitoni L, Ha¨nggi P, Jung P, Marchesoni F (1998) Stochastic resonance.
Reviews of Modern Physics 70: 223.
57. Van den Broeck C, Parrondo J, Toral R, Kawai R (1997) Nonequilibrium phase
transitions induced by multiplicative noise. Physical Review E 55: 4084.
58. Ridolfi L, D’Odorico P, Laio F (2011) Noise-Induced Phenomena in the
Environmental Sciences. Cambridge University Press.
59. Oksendal B (2003) Stochastic differential equations: an introduction with
applications. Universitext (1979). Springer.
Cancer Stem Cells Enjoy Noise
PLOS ONE | www.plosone.org 12 August 2013 | Volume 8 | Issue 8 | e69131
60. Karlin S, Taylor H (2000) A second course in stochastic processes. Academic
Press.
61. Horsthemke W, Lefever R (1984) Noise-induced transitions: theory and
applications in physics, chemistry, and biology. Springer series in synergetics.
Springer.
62. Kavinoky R, Thoo J (2008) The number of real roots of a cubic equation. The
AMATYC Review 29: 3–8.
63. Braumann CA (2007) Harvesting in a random environment: It or stratonovich
calculus? Journal of Theoretical Biology 244: 424–432.
64. Kloeden P, Platen E (1992) Numerical solution of stochastic differential
equations. Applications of mathematics. Springer-Verlag.
65. Kalialis L, Drzewiecki K, Klyver H (2009) Spontaneous regression of metastases
from melanoma: review of the literature. Melanoma research 19: 275.
66. Da-jinW, Li C, Sheng-zhi K (1994) Bistable kinetic model driven by correlated
noises: Steady-state analysis. Phys Rev E 50: 2496–2502.
67. Atkinson R, Zhang M, Diagaradjane P, Peddibhotla S, Contreras A, et al. (2010)
Thermal enhancement with optically activated gold nanoshells sensitizes breast
cancer stem cells to radiation therapy. Science translational medicine 2: 55ra79–
55ra79.
68. Pietras A (2011) Cancer stem cells in tumor heterogeneity. Advances in Cancer
Research 112: 256.
69. Chow E (2012) Implication of cancer stem cells in cancer drug development and
drug delivery. Journal of Laboratory Automation.
70. Chen J, Li Y, Yu TS, McKay RM, Burns DK, et al. (2012) A restricted cell
population propagates glioblastoma growth after chemotherapy. Nature 488:
522–526.
71. Arens N, West I (2008) Press-pulse: a general theory of mass extinction?
Paleobiology 34: 456–471.
72. Driessens G, Beck B, Caauwe A, Simons BD, Blanpain C (2012) Defining the
mode of tumour growth by clonal analysis. Nature 488: 527–530.
Cancer Stem Cells Enjoy Noise
PLOS ONE | www.plosone.org 13 August 2013 | Volume 8 | Issue 8 | e69131
Appendix to
A possible explanation for the variable
frequencies of cancer stem cells in tumors
Rescale transformations
In this appendix we detail the rescales made throughout the main text. The
general model written in terms of the reactions is
C
k1−−−−⇀↽ −
k′2/Ω2
C + C
P
k3−−−−⇀↽ −
k′4/Ω4
P + P
C
k5⇀ C + P
C
k6⇀ P + P
P
k7⇀ ∅
P
k8⇀ C (1)
Using the law of mass action we have C˙ = k1C
(
1− CΩC
)
− k6C + k8P
P˙ = k3P
(
1− PΩP
)
+ k9C − k10P
(2)
with k9 ≡ k5 + 2k6, k10 ≡ k7 + k8, ΩC ≡ k1k2 , ΩP ≡ k3k4 and k2 ≡ k′2/Ω2,
k′4 ≡ k′4/Ω4. Using the reescale C ≡ ΩCx and P ≡ ΩP y :{
x˙ = k1x (1− x)− k6x+ k8ΩPΩC y
y˙ = k3y (1− y) + k9ΩCΩP x− k10y
(3)
Using t ≡ k6t′ and Ω ≡ ΩPΩC :{
dx
dt′ =
k1
k6
x (1− x)− x+ k8Ωk6 y
dy
dt′ =
k3
k6
y (1− y) + k9k6Ωx− k10k6 y
(4)
or {
x′ = Ax(1 − x)− x+By
y′ = Ey(1− y) + Fx−Gy (5)
1
with x′ ≡ dxdt′ , y′ ≡ dydt′ and 
A ≡ k1k6
B ≡ k2k3k8k1k4k6
E ≡ k3k6
F ≡ k1k4k9k2k3k6
G ≡ k10k6
(6)
Gradient system
Starting from (30) and carrying out the transformation C = s1c, P = s2p and
t = s3τ, we can write
dc
dτ = k1s3c
(
1− s1ΩC c
)
− k6s3c+ k8s2s3s1 p
dp
dτ = k3s3p
(
1− s2ΩP p
)
+ k9s1s3s2 c− k10s3p
(7)
Imposing k8s2s3s1 =
k9s1s3
s2
, k6s3 = 1 and s1 = ΩC , we obtain s1 ≡ k1k2 , s2 ≡
ΩC
√
k9
k8
and s3 =
1
k6
.
In this way we obtain
dc
dτ =
k1
k6
c (1− c)− c+
√
k8k9
k6
p
dp
dτ =
k3
k6
p
(
1− ΩCΩP
√
k9
k8
p
)
+
√
k8k9
k6
c− k10k6 p.
(8)
Potential V (x, y)
We want to write the equation (3) in the form x˙ = −∇V (x, y), where x =
(x(t), y(t))T (T means transpose), ∇ is the nabla operator. Integrating f(x, y)
from (3) with respect to x gives
V (x, y) =
∫
f(x, y)dx = −x
2
2
+
Ax2
2
− Ax
3
3
+Bxy + f0(y). (9)
We must now obtain f0(y). Imposing ∂yV (x, y) = g(x, y), we obtain Bx +
f ′0(y) = g(x, y) and then f0(y) =
∫
[Ey(1−Fy)−Gy]dy = Ey22 − Gy
2
2 − 13EFy3.
This provides the equation (5).
2
100
Capı´tulo 6
O ruı´do e o KISS no nicho das ce´lulas
tronco do caˆncer
Esta introduc¸a˜o e´ baseada nos resultados do artigo [2], publicado em Journal of Theoretical Biology.
No artigo anterior propusemos um modelo estoca´stico 0−dimensional para estudarmos a dinaˆmica po-
pulacional das ce´lulas tronco do caˆncer (CTC). Neste segundo artigo, publicado em Journal of Theoretical
Biology, [2] ampliamos a ana´lise do anterior a uma situac¸a˜o onde as ce´lulas podem se difundir em um
espac¸o 1−dimensional. Mudamos o tratamento das equac¸o˜es diferenciais ordina´rias estoca´sticas para as
equac¸o˜es diferenciais parciais estoca´sticas. Verificamos que um fenoˆmeno ana´logo a` transic¸a˜o induzida
por ruı´do, i.e., uma transic¸a˜o de fase, ocorre no caso 1−dimensional: para valores suficientemente ele-
vados das flutuac¸o˜es estatı´sticas no microambiente celular, a distribuic¸a˜o estaciona´ria de probabilidades
sofre uma transic¸a˜o de fase e se torna bimodal. Alguns estudos associaram esta transic¸a˜o na distribuic¸a˜o
estaciona´ra a uma transic¸a˜o do cara´ter benigno para o maligno dos tumores.
Neste artigo tambe´m esta´vamos especialmente interessados no estudo do tamanho crı´tico mı´nimo ne-
cessa´rio Lc para que uma populac¸a˜o possa se desenvolver quando cercada por um ambiente hostil. Tal
tamanho crı´tico mı´nimo a`s vezes e´ denominado na literatura Kierstead-Skellam-Slobodkin size (KISS).
Estudamos os efeitos da plasticidade celular e do ruı´do do microambiente em Lc, atrave´s dos chamados
diagramas de bifurcac¸a˜o. Concluı´mos mais uma vez que os efeitos das perturbac¸o˜es estatı´sticas podem ser
bene´ficos a`s ce´lulas tronco do caˆncer, diminuindo o tamanho crı´tico mı´nimo necessa´rio ao seu desenvolvi-
mento. Tambe´m estudamos os efeitos do ruı´do combinados com os efeitos da difusa˜o celular. Concluı´mos
que a difusa˜o compete com as flutuac¸o˜es estatı´sticas fazendo com que os efeitos destas (por exemplo,
induc¸a˜o de transic¸a˜o de fase em Pst ) sejam amenizados.
101
The noise and the KISS in the cancer stem cells niche
Renato Vieira dos Santos a,n, Linaena Méricy da Silva b,c
a Departamento de Física, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, CP 702, CEP 31270-901 Belo Horizonte, Minas Gerais, Brasil
b Centro Universitário Metodista Izabela Hendrix, Núleo de Biociências, Rua da Bahia 2020, CEP 30160-012, Belo Horizonte, Minas Gerais, Brasil
c Laboratório de Patologia Comparada, Instituto de Ciências Biológicas, Universidade Federal de Minas Gerais, CP 702, CEP 31270–901 Belo Horizonte, Minas
Gerais, Brasil
H I G H L I G H T S

 Possible explanation for wide variability observed in cancer stem cells frequency.

 Plasticity is necessary for maintenance of cancer stem cell populations.

 Cell population may exhibit a noise-induced transition.
a r t i c l e i n f o
Article history:
Received 15 December 2012
Received in revised form
8 May 2013
Accepted 21 June 2013
Available online 1 July 2013
Keywords:
Cancer stem cells
Stochastic modeling
Minimal patch size
Plasticity
Diffusion
a b s t r a c t
There is a persistent controversy regarding the frequency of cancer stem cells (CSCs) in solid tumors.
Initial studies indicated that these cells had a frequency ranging from 0.0001% to 0.1% of total cells.
Recent studies have shown that this does not seem to be always the case. Some of these studies have
indicated a frequency of 40%. Through a simple population dynamics model, we studied the effects of
stochastic noise and cellular plasticity in the minimal path size of a cancer stem cells population, similar
to what is done in what is sometimes called the Kierstead–Skellam–Slobodkin (KISS) Size analysis. We
show that the possibility of large variations in the results obtained in the experiments may be a
consequence of the different conditions under which the different experiments are submitted,
specifically regarding the effective cell niche size where stem cells are transplanted. We also show the
possibility of a noise induced transition where the stationary probability distribution of the CSC
population can present bimodality.
& 2013 Elsevier Ltd. All rights reserved.
1. Introduction
In recent years there has been increasing evidence for the
Cancer Stem Cell (CSC) hypothesis (Reya et al., 2001; Clarke and
Fuller, 2006; Vermeulen et al., 2008; Dalerba et al., 2007), accord-
ing to which tumor formation is a result of genetic and epigenetic
changes in a subset of stem-like cells, also known as tumor-forming
or tumor-initiating cells (Bomken et al., 2010). Cancer stem cells
were first identified in various leukemias and, more recently, in
several solid tumors such as brain, breast, cervix and prostate
tumors (Dalerba et al., 2007).
It has been suggested that these are the cells responsible for
initiating and maintaining tumor growth. In this paper, we study a
model for tumor growth that assumes the existence of cancer stem
cells (CSCs), or tumor initiating cells.
The conceptual starting point relevant to CSC theory is con-
structed from the known heterogeneity of tumors. We now know
that cells in a tumor are not all identical copies of each other, but
that they display a striking array of characteristics (Denison, 2012;
Tian et al., 2011; Shackleton et al., 2009; Marusyk and Polyak,
2010; Marusyk et al., 2012). CSC theory recognizes this fact and
develops its consequences. And one of the most immediate
implications for clinical practice is that conventional treatments
can generally attack the wrong cell type. The appeal of the CSC
idea can be described by the following analogy: just as killing the
queen bee leads to the demise of the hive, destroying cancer stem
cells, should, in theory, stop a tumor from renewing itself.
Unfortunately, things are never that simple. In the hive, workers
react quickly to the death of the queen by replacing her with a
new one. And there is some evidence (Welte et al., 2010; Rapp
et al., 2008) to suggest that could also happen in tumors due to a
phenomenon known as cell plasticity, which allows normal tumor
cells to turn into cancer stem cells, should the situation call for it.
One goal of this study is to evaluate the possible effects of this
Contents lists available at SciVerse ScienceDirect
journal homepage: www.elsevier.com/locate/yjtbi
Journal of Theoretical Biology
0022-5193/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.jtbi.2013.06.025
n Corresponding author. Tel.: +51 3133737186.
E-mail addresses: econofisico@gmail.com (R.V. dos Santos),
linaena.mericy@gmail.com (L.M. da Silva).
Journal of Theoretical Biology 335 (2013) 79–87
plasticity. Analogies with superorganisms such as bee colonies are
taken much more seriously in Grunewald et al. (2011).
Stem cells in general (the same applies to CSCs) tend to be
found on specific areas of a tissue where one particular micro-
environment, called niche (Lander et al., 2012), promotes the
maintenance of its vital functions. This niche has specialized in
providing factors that prevent differentiation and thus maintain
the stemness of CSCs and, ultimately, the tumor's survival. Stem
cells and niche cells interact with each other via adhesion
molecules and paracrine factors. This complex network of interac-
tions exchanges molecular signals and maintains the unique char-
acteristics of stem cells, namely, pluripotency and self-renewal.
Given the extreme complexity of the cellular microenvironment
in general and of the niche in particular (Iwasaki and Suda, 2009;
Lander et al., 2012), wewill formulate an effective stochastic theory
for the population dynamics of CSCs.We are especially interested in
investigating a controversy related to the frequency with which
CSCs appear in various tumors (Ishizawa et al., 2010; Stewart et al.,
2011; Vargaftig et al., 2011; Sarry et al., 2011; Zhong et al., 2010;
Baker, 2008a,b; Johnston et al., 2010). In the initial version of CSC
theory, it was believed that these cells were a tiny fraction of the
total, ranging from 0.0001% to 0.1% (Schatton et al., 2008; Quintana
et al., 2008). However, more recent studies have shown a strong
dependence on the number of stem cells present in a tumor with
the xenograft experimental model used. In explicit contrast towhat
was previously thought, in Quintana et al. (2008) a CSC proportion
of approximately 25% was observed. Other studies have confirmed
this observation (Kelly et al., 2007; Williams et al., 2007; Schatton
et al., 2008), with the proportion potentially reaching 41% (Boiko
et al., 2010). In Gupta et al. (2009) the authors provide evidence that
this discrepancy may be caused by the possibility of phenotypic
switching between different tumor cells. By phenotypic switching
we mean that a more differentiated cancer cell can, under appro-
priate conditions, de-differentiate into a cancer stem cell. This is the
cellular plasticity mentioned above.
In Zapperi and La Porta (2012), it is suggested that inconsis-
tencies in the numbers of cancer stem cells reported in the
literature can also be explained as a consequence of the different
definitions used by different researchers. Different assays will give
different numbers of cells, which can be orders of magnitude away
from each other.
In this paper we are also interested in knowing what are the
possible effects of cells diffusion in space. For this, we constructed
bifurcation diagrams that show how the population size of CSCs
varies when the size of the niche cells changes. We consider the
effects that the plasticity phenomenon as well as spatio-temporal
noise can have in these diagrams. Finally we studied the effects of
the spatial distribution of cells in stationary probability
distributions.
The paper is organized as follows: in Section 2 we explain the
basic assumptions of our model of CSC population dynamics. In
Section 3 we describe the set of reactions we use in the models. The
effects of inclusion of spatial structure in the analysis are considered
in Section 4. Section 5 closes the paper with conclusions.
2. Assumptions
Mathematical modeling has made significant contributions to
our understanding of the biology of cancer since the pioneering
work of Nordling (1953) and Armitage and Doll (1954), in which
the authors proposed that multiple mutations may explain the
data on the incidence of cancer and its correlation with age (Chen
et al., 2005; Horov et al., 2009). For historical reviews on the
subject, see McElwain and Araujo (2004) and Byrne et al. (2006).
In the model used in this paper, cancer stem cells can perform
three types of divisions, according to Morrison and Kimble (2006):

 Symmetric self-renewal: Cell division in which both daughter
cells have the characteristics of the stem cell mother, resulting
in an expanding population of stem cells.

 Symmetric differentiation: A stem cell divides into two prog-
enitor cells.

 Asymmetric self-renewal: A cancer stem cell (denoted by C) is
generated and a progenitor cell (mature cancer cell, denoted by
P) is also produced.
We developed a simple mathematical model for the stochastic
dynamics of CSCs in which the three division types possess
intrinsic replication rates, which are assumed to be time-
independent. Therefore, besides these division types, we assume
that there is also the possibility of a transformation in which a
progenitor cell can acquire characteristics of stem cells where, for
all practical purposes, we may regard it as having become a de-
differentiated stem cell. In mixed lineage leukemia cells, it was
recently shown that committed myeloid progenitor cells acquire
properties of leukemia stem cells without changing their overall
identity (Leder et al., 2010). These cells do not become stem cells,
but rather develop stem cell like behavior by re-activating a subset
of genes highly expressed in normal hematopoietic stem cells
(Rapp et al., 2008). The biological mechanisms underlying this
transformation are described in Gupta et al. (2009), for example.
As mentioned previously, we refer to this process as cell plasticity.
3. Model
This section describes the basic model investigated in this
paper. It is based on the cell division mechanism and the plasticity
property. We will use the language of stochastic differential
equations (Karlin and Taylor, 2000; Schuss, 2010; Oksendal, 2003).
The model is a natural extension of the one proposed in Turner
et al. (2009). This extension refers to the inclusion of competition
between cells because of the scarcity of resources when popula-
tions become large enough. This new possibility in relation to the
model proposed in Turner et al. (2009) makes the model nonlinear
and prevents that the populations tend to infinity. The model is
described in the next subsection.
3.1. The basic model
We assume that the population dynamics of cancer stem cells
and progenitor cells are governed by the following reactions:
C ⇌
k1
k′2=Ω2
C þ C
P ⇌
k3
k′4=Ω4
P þ P
C,
k5
C þ P
C,
k6
P þ P
P,
k7
∅
P,
k8
C ð1Þ
The first and second reactions, in the forward sense, model cell
proliferation, which occurs at a rate k1 and k3, respectively.
Constants k′2 and k′4 are associated to the reverse process and
describe the intensity of competition between the CSC and
progenitor cells, respectively, and prevents their unlimited expo-
nential growth; Ω2 and Ω4 are constants related to the model's
carrying capacity. The third reaction involving k5 originates from
the asymmetric transformation of CSCs in CSC daughter and
progenitor cell types. The reaction involving the rate k6 is related
R.V. dos Santos, L.M. da Silva / Journal of Theoretical Biology 335 (2013) 79–8780
to a symmetrical division of stem cells, which gives rise to two
progenitor cells. The penultimate reaction is associated with pro-
genitor cell death at rate k7. Finally, k8 is the de-differentiation rate.
All rates have dimension (time)1. The specific unit of time (months,
quarters, years, etc.) will depend on the type and aggressiveness of
the tumor.
Using the law of mass action, we can write
dC
dt
¼ k1Ck2C2k6C þ k8P
dP
dt
¼ k3Pk4P2 þ ðk5 þ 2k6ÞCðk7 þ k8ÞP
8><
>: ð2Þ
with k2≡k′2=Ω2, k4≡k′4=Ω4. Setting ΩC≡k1=k2, ΩP≡k3=k4, k9≡k5 þ
2k6 and k10≡k7 þ k8 and making the substitutions C ¼ΩCx,
P ¼ΩC
ffiffiffiffiffiffiffiffiffiffiffiffi
k9=k2
p
y and t ¼ τ=k6, Eq. (2) can be written as (see
Appendix A)
dx
dτ
¼ Axð1xÞxþ By≡f ðx; yÞ
dy
dτ
¼ Eyð1FyÞ þ BxGy≡gðx; yÞ
8><
>: ð3Þ
with
A≡
k1
k6
B≡
ffiffiffiffiffiffiffiffiffi
k2k9
p
k6
E≡
k3
k6
F≡
ΩC
ΩP
ffiffiffiffiffi
k9
k2
s
G≡
k10
k6
8>>>>>>>>><
>>>>>>>>>:
ð4Þ
As ∂f =∂y¼ ∂g=∂x¼ B, Eq. (3) represents a gradient system (Perko,
2000) with potential Vðx; yÞ given by
Vðx; yÞ ¼ 1
6
ð33Aþ 2AxÞx2Bxyþ 1
6
ð3G3E þ 2EFyÞy2: ð5Þ
As a consequence (Hirsch et al., 2004):
1. The eigenvalues of the linearization of Eq. (3) evaluated at
equilibrium point are real.
2. If ðx0; y0Þ is an isolated minimum of V then ðx0; y0Þ is an
asymptotically stable solution of (3).
3. If ðxðτÞ; yðτÞÞ is a solution of (3) that is not an equilibrium point
then VðxðτÞ; yðτÞÞ is a strictly decreasing function and is perpen-
dicular to the level curves of Vðx; yÞ.
4. There are no periodic solutions of (3).
Fig. 1 shows the potential function Vðx; yÞ. Sufficiently small F
(ΩP≫ΩC) implies large differences in equilibrium populations of C
and P. For parameters A¼ B¼ G¼ 1, E¼3 and F ¼ 0:01,
ðx0; y0Þ ¼ ð8:4;70:6Þ. If we set F ¼ 0:0001, keeping the other para-
meters fixed, we get ðx0; y0Þ ¼ ð82;6710Þ.
3.2. Adiabatic elimination
The proposed model in (1) is in fact a general model of stem
cells and does not even carry any specific characteristic of cancer
stem cells. All properties considered, such as plasticity and
changes in the microenvironment conditions (to be included
later), are also found in stem cell systems of normal tissue. The
features associated with cancer stem cells are related to the large
carrying capacity of progenitor cells when compared with the
carrying capacity of cancer stem cells. This fact is represented
numerically by the choice of model parameters made below,
which results in this discrepancy.
We can write (2) in the form (see Appendix A)
x′¼ A′xð1xÞxþ B′y
y′¼ E′yð1yÞ þ F′xG′y
(
ð6Þ
with x′≡dx=dt′, y′≡dy=dt′, t′≡t=k6 and
A′≡
k1
k6
B′≡k2k3k8
k1k4k6
E′≡
k3
k6
F′≡
k1k4k9
k2k3k6
G′≡k10
k6
8>>>>>>>>><
>>>>>>>>>:
ð7Þ
Fig. 2 shows the numerical solutions of Eqs. (6) (the rescaled
equation) and (2) for the following parameter values: k1 ¼
1k5k6, k2 ¼ 4 1013, k3 ¼ 1, k4 ¼ 1013, k5 ¼ 0:1, k6 ¼ 0:1,
k7¼0.1 and k8 ¼ 0:00001.1 We make the usual assumption
(k1 þ k5 þ k6Þb¼ 1 (Tomasetti and Levy, 2010), where β ≡ 1 is a
general parameter with dimension time(1) required for dimen-
sional consistency in the following analysis. The values for k5 and
k6 are consistent with those estimated in Tomasetti and Levy
(2010). For these parameter values, ΩC≡k1=k2 ¼ 2 1012 and
ΩP≡k3=k4 ¼ 1 1013 (see Appendix A). These are rescaling para-
meters for x and y variables, respectively. Stationary values for P(t)
and C(t) are P1 ¼ 9:6 1012 cells and C1 ¼ 1:8 1012 cells,
respectively. By adjusting the k2 and k4 parameters we can easily
obtain more suitable values for the CSC and progenitor cell
equilibrium populations, according to possible new experimental
results.
By using standard adiabatic elimination methods, one can write
Eq. (6) as
x′¼ A′ xð1xÞ x
A′
þ B′
A′
y
 
ϵy′¼ yð1yÞ þ ϵF′xϵG′y
8><
>: ð8Þ
Fig. 1. Potential Vðx; yÞ from Eq. (5) for parameters A¼ B¼ G¼ 1, E¼3 and F¼0.01.
1 These values correspond to A′¼ 8, B′¼ 5 104, E′¼ 10, F ′¼ 0:6 and G′¼ 1.
R.V. dos Santos, L.M. da Silva / Journal of Theoretical Biology 335 (2013) 79–87 81
where ϵ≡1=E′. If we consider ϵ≪1 (this is equivalent to considering
the progenitor cell division rate sufficiently large) we can perform
adiabatic approximation (Berglund and Gentz, 2006; Gardiner,
2009) in (8) and, setting y′¼ 0, we obtain the following equation2
for x:
x′¼ χμxþ αxð1xÞ ð9Þ
where χ≡B′ð1ϵG′Þ ¼ k8k2ðk3k10Þ=k1k4k6,
μ≡1ϵB′F ′¼ 1k8k9=k3k6 and α≡A′¼ k1=k6. Note that χ can be
positive or negative depending on the magnitudes of k3 and k10.
If we consider ϵ to be small enough with respect to G′, B′ and F ′,
we further simplify and write χ ¼ B′ and μ¼ 1. We can observe that
the plasticity phenomenon (associated with k8) is crucial for the
existence of the constant term χ. For this reason, from now on we
will consider the parameter χ as representing the plasticity
phenomenon in the reduced equation (9).
3.3. The deterministic equation
We will briefly review the deterministic analysis of the pro-
blem. An analytic solution of Eq. (9) is possible. For the initial
condition xð0Þ ¼N0, we get
xðtÞ ¼ 1
2α
δ ffiffiκp tan 1
2
t
ffiffi
κ
p þ arctan 2N0αþ δffiffi
κ
p
   
ð10Þ
with δ≡αμ and κ≡δ24αχ. The physically relevant stable fixed
point is
xn ¼ αμþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
α22αμþ μ2 þ 4αχ
p
2α
: ð11Þ
The x scaled population size dynamics can be thought of as
analogous to the particle moving dynamics in an effective poten-
tial V0ðxÞ, seeking its minimum point, with V0ðxÞ≡
R
ζðxÞ dx with
ζðxÞ ¼ χ þ δxαx2 from (9). Thus, V0 is given by the cubic
polynomial,
V0ðxÞ ¼
x3α
3
 δx
2
2
xχ:
We see from (11) that by increasing either χ or δ, the minimum xn
of V0 moves to the right in the potential, thus favoring the tumor
stem cell population. Such behavior is, of course, expected, since
an increase of χ means an increase of the frequency in which the
induced plasticity mechanism occurs, and an increase of δ is an
increase of the symmetric stem cell renewal rate, both of which
increase the population.
4. Possible consequences of a spatial structure
Traditionally, anti-tumor treatments have targeted the cells
directly, removing them with surgery or killing them with radia-
tion. Since these are local treatment methods, they often are not
effective in meeting their objectives. The tumors may recur
because not all cells were killed or because some cells escaped
the primary tumor region where the treatments worked. Since
cells compete and/or cooperate with nontumor cells and between
themselves, these interactions may be better conceptualized as an
evolving ecosystem (Pienta et al., 2008; Kareva, 2011).
One of the possible consequences of this way of seeing the
disease is that the destruction of the tumor microenvironment can
be much more effective than just extracting or killing the cells that
live in it. A prime example of this situation comes from paleontol-
ogy: studies analyzing the conditions that preceded mass extinc-
tions suggest that they occurred more frequently and were more
destructive when pulses of disturbances that cause extensive
mortality were accompanied by perturbative pressures such as
climate change. This sequence weakened and destabilized popula-
tions for several generations (Arens and West, 2008).
Motivated by these considerations, it seems promising to
consider mathematical techniques originating in mathematical
ecology, a well-developed branch of applied mathematics
(Cantrell and Cosner, 2003; Petrovskii and Li, 2005). We are now
interested in the possible effects from the incorporation of diffu-
sion in the model. For this, let uðx; tÞ be a population of CSCs at
position x at time t that lives in a one-dimensional domain of
length L. By adding a diffusion term in Eq. (9), we get
∂u
∂t′
¼D ∂
2u
∂t′2
þ χμuþ αuð1uÞ≡D ∂
2u
∂t′2
þ f ðuÞ ð12Þ
where D is the cell diffusion coefficient and f ðuÞ ¼ χμuþ αuð1uÞ.
This is the deterministic partial differential equation that we will
consider first. Later we will consider a stochastic version. This is a
reaction diffusion equation that is typical in the population
dynamics of species that interact and disperse.
Eq. (12) with χ ¼ 0 is the famous Fisher (1937) equation. In this
equation we analyze the effect of plasticity represented by the
parameter χ on the patch size to sustain a population, similar to
what is sometimes called the Kierstead–Slobodkin–Skellam (KISS)
size (Skellam, 1951; Kierstead and Slobodkin, 1953). The main
motivation for performing this type of analysis is related to the
experimental results obtained in Quintana et al. (2008), where
xenografts in immunosuppressed mice sustained surprisingly high
populations of cancer stem cells. The idea here, therefore, is to
identify some phenomenon related to the size of the CSC niche
that may justify this result. The question is: What is the effect of
transplanting a cancer stem cell population to an environment
where, in theory, they will have more space to live and proliferate?
To formulate the problem mathematically, we can imagine that
there is a finite domain available for the cells to develop (their
niche). Beyond a certain boundary (i.e. outside the niche), there
0 1 2
0
0.6
1.2
0 25 50
Fig. 2. Top: numerical solution for rescaled Eq. (6). x(t) and y(t) represent the
rescaled population of cancer stem cells and progenitor cells, respectively. Bottom:
numerical solution for Eq. (2). C(t) and P(t) represent the population of cancer stem
cells and progenitor cells, respectively. P1 and C1 represent the limits of C(t) and
P(t) when t-1, respectively. Parameters values: k1 ¼ 1k5k6, k2 ¼ 4 1013,
k3 ¼ 1, k4 ¼ 1013, k5 ¼ 0:1, k6 ¼ 0:1, k7¼0.1 and k8 ¼ 0:00001. P1 ¼ 9:6 1012 and
C1 ¼ 1:8 1012. C1=P1 ¼ 0:1875.
2 Expanding in Taylor series up to first order in ϵ.
R.V. dos Santos, L.M. da Silva / Journal of Theoretical Biology 335 (2013) 79–8782
are restrictions (e.g., absence of signaling to support the cancer
stem cell phenotype, normoxic conditions incompatible with the
state of CSCs, adverse pH conditions, etc.) which make the survival
of cancer stem cells unsustainable. Outside the niche, these cells
have a tendency to differentiate into progenitor cells. Thinking of
the niche as a linear domain of length L, our problem can be
mathematically formulated as a boundary value problem with
Dirichlet conditions given by (12) and
uð0; t′Þ ¼ uðL; t′Þ ¼ 0:
Following Méndez et al. (2008), the population density at the
steady state is given by uðx;1Þ≈um sin ðπx=LÞ, where um is the
maximum population density at steady state for a given patch
size L. In Méndez et al. (2008), for a function f ðuÞ in the form
f ðuÞ ¼ a1uþ a2u2 þ a3u3, an approximation to um is found as the
real solution to the equation Φðum; LÞ ¼ 0 with
Φðum; LÞ≡um
3a3
4
u2m þ
8a2
3π
um þ a1D
π
L
	 
2 
: ð13Þ
Allowing the possibility of plasticity (represented by χ), we
insert a term a0 in the f(u) function so that f ðuÞ ¼ a0þ
a1uþ a2u2 þ a3u3. By performing the calculations as in Méndez
et al. (2008), we obtain the new function
Φχðum; LÞ ¼
4a0
π
þ um
3a3
4
u2m þ
8a2
3π
um þ a1D
π
L
	 
2 
: ð14Þ
By solving equation Φχðum; LÞ ¼ 0, we get three solutions that,
when placed on the same figure, make up what we call a
bifurcation diagram (if χ ¼ 0).3 For a0 ¼ χ, a1 ¼ αμ, a2 ¼α and
a3 ¼ 0, we consider the case of Eq. (12).
Fig. 3 shows the bifurcation diagram for χ ¼ 0 (blue curve) and
the curves umðLÞ for χ ¼ 0:1 (dotted red curve) and χ ¼ 0:01 (black
dashed curve). We find that the inclusion of plasticity allows small
cancer stem cell populations to survive even in very small niches.
Above a certain approximate critical minimum value for patch size
(KISS size Lc ¼ π
ffiffiffiffiffiffiffiffiffiffiffi
D=a1
p
), the population undergoes an abrupt
increase in its size. If a small cell niche is abruptly increased to a
value significantly greater than Lc, the CSC population will be
absurdly high. This may have been the case for the xenograft
cancer stem cells in immunosuppressed mice reported in Quintana
et al. (2008). This may also be an answer to the question
raised above.
4.1. Effect of noise on the bifurcation diagram
4.1.1. Noise in the cancer stem cell niche
Cells growing in a tissue are not alone: they are constantly
communicating with one another by sending signals through
tissue that are picked up and transmitted by other cells in the
medium. When thousands of cells are together, there are hundreds
of thousands of these signals present every minute, all competing
to be heard. All this complexity induces stochastic fluctuations in
population dynamics that will hereafter be called noise.
A growing body of evidence indicates that noise is generally not
detrimental to biological systems, but can be employed to gen-
erate behavioral diversity (Samoilov et al., 2005; Fange and Elf,
2006). Mechanisms involving noise are important in the develop-
ment of organisms (Arias and Hayward, 2006; El-Samad and
Khammash, 2006), a fact supported by experiments showing that
noise is down-regulated in embryonic stem cells (Zwaka, 2006)
and that fluctuations of the Nanog transcription factor predispose
these cells towards differentiation (Chambers et al., 2007; Kalmar
et al., 2009). In Hoffmann et al. (2008) it was suggested that the
regulation of noise can be an effective strategy in stem cell
differentiation. The results of the present paper suggest that high
levels of noise can stimulate the development of cancer stem cells.
4.1.2. Modeling noise in a spatial environment
Wewill now reformulate the population dynamics in terms of a
stochastic reaction–diffusion equation and reduce it to a determi-
nistic equation that incorporates the systematic noise contribu-
tions (Santos and Sancho, 2001). Let us first formulate the problem
in a general way and then use our model as an example.
Consider the following stochastic partial differential equa-
tion (SPDE) in the Stratonovich interpretation, with both additive
and multiplicative noises:
∂ϕ
∂t
¼D∇2ϕþ f ðϕÞ þ ϵ1=2gðϕÞηðx; tÞ þ ξðx; tÞ: ð15Þ
In the above equation, ϵ is an explicit measure of the noise strength
given by ηðx; tÞ, ϕðx; tÞ is a field (scalar or vector) that describes the
state of the system (the number of CSCs in our context) at a spatial
location x at time t, and D is the diffusion coefficient. The additive
noise ξðx; tÞ is Gaussian and white in both space and time, with zero
mean and correlations given by
〈ξðx; tÞξðx′; t′Þ〉¼ 2γ2δðxx′Þδðtt′Þ:
The multiplicative noise ηðx; tÞ is Gaussian, with zero mean and
correlation
〈ηðx; tÞηðx′; t′Þ〉¼ 2cðjxx′jÞδðtt′Þ
with cðjxx′jÞ as the spatial correlation function. A crucial feature of
(15) is that while ηðx; tÞ has zero mean, our new noise term gðϕÞηðx; tÞ
does not. If gðϕÞ is constant, Eq. (15) has only additive noise. In our
case however, noise is coupled to the system through function g.
4.1.3. Effective deterministic model
As mentioned above, the new noise term has nonzero mean.
We define it as follows:
ϵ1=2〈gðϕÞηðx; tÞ〉≡Ψ ðϕÞ: ð16Þ
Adding and subtracting Ψ ðϕÞ in (15) lead to an equivalent equation,
but with zero mean noise term R
∂ϕ
∂t
¼D∇2ϕþ f ðϕÞ þ Ψ ðϕÞ þ Rðϕ; x; tÞ ð17Þ
where
Rðϕ; x; tÞ ¼ ϵ1=2gðϕÞηðx; tÞΨ ðϕÞ þ ξðx; tÞ: ð18Þ
Rðϕ; x; tÞ is related to the nonsystematic noise effect. This effect will
0 1 2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
L
u m
Bifurcation Diagram
Fig. 3. Bifurcation diagram obtained from Φχ ðum; LÞ ¼ 0 with Φχ ðum; LÞ given by
Eq. (14) with χ ¼ 0:0 (blue, thick line), χ ¼ 0:1 (red, dotted line), χ ¼ 0:01 (black,
dashed line), μ¼ 1, α¼ 8. (For interpretation of the references to color in this figure
caption, the reader is referred to the web version of this paper.)
3 Naturally, the name bifurcation diagram only makes sense in the absence of
plasticity. This is due to the critical nature of Lc, which exists only for χ ¼ 0. When
χ≠0, the equivalent in figures will be called umðLÞ curves.
R.V. dos Santos, L.M. da Silva / Journal of Theoretical Biology 335 (2013) 79–87 83
be neglected (Santos and Sancho, 2001), thus leading to an
effective deterministic equation
∂ϕ
∂t
¼D∇2ϕþ f ðϕÞ þ Ψ ðϕÞ ð19Þ
with an effective reaction term f ðϕÞ þ Ψ ðϕÞ. Ψ ðϕÞ can be calculated
using Novikov's (1965) theorem producing
ϵ1=2〈gðϕÞηðx; tÞ〉¼ ϵcð0Þ〈g′ðϕÞgðϕÞ〉: ð20Þ
The deterministic effective model is written as
∂ϕ
∂t
¼D∇2ϕþ f ðϕÞ þ ϵcð0Þg′ðϕÞgðϕÞ: ð21Þ
4.1.4. Application to our model
Considering the inclusion of a Gaussian noise of intensity s in
parameter α,4 (by the transformation α-αþ sWðtÞ, W(t) is a
Wiener process Oksendal, 2003) we get a model corresponding
to Eq. (15) with f ðϕÞ ¼ χ þ ðαμÞϕαϕ2 (a polynomial of degree
two) and gðϕÞ ¼ ϕð1ϕÞ. Therefore, the effective deterministic
model is given by
∂ϕ
∂t
¼D∇2ϕþ χ þ ðαμþ sÞϕðαþ 3sÞϕ2 þ 2sϕ3 ð22Þ
or
∂ϕ
∂t
¼D∇2ϕþ χ þMϕNϕ2 þ Pϕ3 ð23Þ
with M≡αμþ s, N≡αþ 3s, P≡2s and s≡ϵcð0Þ. Besides the renor-
malization in parameters μ and α, the degree of the polynomial
function to the right of (22) is lifted from two to three. The
systematic contributions of the noise in the proliferation rate give
rise to a cubic term in the effective reaction function. Conse-
quently, the validity range of Eq. (23) is restricted to sufficiently
small noise strengths or small densities. This does not affect our
conclusions, since we are interested in the conditions for extinc-
tions, i.e., situations where the density is indeed small. At higher
densities, the model needs to be modified to include higher-order
saturation effects.
It is interesting to observe the effect of s on the effective
potential Veff ðϕÞ associated with the model without diffusion,
obtained from Veff ðϕÞ ¼ 
R ½f ðϕÞ þ Ψ ðϕÞ dϕ.
Fig. 4 shows this effect. Curves in blue, black and red (thick,
dashed and dotted, respectively) have s¼ 0;5 and 10, respectively.
An increase in noise decreases the equilibrium population repre-
sented by the minimum of the potential, but this decrease is
accompanied by the possibility of the population falling into the
hole on the right with no minimum. And the larger the noise
intensity, the more likely this is to occur. Cancer stem cells
enjoy noise.
The bifurcation diagram: We can now use Eq. (14) to construct
the bifurcation diagram corresponding to (23). In this case, we put
a0 ¼ χ, a1 ¼M, a2 ¼N and a3 ¼ P with α¼ 8, μ¼ 1.
Fig. 5 shows the bifurcation diagrams for χ ¼ 0 (top of the
figure) and curves um(L) for χ ¼ 0:1 (bottom). We see clearly that
the noise helps cancer stem cells to survive in very small niches.
The minimal patch size Lc ¼ π
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
D=ðαμþ sÞ
p
needed to sustain cell
life is reduced with noise. The price to pay is related to lower
values of its stationary population.
4.2. The effect of diffusion in the stationary probability distribution
In this subsection we estimate the effects induced by diffusion
on the stationary distribution. Let us consider the tumor as a
spatially continuous medium as in the previous section, described
by field variables obeying partial differential equations. We con-
sider the reaction–diffusion equations
∂ϕðx; tÞ
∂t
¼ f ðϕðx; tÞÞ þ D∇2ϕðx; tÞ ð24Þ
where ϕðx; tÞ is a field (scalar or vector) that describes the state of
the system at a spatial location x at time t. A discretization
procedure is commonly used to transform the continuous partial
differential equation to be analyzed into a set of coupled ordinary
differential equations, after approximating the continuous space
by a lattice (García-Ojalvo and Sancho, 1999). In the case of
0.0 0.5 1.0 1.5 2.0
1.0
0.5
0.0
0.5
1.0
x
V e
ff
Fig. 4. Effect of s on Veff with χ ¼ 0, α¼ 8, μ¼ 1. s¼ 0 (blue, thick line), s¼ 5 (black,
dashed line), s¼ 10 (red, dotted line). An increase in the noise intensity decreases
the population equilibrium but facilitates the escape of the cells to a situation
where population growth is uncontrolled. (For interpretation of the references to
color in this figure caption, the reader is referred to the web version of this paper.)
0 1 2 3 4 5 6
L
0.0
0.2
0.4
0.6
0.8
1.0
u m
Bifurcation Diagram
L
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
u m
Bifurcation Diagram
Fig. 5. Bifurcation diagram with χ ¼ 0:0 (top) and χ ¼ 0:1 (bottom), s¼ 0:0001
(blue, thick line), s¼ 5 (red, dotted line), s¼ 10 (black, dashed line), μ¼ 1, α¼ 8.
Noise enables the survival of CSCs in very small niches. (For interpretation of the
references to color in this figure caption, the reader is referred to the web version of
this paper.)
4 We consider the inclusion of noise in this parameter by its essential
importance in the population dynamics, since this parameter is associated with
its nonlinear character.
R.V. dos Santos, L.M. da Silva / Journal of Theoretical Biology 335 (2013) 79–8784
Eq. (24), for example, assuming a regular Cartesian lattice, the
discretization leads to
dϕðtÞ
dt
¼ f ðfϕigÞ þ
D
Δx2
∑
j∈nðiÞ
ðϕjϕiÞ ð25Þ
where the sum term, which runs over the set of nearest neighbors
of i, represents a possible choice for the discrete version of the
Laplace operator and Δx denotes the lattice spacing. The relation
between the discretized field and the real one is ϕiðtÞ ¼ ϕðiΔ; tÞ,
where i¼ ði1; i2;…; idÞ and d is the space dimension.
A lattice will be used so that the state of the system is described
by a set of scalar variables fxig, i¼ 1;…; Ld defined on a d-
dimensional cubic lattice with lattice points i. Suppose that the
dynamics of the variables xi can be described by the following
stochastic differential equation in the Stratonovich sense:
_xi ¼ f ðxiÞ þ gðxiÞξi
D
2d
∑
j∈nðiÞ
ðxixjÞ: ð26Þ
n(i) is the set of the nearest 2d neighbors of site i, and fξiðtÞg are
Gaussian white noises in time and space with zero mean and an
autocorrelation function given by
〈ξiðtÞξjðt′Þ〉¼ s2δijδðtt′Þ
and D is the diffusion coefficient. The functions f ðxiÞ and gðxiÞ are
f ðxiÞ ¼ χμxi þ αxið1xiÞ and gðxiÞ ¼ xið1xiÞ. Following Van den
Broeck et al. (1994), and using a mean-field approximation, the
stationary probability distribution at site i is given by
PstðxÞ ¼ 1Z exp
2
s2
Z x
0
dy
f ðyÞs
2
2
gðyÞg′ðyÞDðyEðyÞÞ
gðyÞ2
2
664
3
775 ð27Þ
where Z is a normalization constant and
EðyÞ ¼ 〈vijvj〉¼
Z
yiPstðyjjyiÞ dyj ð28Þ
represents the steady state conditional average of yj at neighboring
sites j∈nðiÞ, given the value yi at site i. Using the Weiss mean-field
approximation, neglecting the fluctuation in the neighboring sites,
i.e., EðyÞ ¼ 〈x〉, independent of y, and imposing the self-consistent
requirement, we obtain
〈x〉¼
Z 1
1
dx xPstðxÞ ¼ Fð〈x〉Þ: ð29Þ
We are interested in the effect of the diffusion coefficient D in
the stationary probability distribution of the site i. The maxima of
Pst(x) are obtained from f ðyÞðs2=2ÞgðyÞg′ðyÞDðyEðyÞÞ ¼ 0, or
x3 þ x
2ð2α3s2Þ
2s2
þ xð2D2αþ 2μþ s
2Þ
2s2
Dmþ χ
s2
¼ 0 ð30Þ
where we put EðyÞ≡m. We see that D40 raises the coefficient of
the linear term in x and the constant term. For the cubic equation
x3 þ Bx2 þ Cxþ F ¼ 0, the condition for having three real roots is
given by (Kavinoky and Thoo, 2008)
Δ≡
q2
4
þ p
3
27
o0 ð31Þ
with p¼ CB2=3 and q¼ 2B3=27BC=3þ F . For 3s242α ðBo0Þ
and 0omo1, an increase of C and F increases the value of Δ so
that condition (31) is more difficult to achieve. In Fig. 6 (lower
row) we see that an increase in the diffusion constant D has the
effect of hampering the transition from a unimodal to a bimodal
distribution. There is competition between s and D. Fig. 6 (top left)
shows the effect of increased s for D¼0.1 and Fig. 6 (top right)
shows the effect of χ for s¼ 1:5 and D¼1.
We see in Fig. 6 (top left) that for sufficiently large values of s
bistability can occur. This bistable state can lead to the coexistence
of two separate phases in space. In Zhong et al. (2008) the authors
show that this type of bistability can be associated with noninfil-
trative growth of a benign tumor, a case that corresponds to small
noise, as well as an infiltrative type of malignant growth corre-
sponding to intense noise. While increases in s stimulate bistability,
increases in D discourage it, as shown in Fig. 6 (bottom row).
5. Conclusion
We proposed a model to describe population dynamics of CSCs.
Our analysis allows us to address a controversy related to the
frequency of such cells in tumors. Initially, it was thought that
these cells were relatively rare, comprising at most ∼1% of the
0.0 0.2 0.4 0.6 0.8 1.0
x0.0
0.5
1.0
1.5
2.0
2.5
3.0
Pst
1.5
1.0
0.5
0.0 0.2 0.4 0.6 0.8 1.0
x0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.04
0.08
0.15
Pst
0.0 0.2 0.4 0.6 0.8 1.0
x0
1
2
3
4
5
6
D 1.0
D 0.5
D 0.1
Pst
0.0 0.2 0.4 0.6 0.8 1.0
x0.0
0.5
1.0
1.5
2.0
2.5
3.0
D 1.0
D 0.5
D 0.1
Pst
Fig. 6. Top left: effect of noise strength on Pst for parameters α¼ 0:04, μ¼ 0:06, D¼0.1 and χ ¼ 0:04. Top right: effect of χ on Pst for parameters α¼ 0:04, μ¼ 0:06, D¼1 and s¼ 1:5.
Bottom left: effect of D on Pst for parameters α¼ 0:04, μ¼ 0:06, χ ¼ 0:04 and s¼ 0:5. Bottom right: effect of D on Pst for parameters α¼ 0:04, μ¼ 0:06, χ ¼ 0:04 and s¼ 1:5.
R.V. dos Santos, L.M. da Silva / Journal of Theoretical Biology 335 (2013) 79–87 85
cancer cell population. More recent experiments, however, suggest
that the CSC population need not be small.
When considering the spread possibility of CSCs, we estimate
the conditions to support themselves in a niche with hostile
boundary conditions. Without plasticity, there is a threshold of
the niche size below which the population cannot be sustained.
With plasticity, this threshold is lost and cells can survive in niche,
even in small populations. The inclusion of noise in case of no
plasticity decreases the minimum required niche size, conspiring
again in favor of CSCs.
We briefly considered a simplified model with spatial distribu-
tion in a lattice. The possibility of a bimodal stationary probability
distribution was observed. Using mean-field theory, we demon-
strated that diffusion (D) competes with noise (s) in the construc-
tion of this bimodality. We showed that the discrepancy observed
in the frequency of these cells is entirely consistent with the
original hypothesis of the existence of cancer stem cells, as long as
favorable conditions related to the complexity of the microenvir-
onment are met.
Appendix A. Rescale transforms
In this appendix we detail the rescales made throughout the
main text. The first refers to Eq. (6) and the second refers to Eq. (3).
The general model written in terms of the reactions is
C ⇌
k1
k′2=Ω2
C þ C
P ⇌
k3
k′4=Ω4
P þ P
C,
k5
C þ P
C,
k6
P þ P
P,
k7
∅
P,
k8
C ðA:1Þ
Using the law of mass action we have
_C ¼ k1C 1
C
ΩC
 
k6C þ k8P
_P ¼ k3P 1
P
ΩP
 
þ k9Ck10P
8>><
>>:
ðA:2Þ
with k9≡k5 þ 2k6, k10≡k7 þ k8, ΩC≡k1=k2, ΩP≡k3=k4 and k2≡k′2=Ω2,
k4≡k′4=Ω4. Using the rescale C≡ΩCx and P≡ΩPy
_x ¼ k1xð1xÞk6xþ
k8ΩP
ΩC
y
_y ¼ k3yð1yÞ þ
k9ΩC
ΩP
xk10y
8>><
>>:
ðA:3Þ
Using t≡k6t′ and Ω≡ΩP=ΩC
dx
dt′
¼ k1
k6
xð1xÞxþ k8Ω
k6
y
dy
dt′
¼ k3
k6
yð1yÞ þ k9
k6Ω
xk10
k6
y
8>><
>>:
ðA:4Þ
or
x′¼ Axð1xÞxþ By
y′¼ Eyð1yÞ þ FxGy
(
ðA:5Þ
with x′≡dx=dt′, y′≡dy=dt′ and
A≡
k1
k6
B≡
k2k3k8
k1k4k6
E≡
k3
k6
F≡
k1k4k9
k2k3k6
G≡
k10
k6
8>>>>>>>>><
>>>>>>>>>:
ðA:6Þ
A.1. Gradient system
Starting from (A.2) and carrying out the transformation S¼ s1c,
P ¼ s2p and t ¼ s3τ, we can write
dc
dτ
¼ k1s3c 1
s1
ΩC
c
 
k6s3c þ
k2s2s3
s1
p
dp
dτ
¼ k3s3p 1
s2
ΩP
p
 
þ k9s1s3
s2
ck10s3p
8>><
>>:
ðA:7Þ
Imposing k2s2s3=s1 ¼ k9s1s3=s2, k6s3 ¼ 1 and s1 ¼ΩC , we obtain
s1≡k1=k2, s2≡ΩC
ffiffiffiffiffiffiffiffiffiffiffiffi
k9=k2
p
and s3 ¼ 1=k6.
In this way we obtain
dc
dτ
¼ k1
k6
cð1cÞc þ
ffiffiffiffiffiffiffiffiffi
k2k9
p
k6
p
dp
dτ
¼ k3
k6
p 1ΩC
ΩP
ffiffiffiffiffi
k9
k2
s
p
 !
þ
ffiffiffiffiffiffiffiffiffiffi
k2k9
p
k6
ck10
k6
p
8>>><
>>>:
ðA:8Þ
References
Arens, N., West, I., 2008. Paleobiology 34, 456–471.
Arias, A.M., Hayward, P., 2006. Nat. Rev. Genet. 7, 34–44.
Armitage, P., Doll, R., 1954. Br. J. Cancer 8, 1–12.
Baker, M., 2008a. Nature 456, 553.
Baker, M., 2008b. Nature Reports Stem Cells.
Berglund, N., Gentz, B., 2006. Noise-Induced Phenomena in Slow–Fast Dynamical
Systems: A Sample-Paths Approach, Probability and its Applications. Springer.
Boiko, A., Razorenova, O., van de Rijn, M., Swetter, S., Johnson, D., Ly, D., Butler, P.,
Yang, G., Joshua, B., Kaplan, M., et al., 2010. Nature 466, 133–137.
Bomken, S., Fišer, K., Heidenreich, O., Vormoor, J., 2010. Br. J. Cancer 103, 439–445.
Byrne, H.M., Alarcon, T., Owen, M.R., Webb, S.D., Maini, P.K., 2006. Philos. Trans. Ser
A: Math. Phys. Eng. Sci. 364, 1563–1578.
Cantrell, R., Cosner, C., 2003. Spatial Ecology via Reaction–Diffusion Equations,
Wiley Series in Mathematical and Computational Biology. John Wiley & Sons.
Chambers, I., Silva, J., Colby, D., Nichols, J., Nijmeijer, B., Robertson, M., Vrana, J.,
Jones, K., Grotewold, L., Smith, A., 2007. Nature 450, 1230–1234.
Chen, Y.Q., Jewell, N.P., Lei, X., Cheng, S.C., 2005. Biometrics 61, 170–178.
Clarke, M.F., Fuller, M., 2006. Cell 124, 1111–1115.
Dalerba, P., Cho, R.W., Clarke, M.F., 2007. Annu. Rev. Med. 58, 267–284.
Denison, T.A., Bae, Y.H., 2012. J. Control. Release, http://dx.doi.org/10.1016/j.jcon-
rel.2012.04.014, in press.
El-Samad, H., Khammash, M., 2006. Biophys. J. 90, 3749.
Fange, D., Elf, J., 2006. PLoS Comput. Biol. 2, e80.
Fisher, R., 1937. Ann. Hum. Genet. 7, 355–369.
García-Ojalvo, J., Sancho, J., 1999. Noise in Spatially Extended Systems: With 120
Illustrations, Institute for Nonlinear Science Series. Springer-Verlag.
Gardiner, C., 2009. Stochastic Methods: A Handbook for the Natural and Social
Sciences, Springer Series in Synergetics. Springer.
Grunewald, T., Herbst, S., Heinze, J., Burdach, S., 2011. J. Transl. Med. 9, 79.
Gupta, P., Chaffer, C., Weinberg, R., 2009. Nat. Med. 15, 1010–1012.
Hirsch, M., Smale, S., Devaney, R., 2004. Differential Equations, Dynamical Systems,
and an Introduction to Chaos, Pure and Applied Mathematics. Academic Press.
Hoffmann, M., Chang, H.H., Huang, S., Ingber, D.E., Loeffler, M., Galle, J., 2008. PLoS
One 3, e2922.
Horov, I., Pospsil, Z., Zelinka, J., 2009. J. Theor. Biol. 258, 437–443.
Ishizawa, K., Rasheed, Z., Karisch, R., Wang, Q., Kowalski, J., Susky, E., Pereira, K.,
Karamboulas, C., Moghal, N., Rajeshkumar, N., et al., 2010. Cell Stem Cell 7,
279–282.
Iwasaki, H., Suda, T., 2009. Cancer Sci. 100, 1166–1172.
R.V. dos Santos, L.M. da Silva / Journal of Theoretical Biology 335 (2013) 79–8786
Johnston, M., Maini, P., Jonathan Chapman, S., Edwards, C., Bodmer, W., 2010.
J. Theor. Biol. 266, 708–711.
Kalmar, T., Lim, C., Hayward, P., Muñoz-Descalzo, S., Nichols, J., Garcia-Ojalvo,
J., Arias, A.M., 2009. PLoS Biol. 7, e1000149.
Kareva, I., 2011. Transl. Oncol. 4, 266.
Karlin, S., Taylor, H., 2000. A Second Course in Stochastic Processes. Academic Press.
Kavinoky, R., Thoo, J., 2008. AMATYC Rev. 29, 3–8.
Kelly, P., Dakic, A., Adams, J., Nutt, S., Strasser, A., 2007. Science 317, 337.
Kierstead, H., Slobodkin, L.B., 1953. J. Mar. Res. 12, 141.
Lander, A., Kimble, J., Clevers, H., Fuchs, E., Montarras, D., Buckingham, M., Calof, A.,
Trumpp, A., Oskarsson, T., 2012. BMC Biol. 10, 19.
Leder, K., Holland, E., Michor, F., 2010. PloS One 5, e14366.
Marusyk, A., Polyak, K., 2010. Biochim. Biophys. Acta (BBA): Rev. Cancer 1805,
105–117.
Marusyk, A., Almendro, V., Polyak, K., 2012. Nat. Rev. Cancer, http://dx.doi.org/10.
1038/nrc3261, in press.
McElwain, S., Araujo, R., 2004. Bull. Math. Biol. 66, 1039–1091. (for more informa-
tion, please refer to the journal's website (see hypertext link) or contact the
author).
Méndez, V., Campos, D., et al., 2008. Phys. Rev. Ser. E 77, 22901.
Morrison, S., Kimble, J., 2006. Nature 441, 1068–1074.
Nordling, C.O., 1953. Br. J. Cancer 7, 68–72.
Novikov, E., 1965. Sov. Phys.—JETP 20, 1290–1294.
Oksendal, B., 2003. Stochastic Differential Equations: An Introduction with Applica-
tions, Universitext (1979). Springer.
Perko, L., 2000. Differential Equations and Dynamical Systems, Texts in Applied
Mathematics. Springer.
Petrovskii, S., Li, B., 2005. Exactly Solvable Models of Biological Invasion, Mathe-
matical Biology and Medicine. Taylor & Francis.
Pienta, K., McGregor, N., Axelrod, R., Axelrod, D., 2008. Transl. Oncol. 1, 158.
Quintana, E., Shackleton, M., Sabel, M., Fullen, D., Johnson, T., Morrison, S., 2008.
Nature 456, 593–598.
Rapp, U., Ceteci, F., Schreck, R., et al., 2008. Cell Cycle—Landes Biosci. 7, 45.
Reya, T., Morrison, S.J., Clarke, M.F., Weissman, I.L., 2001. Nature 414, 105–111.
Samoilov, M., Plyasunov, S., Arkin, A.P., 2005. Proc. Nat. Acad. Sci. USA 102,
2310–2315.
Santos, M., Sancho, J., 2001. Phys. Rev. E 64, 016129.
Sarry, J., Murphy, K., Perry, R., Sanchez, P., Secreto, A., Keefer, C., Swider, C.,
Strzelecki, A., Cavelier, C., Récher, C., et al., 2011. J. Clin. Invest. 121, 384.
Schatton, T., Murphy, G., Frank, N., Yamaura, K., Waaga-Gasser, A., Gasser, M., Zhan, Q.,
Jordan, S., Duncan, L., Weishaupt, C., et al., 2008. Nature 451, 345–349.
Schuss, Z., 2010. Theory and Applications of Stochastic Processes: An Analytical
Approach, Applied Mathematical Sciences. Springer.
Shackleton, M., Quintana, E., Fearon, E., Morrison, S., 2009. Cell 138, 822–829.
Skellam, J., 1951. Biometrika 38, 196–218.
Stewart, J., Shaw, P., Gedye, C., Bernardini, M., Neel, B., Ailles, L., 2011. Proc. Natl.
Acad. Sci. 108, 6468.
Tian, T., Olson, S., Whitacre, J., Harding, A., 2011. Integr. Biol. 3, 17–30.
Tomasetti, C., Levy, D., 2010. Proc. Natl. Acad. Sci. 107, 16766–16771.
Turner, C., Stinchcombe, A.R., Kohandel, M., Singh, S., Sivaloganathan, S., 2009. Cell
Prolif. 42, 529–540.
Van den Broeck, C., Parrondo, J., Toral, R., 1994. Phys. Rev. Lett. 73, 3395–3398.
Vargaftig, J., Taussig, D., Griessinger, E., Anjos-Afonso, F., Lister, T., Cavenagh, J.,
Oakervee, H., Gribben, J., Bonnet, D., 2011. Leukemia, http://dx.doi.org/10.1038/
leu.2011.250, in press.
Vermeulen, L., Sprick, M.R., Kemper, K., Stassi, G., Medema, J.P., 2008. Cell Death
Differ. aop, http://dx.doi.org/10.1038/cdd.2008.20, in press.
Welte, H.R.L., Yvonne, Adjaye, James, Regenbrecht, C., 2010. Cell Commun. Signal. 8, 6.
Williams, R., Den Besten, W., Sherr, C., 2007. Genes Dev. 21, 2283.
Zapperi, S., La Porta, C.A.M., 2012. Scientific Reports 2.
Zhong, W., Shao, Y., Li, L., Wang, F., He, Z., 2008. Europhys. Lett. 82, 20003.
Zhong, Y., Guan, K., Zhou, C., Ma, W., Wang, D., Zhang, Y., Zhang, S., 2010. Cancer
Lett. 292, 17–23.
Zwaka, T.P., 2006. Cell 127, 1301–1302.
R.V. dos Santos, L.M. da Silva / Journal of Theoretical Biology 335 (2013) 79–87 87
Capı´tulo 7
Discreteza induzindo coexisteˆncia
Esta introduc¸a˜o e´ baseada nos resultados do artigo [6], publicado em Physica A.
Um princı´pio fundamental da ecologia teo´rica e´ o “princı´pio da exclusa˜o competitiva” (PEC) [59]. De
acordo com este princı´pio, duas espe´cies semelhantes que competem por um recurso limitado na˜o podem
coexistir. Uma das espe´cies sera´ necessariamente levada a` extinc¸a˜o. Este princı´pio e´ apoiado por muitos
modelos matema´ticos, o mais famoso dos quais e´ o modelo de equac¸o˜es diferenciais de Lotka e Volterra
para duas espe´cies concorrentes. No entanto, o princı´pio da exclusa˜o competitiva na˜o parece ocorrer na
natureza, onde a alta biodiversidade e´ comumente observada. Esta contradic¸a˜o e´ conhecida como o para-
doxo da biodiversidade. Por causa dos longos debates envolvidos, este paradoxo e´ um problema central
da ecologia teo´rica. Va´rias explicac¸o˜es diferentes do paradoxo da biodiversidade foram apresentados, no
entanto, os debates a respeito continuam.
Neste artigo [6] apresento um modelo onde a coexisteˆncia de duas espe´cies competitivas e´ induzida pela
discreteza das interac¸o˜es. Este resultado se contrapo˜e ao princı´pio da exclusa˜o competitiva. Mapeamos
o conjunto de interac¸o˜es entre as espe´cies formulado em termos de reac¸o˜es, que e´ a descric¸a˜o elementar
do modelo, em um sistema de equac¸o˜es diferenciais parciais estoca´sticas que foram resolvidas numeri-
camente. Estas equac¸o˜es incorporam a estocasticidade intrı´nseca (ruı´do demogra´fico) da dinaˆmica em
uma aproximac¸a˜o contı´nua obtida da representac¸a˜o de Poisson [60]. A influeˆncia de um ruı´do ambiental,
entendindo como sendo associado a`s variac¸o˜es do ambiente, tambe´m e´ investigada e seus efeitos sa˜o com-
parados aos efeitos do ruı´do demogra´fico. Tambe´m foram estudados os efeitos das constantes de difusa˜o
na dinaˆmica populacional. As principais concluso˜es foram:
1. Ruı´do ambiental favorece as populac¸o˜es densas, a`s custas das menos densas, ratificando o PEC.
2. Ruı´do demogra´fico, por outro lado, favorece as populac¸o˜es menos densas em detrimento das mais
densas, contrariando o PEC.
3. As espe´cies mais lentas sempre sofrem os efeitos mais delete´rios das flutuac¸o˜es estatı´sticas em um
meio homogeˆneo.
Estes resultados, em particular o segundo, nos levam inevitavelmente a associar a alta diversidade obser-
vada na natureza com a discreteza das interac¸o˜es oriunda da finitude das populac¸o˜es.
111
Physica A 392 (2013) 5888–5897
Contents lists available at ScienceDirect
Physica A
journal homepage: www.elsevier.com/locate/physa
Discreteness inducing coexistence
Renato Vieira dos Santos ∗
Departamento de Física, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, CP 702, CEP 30161-970, Belo Horizonte,
Minas Gerais, Brazil
h i g h l i g h t s
• We proposed a simple model that seems to violate the competitive exclusion principle.
• Demographic noise favors less dense populations at the expanse of the denser ones.
• The discontinuous character of interactions can induce coexistence.
• The slower species suffers deleterious effects of noise in a homogeneous medium.
a r t i c l e i n f o
Article history:
Received 15 May 2013
Received in revised form 28 June 2013
Available online 5 August 2013
Keywords:
Additive noise and demographic
stochasticity
Diffusion
Competition model
Monte Carlo simulation
a b s t r a c t
Consider two species that diffuse through space. Consider further that they differ only in
initial densities and, possibly, in diffusion constants. Otherwise they are identical. What
happens if they competewith each other in the same environment?What is the influence of
the discrete nature of the interactions on the final destination? And what are the influence
of diffusion and additive fluctuations corresponding to randommigration and immigration
of individuals? This paper aims to answer these questions for a particular competition
model that incorporates intra and interspecific competition between the species. Based
on mean field theory, the model has a stationary state dependent on the initial density
conditions. We investigate how this initial density dependence is affected by the presence
of demographic multiplicative noise and additive noise in space and time. There are three
main conclusions: (1) Additive noise favors denser populations at the expense of the less
dense, ratifying the competitive exclusion principle. (2) Demographic noise, on the other
hand, favors less dense populations at the expense of the denser ones, inducing equal
densities at the quasi-stationary state, violating the aforementioned principle. (3) The
slower species always suffers the more deleterious effects of statistical fluctuations in a
homogeneous medium.
© 2013 Elsevier B.V. All rights reserved.
1. Introduction
The competitive exclusion principle is one of the most fundamental principles in population biology [1]. Two species
competing for a resource cannot coexist and one of the species must disappear. This principle is supported by many
mathematical models, of which the Lotka–Volterra model for two competing species is the most famous. This principle
has been verified in some experiments [2]. Nevertheless, discussions about the veracity of the principle persist due to the
unequivocal presence of coexisting species in many ecosystems. In this paper we propose a simple toy model to study the
competition between two species from the stochastic point of view. We are particularly interested in the effects of additive
and demographic multiplicative noise in population dynamics.
∗ Tel.: +55 513133737186.
E-mail address: econofisico@gmail.com.
0378-4371/$ – see front matter© 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.physa.2013.07.058
R.V. dos Santos / Physica A 392 (2013) 5888–5897 5889
The dynamics of stochastic systems with agents who interact via reaction rules or have their dynamics dictated by
transition rate functions can be accurately modeled by master equations.1 Such equations are widely used in various fields
of science, from physics, chemistry [7–9] and applied mathematics [5], to biology [3,10], ecology and epidemiology [11,12],
to the social and economic sciences [13].
Stochastic models have been developed for the analysis of two distinct types of noise, internal and external [7,8,14]. In
the field of quantitative ecology, such designations are commonly referred to as demographic and environmental noises,
respectively [15]. External or environmental noises are the fluctuations created in an otherwise deterministic system by
the application of an external random force, whose stochastic properties are supposed to be known a priori. This type of
noise is modeled primarily by stochastic differential equations (SDE) [16,17] or Langevin equations [18,7,8]. Internal or
demographic noise is modeled by master equations and caused by the fact that the system itself consist of discrete particles
and it is inherent of the very mechanism through which the process evolves. Discrete systems of interacting particles often
exhibit notable internal fluctuations [19].
Demographic stochasticity is known to be important in population dynamics [20]. The inclusion of stochasticity into non-
linear mathematical models affects the mean dynamics. We can cite as example the ability of demographic stochasticity
to excite macroscopic-scale coherent oscillations, known as quasi-cycles [21]. An extension to the spatial case where
spatiotemporal patterns are induced by demographic noise was also observed [22]. Generation [23] and exacerbation [24]
of the Allee effect are also consequences of the discrete character of the interactions associated with demographic noise,
which can even induce survival when mean field theory indicates extinction [19]. The importance of this intrinsic noise in
the microscopic dynamics of cellular systems has also been studied intensively in recent years [25], causing the appearance
of a flurry of papers in this area [26]. These facts indicate a rather large effervescence in research associated with stochastic
phenomena related to the inevitable discrete character of the interactions.
In this paper we study a model that in mean field theory exhibits dependence on initial conditions in steady states for
two populations of species A and B in competition. We start by studying the model considering that the species are well
mixed. Later we consider the possibility that the species may spread in a 1-dimensional space. First, we investigate the
effects of the presence of additive environmental2 noise in the relevant ordinary and partial differential equations systems.
In a second step, we will investigate the effects of demographic intrinsic noise arising from discrete interactions between
agents, as described by the master equation. The study is based on numerical techniques to obtain solutions of relevant
stochastic differential equations, obtained as an approximation of the stochastic process as described by themaster equation.
All simulations were done using the software XMDS2 [27].
This paper is organized as follows: In Section 2 we describe the model. Approximation methods for master equations
via diffusion processes are briefly discussed. Section 3 shows the results for the 0-dimensional case. Results for the
1-dimensional case are shown in Section 4. Diffusion effects are discussed in Sections 5 and 6 closes the paper with further
discussion and conclusions. In the Appendix we present the code for one of the XMDS2 simulations.
2. Model
The model proposed may be described by the following sequence of reactions:
A
α
⇀ A+ A B β⇀ B+ B
A+ A α
′
⇀ A B+ B β
′
⇀ B
A+ B ζ⇀ B A+ B ξ⇀ A.
(1)
The first couple of reactions in the first row refer to the birth processes of both species. The second pair describes intraspecific
competition and lastly we have interspecific competition. α (β), α′ (β ′) and ζ (ξ ) are parameters associated with these
reactions to species A (B) in that order.
The values used for all parameters are set to 1. Only for this parameter value is the steady state in mean-field theory
dependent on the initial values of the A and B populations.Wewonderedwhat the effects of including noise in this particular
case are. Also, this hypothesis is not as restrictive as it sounds: if α and β (with α = β) are small enough for us to speak of
criticality and the renormalization group methods can be used, one can show that if ζ = ξ and the dimension d ≠ 0 of the
space is smaller than the critical dimension dc(d < dc = 4), consecutive iterations of the renormalization group flow will
naturally lead to the condition α′ = β ′ ≡ u∗ ∝ 2ϵ/3, with ϵ ≡ 4− d [28]. In short, we have that the species should be very
similar: they have the same low rate of reproduction (α = β & 0) and equal interspecific competition rates (ξ = ζ ), and
live on the line, in the surface, or in space.
1 Also known as chemical master equations in the field of systems biology [3,4] as well as Kolmogorov forward equations in the mathematical literature
[5,6].
2 As the additive noise can be interpreted as arising from random fluctuations in immigration and emigration of individuals, and interpreting these
processes as external to the main system, for simplicity we use the term environmental noise to designate such fluctuations.
5890 R.V. dos Santos / Physica A 392 (2013) 5888–5897
2.1. From master equation to Itô equation
To evaluate the solution of the master equation, a number of approximation techniques have been proposed in the
literature, such as the diffusion approximation by a Fokker–Planck equation [8], the system-size expansion method of
van Kampen [7], the path integral field theory from Martin–Siggia–Rose–Janssen–de Dominicis [29–31], the Doi–Peliti
field theory [32–34] and the Poisson representation [35,8].3 The latter is particularly simple and will be used here. This
method consists of obtaining a Fokker–Planck equation starting directly from reactions that describe the model. From this
Fokker–Planck equation we can infer the associated Langevin equation or stochastic differential equation. Further details in
Ref. [8].
3. 0-dimensional case
First, let us look at the simplest possible case where species are considered well mixed, which is equivalent to studying
the 0-dimensional model.4 We are interested in the case where all parameters in the reactions (1) are equal to 1. Under
these conditions, we have
φ˙ = φ − φ2 − φψ
ψ˙ = ψ − ψ2 − ψφ, (2)
where the dot denotes a time derivative. Eq. (2) can be solved analytically. By settingΦ ≡ φ+ψ andΨ ≡ φ−ψ, they can
be written as
Φ˙ = Φ − Φ2
Ψ˙ = Ψ − ΨΦ. (3)
SolvingΦ equation and then solving for Ψ , we get
Φ(t) = e
t (Φ0 + Ψ0)
1+ (et − 1)Φ0 + (et − 1)Ψ0 (4)
and
Ψ (t) = e
t (Φ0 − Ψ0)
1+ (et − 1)Φ0 + (et − 1)Ψ0 (5)
where we useΦ(0) = Φ0 + Ψ0 and Ψ (0) = Φ0 − Ψ0. In the original variables:
φ(t) = e
tφ0
1− φ0 − ψ0 + et (φ0 + ψ0) (6)
and
ψ(t) = e
tψ0
1− φ0 − ψ0 + et (φ0 + ψ0) (7)
with initial conditions φ(0) = φ0 and ψ(0) = ψ0. For t →∞we obtain the stationary states
lim
t→∞φ(t) =
φ0
φ0 + ψ0
lim
t→∞ψ(t) =
ψ0
φ0 + ψ0
(8)
with limt→∞[φ(t)+ ψ(t)] = 1. Therefore the steady states depend on the initial composition of the system with constant
sum= 1.
3.1. Noise in 0-dimensional model
When Eq. (2) are subjected to the action of environmental additive noise, we get:
φ˙ = φ − φ2 − φψ + ση1
ψ˙ = ψ − ψ2 − ψφ + ση2, (9)
where η1 and η2 arewhite noise stochastic processwith statistics ⟨η1(t)⟩ = ⟨η2(t)⟩ = 0 and ⟨η1(t)η1(t ′)⟩ = ⟨η2(t)η2(t ′)⟩ =
δ(t − t ′).
3 It was later shown that these last two methods are equivalent [36].
4 We can also imagine that species interact in the complete graph, in the way that ‘‘everyone interacts with everyone’’ on the lattice.
R.V. dos Santos / Physica A 392 (2013) 5888–5897 5891
(a) Additive noise 0-dimensional model. Average of
1× 103 Monte Carlo realizations with σ = 0.1. Black
dashed lines represent the average population, shown
with its red dotted lines representing 1 standard
deviation from the mean.
(b) Demographic noise 0-dimensional model. Average of
1× 103 Monte Carlo realizations with σ = 0.07. Black
dashed lines represent the average population, shown
with its red dotted lines representing 3 standard
deviations from the mean.
Fig. 1. 0-dimensional SDE numerical simulations. The superior black curve represents mean ⟨φ⟩ population and the inferior black curve represents ⟨ψ⟩
population. Superior green dashed line is the sum ⟨φ⟩ + ⟨ψ⟩ and inferior blue dashed line is the difference ⟨φ⟩ − ⟨ψ⟩. (For interpretation of the references
to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 1 shows the results of simulations for the initial conditions φ(0) = 0.6 and ψ(0) = 0.4. The upper dashed green
line indicates the population sum, always constant and equal to 1. The lower blue dashed line indicates the difference in
densities. We see that environmental fluctuations tend to quickly eliminate the scarcer species. Environmental fluctuations
induce deleterious effects on less numerous populations favoring the more abundant species. We see from the red dotted
lines, which represent 1 standard deviation from the mean, that fluctuations increase very fast, which inevitably leads to
the extinction of both species. How fast this happens, will naturally depend on the noise intensity. The system reaches the
absorbing state and talk about averages no longer makes sense.
Demographic noise is considered in the following equations, interpreted in the Itô sense [8]5:
φ˙ = φ − φ2 − φψ + σx(1− x)η1
ψ˙ = ψ − ψ2 − ψφ + σy(1− y)η2. (10)
η1 and η2 have the same statistical properties as before. When we observe Fig. 1(b) with the simulation results, we can
draw identical conclusions as in the case of the additive noise. Note that the dotted red lines indicate bands of 3 standard
deviations from themean. The final conclusion of this section is that both environmental and demographic noise benefit the
larger population when in competition with a less numerous population.
3.2. Comparison with the phenomenon of genetic drift
Let us make a comparison between the results obtained above with the phenomenon of genetic drift [37]. Wewill briefly
discuss the phenomenon of natural selection and genetic drift.
The core difference between Natural Selection (NS) and Genetic (or allelic) Drift (GD) is the cause. Both are methods of
genetic change in a population. However, one happens randomly (GD) while the other is a direct response to and environ-
mental challenge (NS). Some examples:
Natural Selection is the mechanism by which a species changes in response to an environmental challenge. Imagine a
population of brown rabbits in a field. They are breeding and being eaten occasionally by foxes until over time. Suppose
that suddenly the environment changes so that the field is covered in snow. Now, the brown rabbits are highlighted and the
foxes have an easy time of hunting them. Consequently, the number of brown rabbits decreases dramatically and they are
threatened with extinction. White rabbits which used to be caught and eaten quickly before the snow came, are nowmuch
better adapted. As such, they are more likely to survive, breed and pass on their white genetic make up and hence more
white rabbits are born—they are naturally selected by the snow and the foxes, which constitute its environment.
GD is also a change in genetic make up of a population. However it is not stimulated by the environment. Imagine our
population of rabbits again. 50% of them have blue eyes and 50% have green. The eye color makes little difference to their
survival chances and is just a natural variation. A newborn rabbit will statistically have a 50% chance of blue eyes and 50%
chance of green eyes.
In a big population, the proportion of blue to green is likely to stay at or around 50%. However, that is not the case in a
small population. Imagine there are now only 20 rabbits: 10 with blue eyes, 10 with green. Purely by chance, some of these
rabbits will not breed, or some breed more often. Let us say – by chance – one green-eyed rabbit gets run over and does
not breed. There are 10 blues and 9 greens. That means that there are now 53% blues and 47% greens. These proportions
5 A factor
√
2 was absorbed in the constant σ .
5892 R.V. dos Santos / Physica A 392 (2013) 5888–5897
will now have a greater impact on the subsequent generation since there are more blues, there will be a greater chance
of blues appearing in the next generation and less chance of greens. This phenomenon is analogous to the phenomenon of
environmental noise favoring more dense species and disfavoring the less dense shown above.
4. 1-dimensional case
4.1. Environmental noise
In this subsectionwe showsimulations results for the 1-dimensional casewith environmental noise.We are considering a
1-dimensional latticewith 128 sites uniformly distributed in the interval (−1, 1). Fig. 2 shows how the population dynamics
can change with variations in noise intensity. The stochastic partial differential equations (SPDE) for this case are:
∂φ
∂t
= Dφ∇2φ + φ − φ2 − φψ + η1, (11)
∂ψ
∂t
= Dψ∇2ψ + gψ − ψ2 − ψφ + η2 (12)
with the following noise properties: ⟨η1(x, t)⟩ = ⟨η2(x, t)⟩ = 0 and
⟨η1(x, t)η1(x′, t ′)⟩ = σ 2δd(x− x′)δ(t − t ′) (13)
⟨η2(x, t)η2(x′, t ′)⟩ = σ 2δd(x− x′)δ(t − t ′). (14)
The diffusion constants are Dφ and Dψ . From now on until the penultimate section we will consider Dφ = Dψ = 1.
In Fig. 2(a) we have as populations change when the noise intensity is low. We see that, within statistical fluctuations,
the densities remain invariant in time. Fig. 2(b) shows the case of larger noise intensities.We see that the smaller population
is more affected, while the larger population is once more benefited.
4.2. Demographic noise
The SPDE involving demographic noise interpreted in Itô sense are given by [8]
∂φ
∂t
= ∇2φ + φ − φ2 − φψ + η1, (15)
∂ψ
∂t
= ∇2ψ + gψ − ψ2 − ψφ + η2 (16)
with the following noise properties: ⟨η1(x, t)⟩ = ⟨η2(x, t)⟩ = 0 and
⟨η1(x, t)η1(x′, t ′)⟩ = σ 2φ(x, t)(1− φ(x, t))δd(x− x′)δ(t − t ′) (17)
⟨η2(x, t)η2(x′, t ′)⟩ = σ 2ψ(x, t)(1− ψ(x, t))δd(x− x′)δ(t − t ′), (18)
where a factor 2 has been incorporated into the constant σ 2.
Fig. 3 shows the simulation results in one dimension using periodic boundary conditions for the case involving
environmental and demographic noise. In both figures the same initial conditions were used, given by φ(x, 0) = 0.9 and
ψ(x, 0) = 0.1.We see that the effect of demographic noise is very different from environmental noise. The discrete nature of
the interactions that induces demographic noise has the effect of equalizing population densities. This fact is in full contrast
with the principle of competitive exclusion, which roughly states that complete competitors cannot exist [1].
The reason for this violation can be the formation of clusters in which the nonlinear effects associated with intraspecific
competition intensifies. In this case, the initially more massive species would aggregate in denser clusters and therefore tend
to suffer more severe consequences. More detailed numerical simulations in higher dimensionality should be conducted for further
information.
5. Diffusion effects
In this section we show the effects that different values for the diffusion constants have on the dynamics. Dispersion is
an important strategy that allows organisms to locate and exploit favorable habitats. Studies have already been done with
the intention of studying the effects of diffusion in competition models. In Ref. [38] the authors found that species with a
low dispersal rate always drive a competing species to extinction in a deterministic continuousmodel with a heterogeneous
growth rate. A slowly dispersing species exploits favorable environments without wastefully exploring the landscape. We
are talking about the so-called cost of migration.
Hamilton and May [39], on the other hand, considered a stochastic model in which the fast species may be preferred.
In Ref. [40] the authors consider explicitly the discrete character of interactions in an agent-based model and found that
R.V. dos Santos / Physica A 392 (2013) 5888–5897 5893
(a) 1-dimensional case with σ = 0.05.
N = 2× 104, 1× 103 Monte Carlo samples. Black dashed
lines represent the average population, shown with its
red dotted lines (not easily discernible in the scale
displayed) representing 1 standard deviation from the
mean.
(b) 1-dimensional case with σ = 0.15.
N = 2× 104, 1× 103 Monte Carlo samples. Black dashed
lines represent the average population, shown with its
red dotted lines representing 1 standard deviation from
the mean.
Fig. 2. 1-dimensional SPDE numerical simulations with additive noise. Lattice has 128 sites, uniformly distributed in interval (−1, 1). The superior black
curve represents mean ⟨φ⟩ population and the inferior black curve represents ⟨ψ⟩ population. Superior green dashed line is the sum ⟨φ⟩+⟨ψ⟩ and inferior
blue dashed line is the difference ⟨φ⟩ − ⟨ψ⟩. Homogeneous initial conditions are φ(x, 0) = 0.6, ψ(x, 0) = 0.4. Diffusion constants are unitary. (For
interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
(a) Environmental noise 1-dimensional model. Average
of 1× 103 Monte Carlo realizations with σ = 0.1. Black
dashed lines represent the average population, shown
with its red dotted lines representing 1 standard
deviation from the mean.
(b) Demographic noise 1-dimensional model. Average of
1× 103 Monte Carlo realizations with σ = 0.2. Black
dashed lines represent the average population, shown
with its red dotted lines representing 10 standard
deviations from the mean.
Fig. 3. 1-dimensional SPDE numerical simulations. Lattice has 128 sites, uniformly distributed in interval (−1, 1). The superior black curve represents
mean ⟨φ⟩ population and inferior black curve represents ⟨ψ⟩ population. Superior green dashed line is the sum ⟨φ⟩ + ⟨ψ⟩ and inferior blue dashed line
is the difference ⟨φ⟩ − ⟨ψ⟩. Homogeneous initial conditions are φ(x, 0) = 0.9, ψ(x, 0) = 0.1. Diffusion constants are unitary. (For interpretation of the
references to colour in this figure legend, the reader is referred to the web version of this article.)
the faster species could drive the slower one to extinction if stochastic fluctuations were large. In essence they showed
that discreteness favors faster dispersers. In all the models mentioned above the medium in which the species lives is
heterogeneous in birth rate, i.e., it depends on the position. There is a competition between environment heterogeneity and
interaction discreteness. The former tends to favor the slow and the latter the fast species.
Fig. 4 presents simulation results for the case of environmental noise. Fig. 4(a) corresponds to the case with equal values
for diffusion constants (Dφ = Dψ = 1) and, on the right, Fig. 4(b) shows the case with different values (Dφ = 1,Dψ = 0.1).
All other parameterswere held constant.We see that reducing the diffusion constantDψ brings bad consequences for species
B. For the same time interval and noise intensity, it moves more quickly to extinction. The reason for this is simple: fast
dispersers smooth out fluctuations by diffusion, putting slower competitors at a disadvantage. This is due to the non-linear
death rates compared to the linear birth rates in the equations. Suppose that the density of the slow species is given by
ψ(x) = ψ0 + η(x, t) where ⟨η⟩ = 0 and statistical properties (13) and (14) are valid. The total number of births in time T
is independent of the fluctuations,
births =
 T
0
dt
 L
0
ψ(x)dx = LTψ0. (19)
Nevertheless, the total number of deaths is
deaths =
 T
0
dt
 L
0
ψ(x)2dx = LT ψ0 + σ 2 . (20)
5894 R.V. dos Santos / Physica A 392 (2013) 5888–5897
(a) Environmental noise 1-dimensional model. Average
of 100 Monte Carlo realizations with σ = 0.15 and
diffusion coefficients Dφ = 1 and Dψ = 1. Black dashed
lines represent the average population, shown with its
red dotted lines representing 1 standard deviations from
the mean.
(b) Environmental noise 1-dimensional model. Average
of 100 Monte Carlo realizations with σ = 0.15 and
diffusion coefficients Dφ = 1 and Dψ = 0.1. Black dashed
lines represent the average population, shown with its
red dotted lines representing 1 standard deviations from
the mean.
Fig. 4. 1-dimensional SPDEnumerical simulations. Lattice has 128 sites, uniformly distributed in interval (−1, 1). The superior black curve representsmean
⟨φ⟩ population and inferior black curve represents ⟨ψ⟩ population. Inferior blue dashed line is the difference ⟨φ⟩−⟨ψ⟩. Homogeneous initial conditions are
φ(x, 0) = 0.6, ψ(x, 0) = 0.4. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
(a) Demographic noise 1-dimensional model. Average of
10 Monte Carlo realizations with σ = 0.2 and diffusion
coefficients Dφ = 1 and Dψ = 1. Black dashed lines
represent the average population, shown with its red
dotted lines representing 10 standard deviations from
the mean.
(b) Demographic noise 1-dimensional model. Average of
10 Monte Carlo realizations with σ = 0.2 and diffusion
coefficients Dφ = 1 and Dψ = 0.1. Black dashed lines
represent the average population, shown with its red
dotted lines representing 10 standard deviations from
the mean.
Fig. 5. Numerical simulations 1-dimensional SPDE. Lattice has 128 sites, uniformly distributed in interval (−1, 1). The superior black curve represents
mean ⟨φ⟩ population and inferior black curve represents ⟨ψ⟩ population. Inferior blue dashed line is the difference ⟨φ⟩ − ⟨ψ⟩. Homogeneous initial
conditions are φ(x, 0) = 0.6, ψ(x, 0) = 0.4. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version
of this article.)
Fluctuations increase the death rate. As mentioned above, we expect the fluctuations of fast dispersers to be smaller, so the
slow species is driven to extinction.
Fig. 5 shows the results for demographic noise. On the left is the case for equal diffusion constants and different values
for diffusion constants are shown on the right, as before. The deleterious effect for the slow population is now present as
an increase in fluctuations, shown in blue, compared with the fluctuations of the faster species, shown in yellow. This fact
confirms the statement made above that diffusion smoothes out fluctuations. The species also tend to equalize, but the
slower species now has a greater extinction probability.
6. Discussion and conclusion
Many theoretical models predict the competitive exclusion possibility. However, for reasons still poorly understood,
competitive exclusion is not easily seen in nature and many biological systems seem to violate this principle. A well-known
example is the paradox of the plankton [41]: how is it possible for a number of species to coexist in a relatively isotropic or
unstructured environment all competing for the same sorts of resources? The problem is particularly acute because there is
adequate evidence from enrichment experiments that natural waters, at least in the summer, present an environment that is
strikingly nutrient deficient, so that competition is likely to be severe. According to the competitive exclusion principle, only
a small number of plankton species should be able to coexist on these resources. Nevertheless, large numbers of plankton
species coexist within small areas of open sea.
The achieved results (only for d = 1) seems to shed some light on this apparent paradox: The discontinuous character
of interactions can induce coexistence. At least for situations that fit the assumptions of the proposed model. It is worth
R.V. dos Santos / Physica A 392 (2013) 5888–5897 5895
remembering the observation about the hypothesis of many similarities between species. According to the renormalization
group ideas, it is enough that the species have equal birth and interspecific competition rates and live in 0 < d < 4 for the
results to be valid. Lastly, we believe that other less restrictive models in which the discrete character of the interactions are
taken into account can show more vigorously the results presented here.
Acknowledgment
This work is supported by CNPq, Brazil.
Appendix. Code for demographic noise simulation
The stochastic differential equations in XMDS2 must be written in the Stratonovich formulation [27]. All stochastic
equations shown in this article are in the Itô representation. Therefore, we need to transform Itô equations in Stratonovich
equations. To do this wemust add the term− 12g(φ)g(φ)′ = − σ4 (1−2φ)with g(φ) ≡ σ
√
φ(1− φ) in the Itô equations [8].
The XMDS2 code for the demographic noise simulation is:
<?xml version="1.0" encoding="UTF-8"?>
<simulation xmds-version="2">
<name>DIC</name>
<author>Renato Vieira dos Santos</author>
<description>
Discreteness Induced Coexistence
</description>
<geometry>
<propagation_dimension> t </propagation_dimension>
<transverse_dimensions>
<dimension name="x" lattice="128" domain="(-1, 1)" />
</transverse_dimensions>
</geometry>
<driver name="mpi-multi-path" paths="1000" />
<features>
<auto_vectorise />
<validation kind="none"/>
<benchmark />
<error_check />
<bing />
<fftw />
<globals>
<![CDATA[
const real alpha = 1.0;
const real beta = 1.0;
const real sigma = 0.2;
const real g_A = 1.0;
const real g_B = 1.0;
const real h_A = 1.0;
const real h_B = 1.0;
const real D_A = 1.0;
const real D_B = 1.0;
]]>
</globals>
</features>
<noise_vector name="noiseEvolution" dimensions="x" kind="wiener" type="real"
method="dsfmt" seed="314 159 276">
<components>p_1 p_2</components>
</noise_vector>
<vector name="main" initial_basis="x" type="complex">
5896 R.V. dos Santos / Physica A 392 (2013) 5888–5897
<components>phi psi</components>
<initialisation>
<![CDATA[
phi = 0.9 ;
psi = 0.1 ;
]]>
</initialisation>
</vector>
<sequence>
<integrate algorithm="SI" iterations="3" interval="10" steps="100000">
<samples>100</samples>
<operators>
<operator kind="ip" constant="yes">
<operator_names>L</operator_names>
<![CDATA[
L = -kx*kx;
]]>
</operator>
<dependencies>noiseEvolution</dependencies>
<integration_vectors>main</integration_vectors>
<![CDATA[
dphi_dt=L[phi]+(alpha*phi-g_A*pow(phi,2)-h_A*phi*psi
+sigma*pow(g_A*phi*(1-phi),0.5)*p_1-0.25*sigma*g_A*(1-2*phi));
dpsi_dt=L[psi]+(beta*psi-g_B*pow(psi,2)-h_B*phi*psi
+sigma*pow(g_B*psi*(1-psi),0.5)*p_2-0.25*sigma*g_B*(1-2*psi));
]]>
</operators>
</integrate>
</sequence>
<output>
<sampling_group basis="x" initial_sample="yes">
<moments>phiR psiR sdphi sdpsi</moments>
<dependencies>main</dependencies>
<![CDATA[
phiR = Re(phi);
psiR = Re(psi);
sdphi = Re(pow(phi,2));
sdpsi = Re(pow(psi,2));
]]>
</sampling_group>
</output>
</simulation>
References
[1] G. Hardin, et al., Science 131 (1960) 1292–1297.
[2] G. Gause, The Struggle for Existence, Dover Publications, 2003.
[3] M. Ullah, O. Wolkenhauer, Stochastic Approaches for Systems Biology, in: Interdisciplinary Applied Mathematics Series, Springer, 2011.
[4] J. Feng, W. Fu, F. Sun, Frontiers in Computational and Systems Biology, Computational Biology, Springer, 2010.
[5] S. Karlin, H. Taylor, A Second Course in Stochastic Processes, Academic Press, 2000.
[6] W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, John Wiley & Sons, 2008.
[7] N. Kampen, Stochastic Processes in Physics and Chemistry, North-Holland personal library, Elsevier, 2007.
[8] C. Gardiner, Stochastic Methods: a Handbook for the Natural and Social Sciences, in: Springer Series in Synergetics, Springer, 2009.
[9] K. Jacobs, Stochastic Processes for Physicists: Understanding Noisy Systems, Cambridge University Press, 2010.
[10] J. Freund, T. Pöschel, Stochastic Processes in Physics, Chemistry, and Biology, in: Lecture Notes in Physics, Springer, 2000.
[11] C. Mode, C. Sleeman, Stochastic Processes in Epidemiology: HIV/AIDS, Other Infectious Diseases, and Computers, World Scientific, 2000.
[12] N. Bailey, The Elements of Stochastic Processes with Applications to the Natural Sciences, in: Wiley Series in Probability and Mathematical Statistics:
Applied Probability and Statistics, Wiley, 1964.
[13] D. Helbing, Quantitative Sociodynamics: Stochastic Methods and Models of Social Interaction Processes, Springer, 2011.
[14] O. Ovaskainen, B. Meerson, Trends in Ecology & Evolution 25 (2010) 643–652.
[15] D. Gravel, F. Guichard, M. Hochberg, Ecology Letters 14 (2011) 828–839.
[16] Z. Schuss, Theory and Applications of Stochastic Processes: an Analytical Approach, in: Applied Mathematical Sciences, Springer, 2010.
[17] B. Oksendal, Stochastic Differential Equations: an Introduction with Applications, in: Universitext (1979), Springer, 2003.
R.V. dos Santos / Physica A 392 (2013) 5888–5897 5897
[18] H. Risken, The Fokker–Planck Equation: Methods of Solutions and Applications, second ed., in: Springer Series in Synergetics, Springer, 1989, third
printing edition, 1996.
[19] N. Shnerb, Y. Louzoun, E. Bettelheim, S. Solomon, Proceedings of the National Academy of Sciences 97 (2000) 10322.
[20] A. Black, A. McKane, Trends in Ecology & Evolution (2012).
[21] A. McKane, T. Newman, Physical Review Letters 94 (2005) 218102.
[22] T. Butler, N. Goldenfeld, Physical Review E 84 (2011) 011112.
[23] R. Lande, Oikos (1998) 353–358.
[24] B. Dennis, Oikos 96 (2002) 389–401.
[25] J. Paulsson, Nature 427 (2004) 415–418.
[26] M. Thattai, A. Van Oudenaarden, Proceedings of the National Academy of Sciences 98 (2001) 8614–8619.
[27] G. Dennis, J. Hope, M. Johnsson, Computer Physics Communications (2012).
[28] H. Janssen, Journal of Statistical Physics 103 (2001) 801–839.
[29] P.C. Martin, E.D. Siggia, H.A. Rose, Physical Review A 8 (1973) 423–437.
[30] C. de Dominicis, Le Journal de Physique Colloques 37 (1976) 247–253.
[31] H. Janssen, Zeitschrift für Physik B Condensed Matter 23 (1976) 377–380.
[32] A. Kamenev, Keldysh and Doi-Peliti techniques for out-of-equilibrium systems, 2001.
[33] M. Doi, Journal of Physics A (Mathematical and General) 9 (1976) 1479.
[34] L. Peliti, Path Integral Approach to Birth–Death Processes on a Lattice, Technical Report, 1985.
[35] C. Gardiner, S. Chaturvedi, Journal of Statistical Physics 17 (1977) 429–468.
[36] M. Droz, A. McKane, Journal of Physics A (Mathematical and General) 27 (1994) L467.
[37] R. Der, C.L. Epstein, J.B. Plotkin, Theoretical Population Biology 80 (2011) 80–99.
[38] J. Dockery, V. Hutson, K. Mischaikow, M. Pernarowski, Journal of Mathematical Biology 37 (1998) 61–83.
[39] W. Hamilton, R. May, 1977.
[40] D.A. Kessler, L.M. Sander, Physical Review E 80 (2009) 041907.
[41] G. Hutchinson, American Naturalist (1961) 137–145.
122
Capı´tulo 8
Conclusa˜o
E´ com muito orgulho que apresento as concluso˜es associadas ao meu trabalho final de formac¸a˜o acadeˆmica,
conquistado com muita luta e perseveranc¸a, realizado concomitantemente com a carreira docente em todos
os nı´veis educacionais e so´cio-econoˆmicos, nas mais diversas instituic¸o˜es de ensino.
Uma das principais perguntas da minha linha de pesquisa pode ser formulada da seguinte forma:
Que tipo de fenoˆmeno emerge nos processos biolo´gicos em geral quando levamos em conta o cara´ter
discreto das interac¸o˜es e a finitude das populac¸o˜es?
Esta e´ a pergunta relevante e e´ interessante conjecturar que a discreteza das interac¸o˜es nos fenoˆmenos
biolo´gicos de modo geral parece ter um impacto profundo nos processos celulares e ecolo´gicos. Esta
afirmativa tem respaldo no fanta´stico desenvolvimento da fı´sica no inı´cio do se´culo XX com o advento
da mecaˆnica quaˆntica, onde o cara´ter discreto de muitas grandezas fı´sicas tiveram tambe´m um enorme
impacto na nossa sociedade.
Muitas quantidades em todo o mundo fı´sico sa˜o contı´nuas e medidas por nu´meros reais: posic¸a˜o, velo-
cidade, concentrac¸a˜o, peso, etc. Em muitas a´reas da cieˆncia, no entanto, percebeu-se que certos padro˜es
complexos podem ser explicados pela suposic¸a˜o de existeˆncia de nı´veis discretos subjacentes que podem
ser descritos utilizando nu´meros inteiros. Em quı´mica, as diversas leis de composic¸a˜o de elementos, tais
como “proporc¸o˜es definidas” e “mu´ltiplas proporc¸o˜es”, conhecidas por volta de 1800, levaram Dalton a
formular a teoria atomista e dar uma explicac¸a˜o simples e elegante para todas essas leis. Por volta de 1900,
Planck, Einstein, Bohr e outros perceberam que os problemas mais assustadores da (enta˜o) fı´sica moderna,
como o espectro de radiac¸a˜o do corpo negro, a dependeˆncia universal da temperatura do calor especı´fico
de so´lidos, a velocidade dos ele´trons ejetados por so´lidos sob radiac¸a˜o (efeito fotoele´trico), poderiam ser
resolvidos elegantemente supondo que a energia e´ quantizada e varia apenas em unidades de inteiros. Em
biologia, a teoria da evoluc¸a˜o de Darwin era incompatı´vel com a teoria da mistura de heranc¸a, ate´ enta˜o
tida como o´bvia. O trabalho de Mendel, e sua redescoberta por de Vries, Correns e Tschermak por volta
de 1900 (uma curiosa coincideˆncia), restaurou a consisteˆncia matema´tica da evoluc¸a˜o com a introduc¸a˜o do
conceito de genes como o quantum de informac¸a˜o da heranc¸a gene´tica.
Estes sa˜o apenas alguns exemplos onde os padro˜es complexos poderiam ser explicados simplesmente
pela suposic¸a˜o de um nı´vel discreto subjacente. A hipo´tese de descontinuidade e, especialmente, suas
consequeˆncias, foi em cada um destes casos anti-intuitiva. Os organismos vivos, por outro lado, na˜o ne-
cessitam da hipo´tese de descontinuidade. Mas um fato o´bvio sobre eles e´ que os eventos de nascimento e
morte mudam seu nu´mero apenas por quantidades inteiras.
A mensagem que fica com os treˆs exemplos analisados nos artigos e´ que, como em muitas outras a´reas
da cieˆncia, a natureza discreta da vida tem consequeˆncias importantes, que teˆm sido muitas vezes negligen-
ciadas. Espero que estes resultados ajudem a disseminar na comunidade cientı´fica a necessidade de se levar
em considerac¸a˜o a finitude das populac¸o˜es e a consequente natureza discreta das interac¸o˜es em biologia.
123
124
Apeˆndice A
Mapeamento de Doi
A.1. Introduc¸a˜o
Aqui comec¸a uma sequeˆncia de apeˆndices que visam dar subsı´dios detalhados para a construc¸a˜o de uma
teoria de campo equivalente a um dado processo estoca´stico descrito por uma equac¸a˜o mestra. Estamos
falando da teoria de Doi-Peliti. O objetivo nesta sec¸a˜o e da pro´xima e´ descrever reac¸o˜es locais de criac¸a˜o
e aniquilac¸a˜o de partı´culas se difundindo na rede ou em um meio contı´nuo atrave´s de um mapeamento em
uma integral funcional, passando pela representac¸a˜o de segunda quantizac¸a˜o utilizando a teoria de Doi-
Peliti [61, 33]. Alguns exemplos de espe´cie u´nica sa˜o a reac¸a˜o de aniquilac¸a˜o de pares A+A→ /0, onde /0
denota um componente quimicamente inerte, e a coagulac¸a˜o A+A→A. A propagac¸a˜o da partı´cula difusiva
pode ser modelada como um passeio aleato´rio discreto ou contı´nuo no tempo. Uma reac¸a˜o ocorre quando
uma partı´cula se encontra suficientemente pro´xima de outra, onde esta proximidade, caracterizada pelo
alcance da interac¸a˜o entre as partı´culas, deve ser definida a priori. Na rede, pode-se definir que a reac¸a˜o
ocorre quando as partı´culas se encontram no mesmo sı´tio.
A.1.1. Caracterı´sticas ba´sicas dos sistemas de reac¸a˜o-difusa˜o
Um processo muito importante no estudo das transic¸o˜es de fase de na˜o equilı´brio entre estados ativos e
absorventes e´ o processo descrito pelas reac¸o˜es A+A→ A, A→ A+A e A→ /0, que ocorrem em uma
rede regular d−dimensional. Este processo e´ chamado de Percolac¸a˜o Direcionada (PD) e caracteriza
uma ampla classe de Universalidade. Um ponto crı´tico e´ alcanc¸ado pelo ajuste das taxas de reac¸a˜o A→
( /0,2A). Processos mais gene´ricos de duas espe´cies, por exemplo, A+B→ /0 requerem que tipos diferentes
de partı´culas se aproximem para que a reac¸a˜o ocorra. As espe´cies diferentes de partı´culas podem ou
na˜o ter a mesma constante de difusa˜o. Com tal generalidade disponı´vel, e´ possı´vel construir sistemas
que relaxam para o equilı´brio, bem como sistemas guiados. O primeiro caso, que inclui a Percolac¸a˜o
Direcionada, compreende tipicamente tanto as reac¸o˜es que aumentam o nu´mero de partı´culas como as que
diminuem. Dependendo das combinac¸o˜es apropriadas das taxas de reac¸a˜o, a subsequente competic¸a˜o pode,
no limite termodinaˆmico e em tempos longos, resultar em um estado ‘ativo’, caracterizado por um estado
estaciona´rio de densidade finita de partı´culas, bem como uma situac¸a˜o que evolui para uma rede vazia.
Para reac¸o˜es que requerem a presenc¸a de pelo menos uma das partı´culas, o u´ltimo caso constitui um estado
‘inativo’ ou ‘absorvente’ sem flutuac¸o˜es, estado do qual o sistema na˜o podera´ escapar nunca. A transic¸a˜o
contı´nua de um estado ativo para um estado absorvente e´ ana´loga a uma transic¸a˜o de fase de equilı´brio
de segunda ordem, e de modo similar requer o ajuste das taxas de transic¸a˜o como paraˆmetros de controle
para atingir a regia˜o crı´tica. Como no equilı´brio, universalidade das leis de poteˆncia crı´ticas emergem como
uma consequeˆncia da divergeˆncia do comprimento de correlac¸a˜o ξ, o que induz o surgimento de invariaˆncia
de escala e independeˆncia dos paraˆmetros microsco´picos no regime crı´tico. Por outro lado, sistemas tais
125
Apeˆndice A. Mapeamento de Doi
como a reac¸a˜o de aniquilac¸a˜o de pares A+A→ /0 sa˜o sistemas que necessariamente decaem para o estado
absorvente. Nestes casos, a lei de decaimento assinto´tico e´ que e´ de interesse bem como o comportamento
das func¸o˜es de correlac¸a˜o no regime universal que e´ atingido para valores grandes da varia´vel tempo.
A.1.2. Teoria de Doi
Faremos uma revisa˜o bem detalhada do mapeamento de sistemas de reac¸a˜o-difusa˜o em uma teoria de cam-
pos, elaborando uma definic¸a˜o do sistema atrave´s de uma equac¸a˜o mestra, convertendo esta equac¸a˜o mestra
em uma descric¸a˜o em termos de operadores de criac¸a˜o e aniquilac¸a˜o correspondente, com a subsequente
transposic¸a˜o para uma representac¸a˜o em integral funcional. E´ esta representac¸a˜o final em integral funci-
onal que nos permitira´ utilizar todo o arsenal de te´cnicas disponı´veis em uma teoria de campos, inclusive
Grupo de Renormalizac¸a˜o (GR).
Considere um conjunto de sı´tios na rede denotados por i = 1,2,3, · · · onde cada sı´tio e´ ocupado por
n1,n2,n3, · · · partı´culas. Definimos α = {n1,n2,n3, · · ·} como representando um estado particular da rede
bem como P(α, t) a probabilidade de obter o estado α no tempo t. Processos dinaˆmicos tais como saltos
de um sı´tio para outro, decaimentos, reac¸o˜es etc. ira˜o causar uma mudanc¸a do estado α para o estado β,
descrito por uma equac¸a˜o mestra caracterizada por uma taxa de transic¸a˜o que define a dinaˆmica.
wα→β = taxa de transic¸a˜o de α para β
A dinaˆmica fica totalmente caracterizada atrave´s de uma equac¸a˜o mestra do tipo
d
dt
P(α, t) =∑
β
[
wβ→αP(β, t)−wα→βP(α, t)
]
(A.1)
onde as condic¸o˜es iniciais P(α,0) devem ser especificadas e a condic¸a˜o∑
α
P(α, t) = 1 deve ser satisfeita.
Esta equac¸a˜o e´ parecida com a equac¸a˜o de Schrodinger para uma func¸a˜o de onda de muitas partı´culas ja´
que ela e´ linear em P(α, t) e de primeira ordem na derivada temporal d/dt.
Agora, faremos uso de uma se´rie de exemplos simples de aplicac¸a˜o da equac¸a˜o mestra para que, nas
pro´ximas subsec¸o˜es, mostrarmos como a descric¸a˜o em termos de operadores de criac¸a˜o e aniquilac¸a˜o para
a equac¸a˜o mestra e´ construı´da.
Exemplo 1: A equac¸a˜o mestra para o decaimento A→ /0.
Considere um u´nico sı´tio da rede que conte´m algum nu´mero de partı´culas ideˆnticas. Estas partı´culas
decaem a taxa λ. A taxa para uma transic¸a˜o de n para m partı´culas e´
wn→m =
{
0 para m 6= n−1
nλ para m = n−1
e a equac¸a˜o mestra fica
d
dt
P(n, t) = λ [(n+1)P(n+1, t)−nP(n, t)] . (A.2)
Esta equac¸a˜o mestra possui soluc¸a˜o exata para P(n, t) o que nos permite obter qualquer momento 〈nk〉 ≡
∑
n
nkP(n, t) (k inteiro) para o processo. Em particular, seja ρ(t) = 〈n〉 a me´dia no nu´mero de partı´culas no
tempo t. Enta˜o, sem resolver a equac¸a˜o mestra diretamente:
ρ˙=∑
n
nP˙(n, t) =∑
n
n [λ(n+1)P(n+1, t)−λnP(n, t)]
126
A.1. Introduc¸a˜o
= λ∑
n
n(n+1)P(n+1, t)−λ∑
n
n2P(n, t)
= λ∑
m
(m−1)mP(m, t)−λ∑
n
n2P(n, t)
=−λ∑
m
mP(m, t) =−λρ,
que e´ o resultado que esperamos para o processo de decaimento.
Exemplo 2: A reac¸a˜o A+A→ /0.
Novamente considere um u´nico sı´tio da rede, com a regra que um par de partı´culas pode se aniquilar. As
taxas sa˜o
wn→m =
{
0 para m 6= n−2
n(n−1)λ para m = n−2
e a equac¸a˜o mestra e´
d
dt
P(n, t) = λ [(n+2)(n+1)P(n+2, t)−n(n−1)P(n, t)] . (A.3)
Exemplo 3: Salto entre dois sı´tios da rede
Agora considere dois sı´tios, i = 1,2, com uma taxa Γ de salto do sı´tio 1 para o sı´tio 2.
w(n1,n2)→(m1,m2) =
{
0 para m1 6= n1−1 ou m2 6= n2+1
n1Γ para m1 = n1−1 e m2 = n2+1
onde a equac¸a˜o mestra e´
d
dt
P(n1,n2, t) = Γ [(n1+1)P(n1+1,n2−1, t)−n1P(n1,n2, t)] . (A.4)
Exemplo 4: Difusa˜o
Considere uma cadeia unidimensional de sı´tios da rede i = 1,2, · · · e permita que todas as partı´culas
saltem para a direita ou para a esquerda com taxa Γ. A equac¸a˜o mestra e´
d
dt
P(α, t) = Γ∑
〈i j〉
[
(ni+1)P(ni+1,n j−1, · · · , t)−niP(α, t)
+(n j +1)P(ni−1,n j +1, · · · , t)−n jP(α, t)
]
, (A.5)
onde a soma se estende sobre pares de sı´tios que sa˜o primeiros vizinhos.
Definindo ρ(x, t) =∑
α
niP(α, t) com x = i∆x, e´ possı´vel mostrar que, para ∆x→ 0 a equac¸a˜o (A.5) se
transforma na equac¸a˜o de difusa˜o unidimensional
∂ρ
∂t
= D
∂2ρ
∂x2
onde D≡ Γ(∆x)2 e´ a constante de difusa˜o.
Exemplo 5: Equac¸a˜o Mestra para a reac¸a˜o A+A→ /0 com difusa˜o
O estado do sistema sera´ caracterizado, como sempre, pela probabilidade P(α, t). A dinaˆmica estoca´stica
e´ capturada atrave´s da seguinte equac¸a˜o mestra:
d
dt
P(α, t) =
D
(∆x)2 ∑〈i j〉
[
(ni+1)P(· · · ,ni+1,n j−1, · · · , t)−niP(α, t)
127
Apeˆndice A. Mapeamento de Doi
+(n j +1)P(· · · ,ni−1,n j +1, · · · , t)−n jP(α, t)
]
+λ∑
i
[(ni+2)(ni+1)P(· · · ,ni+2, · · · , t)−ni(ni−1)P(α, t)] . (A.6)
onde a soma se estende sobre os pares de sı´tios primeiros vizinhos. O primeiro termo entre colchetes
representa uma partı´cula saltando do sı´tio i para o sı´tio j, incluindo tanto o fluxo de probabilidade que
“entra” quanto o fluxo de probabilidade que “sai” da configurac¸a˜o α como consequeˆncia do movimento das
partı´culas. O segundo termo corresponde aos saltos do sı´tio j para o sı´tio i. Os fatores multiplicativos n e
n+1 sa˜o um resultado das partı´culas atuando independentemente. Falta especificar apenas a probabilidade
inicial P(α, t = 0). As partı´culas sera˜o distribuidas em cada sı´tio i com a distribuic¸a˜o de Poisson:
P(α,0) =∏
i
(
nni0
ni!
e−n0
)
, (A.7)
onde n0 denota o nu´mero me´dio de partı´culas por sı´tio.
Nas pro´ximas sec¸o˜es aprenderemos como representar estas e outras equac¸o˜es mestras em termos de
operadores de criac¸a˜o e aniquilac¸a˜o. Posteriormente, este passo nos permitira´ obter a formulac¸a˜o em
integral funcional destas equac¸o˜es mestras.
A.2. A representac¸a˜o de Doi
Modelos estoca´sticos cla´ssicos de partı´culas podem ser reescritos em termos de operadores de criac¸a˜o e
aniquilac¸a˜o familiares da mecaˆnica quaˆntica, como mostrado por Doi [61]. Esta representac¸a˜o explora o
fato de que esses processos apenas mudam o nu´mero de ocupac¸a˜o dos sı´tios por um inteiro. Os microesta-
dos para qualquer processo estoca´stico em uma rede e´ dado pelos nu´meros de ocupac¸a˜o {n} ≡ {n1,n2, · · ·}
de cada sı´tio. Seja P({n}, t) a probabilidade dos microestados no tempo t. Como na˜o implementamos
nenhuma restric¸a˜o para ocupac¸a˜o dos sı´tios, introduzimos para cada sı´tio da rede i, j, · · · operadores de
criac¸a˜o aˆ† e aniquilac¸a˜o aˆ sujeitos a`s relac¸o˜es “bosoˆnicas” de comutac¸a˜o
[aˆi, aˆ
†
j ] = δi j, [aˆi, aˆ j] = [aˆ
†
i , aˆ
†
j ] = 0. (A.8)
Estas relac¸o˜es de comutac¸a˜o podem ser realizadas pelas seguintes relac¸o˜es formais
aˆi =
∂
∂a†i
a†i =−
∂
∂aˆi
. (A.9)
A rede vazia e´ caracterizada por aˆi|0〉 = 0 para todo i e em cada sı´tio i definimos o vetor de estado
|ni〉= (aˆ†i )ni |0〉. Para estes estados, temos:
aˆi|ni〉= ni|ni−1〉, aˆ†i |ni〉= |ni+1〉. (A.10)
Podemos agora encapsular toda a informac¸a˜o sobre o estado {n} = (n1,n2, · · ·) da rede no tempo t na
quantidade
|φ(t)〉=∑
{n}
P({n}, t)∏
i
(aˆ†i )
ni |0〉. (A.11)
128
A.2. A representac¸a˜o de Doi
Como consequeˆncia dessa descric¸a˜o de estados em termos de um espac¸o de Fock em completa analogia
com a descric¸a˜o da mecaˆnica quaˆntica de muitos corpos [23], podemos escrever a evoluc¸a˜o temporal da
equac¸a˜o mestra em uma equac¸a˜o do tipo Schrodinger em tempo imagina´rio
d
dt
|φ(t)〉=−Hˆ|φ(t)〉. (A.12)
O motivo para a realizac¸a˜o deste procedimento e´ que ele permite uma descric¸a˜o mais simples da dinaˆmica.
Por exemplo, para o processo A+A→ /0 teremos
Hˆ = Γ∑
〈i j〉
(aˆ†i − aˆ†j)(aˆi− aˆ j)−λ∑
i
(1− (aˆ†i )2)(aˆi)2 (A.13)
com a correspondente soluc¸a˜o formal
|φ(t)〉= e−Hˆt |φ(0)〉. (A.14)
Para entendermos a origem da equac¸a˜o (A.13), seja a equac¸a˜o mestra para o processo A+A→ /0 em um
u´nico sı´tio (A.3) dada por
d
dt
P(n, t) = λ [(n+2)(n+1)P(n+2, t)−n(n−1)P(n, t)] . (A.15)
Multiplicando por |n〉 e somando sobre n, vem:
d
dt
|φ(t)〉= λ∑
n
P(n+2, t)(n+2)(n+1)|n〉−λ∑
n
P(n, t)n(n−1)|n〉
= λ∑
n
P(n+2, t)aˆ2|n+2〉−λ∑
n
P(n, t)(aˆ†)2aˆ2|n〉
= λ(1− (aˆ†)2)aˆ2∑
n
P(n, t)|n〉
= λ(1− (aˆ†)2)aˆ2|φ(t)〉=−Hˆ|φ(t)〉, (A.16)
o que corresponde ao segundo termo do lado direito da equac¸a˜o (A.13).
Por outro lado, se lembrarmos da equac¸a˜o mestra para o processo de salto entre os sı´tios 1 e 2 (A.4)
d
dt
P(n1,n2, t) = Γ [(n1+1)P(n1+1,n2−1, t)−n1P(n1,n2, t)] , (A.17)
multiplicando por |n1,n2〉 e somando em n1 e n2, teremos:
d
dt
|φ(t)〉= Γ ∑
n1,n2
P(n1+1,n2−1)(n1+1)|n1,n2〉−Γ ∑
n1,n2
P(n1,n2, t)n1|n1,n2〉
= Γ ∑
n1,n2
P(n1+1,n2−1, t)aˆ†2aˆ1|n1+1,n2−1〉−Γ ∑
n1,n2
P(n1,n2, t)aˆ
†
1aˆ1|n1,n2〉
= Γ(aˆ†2− aˆ†1)aˆ1|φ(t)〉. (A.18)
129
Apeˆndice A. Mapeamento de Doi
Assim, o Hamiltoniano para o processo de salto do sı´tio 1 para o sı´tio 2 fica dado por
Hˆ1→2 = Γ(aˆ†2− aˆ†1)aˆ1. (A.19)
Se permitirmos o salto reverso a` mesma taxa do sı´tio 2 para o sı´tio 1, ficamos com
Hˆ1↔2 = Γ(aˆ†2− aˆ†1)(aˆ1− aˆ2). (A.20)
Para saltos entre todos os vizinhos da rede, a equac¸a˜o (A.20) se extende para
HˆD =
D
(∆x)2 ∑〈i j〉
(aˆ†j − aˆ†i )(aˆi− aˆ j) (A.21)
que e´ o Hamiltoniano associado a` difusa˜o das partı´culas na rede e corresponde ao primeiro termo do lado
direito da equac¸a˜o (A.13) com Γ≡ D
(∆x)2 . As equac¸o˜es de movimento para P({n}, t) e seus momentos nesta
representac¸a˜o sa˜o ideˆnticas obviamente a`s que seguem diretamente da equac¸a˜o mestra. Pore´m, neste ponto
podemos ver a vantagem do uso do formalismo de Doi: a equac¸a˜o mestra original fica complicada pela
presenc¸a de fatores com n e n2 que esta˜o ausentes na representac¸a˜o de Doi. O formalismo de “segunda
quantizac¸a˜o” fornece o palco natural para a descric¸a˜o de partı´culas independentes que podem variar em
nu´mero, podendo ser criadas e/ou destruı´das.
Mu´ltiplas espe´cies: Em reac¸o˜es de duas espe´cies de partı´culas, por exemplo A+B→ /0, a equac¸a˜o mestra
e´ definida em termos dos nu´meros de ocupac¸a˜o das partı´culas A e B, P({m},{n}, t). Para a representac¸a˜o
de Doi, introduzimos operadores de criac¸a˜o e aniquilac¸a˜o distintos para cada uma das espe´cies: aˆi, aˆ
†
i , bˆi e
bˆ†i e definimos o estado
|φ(t)〉= ∑
{m},{n}
P({m},{n}, t)∏
i
(aˆ†i )
mi(bˆ†i )
ni |0〉. (A.22)
A generalizac¸a˜o para mais de duas espe´cies e´ trivial.
Hamiltonianos de Doi: Em qualquer reac¸a˜o descrita por processos do tipo k1X1 + k2X2 + · · ·+ knXn →
l1Y1+ l2Y2+ · · ·+ lmYm, cada processo contribui com dois termos para Hˆ da forma
(Taxa) [(Reagentes)− (Produto)] (A.23)
onde
• Reagentes = operadores de criac¸a˜o e aniquilac¸a˜o para cada reagente, ordenados normalmente;
• Produto = operador de aniquilac¸a˜o para cada reagente, operador de criac¸a˜o para cada produto, orde-
nados normalmente.
Exemplos:
1. A+A→ /0 : Temos dois reagentes iguais a A que induzem o aparecimento do termo λ(aˆ†)2aˆ2 no
Hamiltoniano, correspondente ao termo Reagentes na equac¸a˜o (A.23). Como na˜o temos nenhum
produto nesta reac¸a˜o, o termo correspondente a` Produto em (A.23) sera´ λaˆ2 de modo que o Hamil-
toniano completo fica
HˆA+A→ /0 = λ[(aˆ†)2aˆ2− aˆ2].
2. A+A→ A : Usando um raciocı´nio ana´logo ao usado acima, teremos
HˆA+A→A = λ[(aˆ†)2aˆ2− aˆ†aˆ2].
3. A→ A+A :
HˆA→A+A = λ[aˆ†aˆ− (aˆ†)2aˆ].
130
A.2. A representac¸a˜o de Doi
4. A+B→C :
HˆA+B→C = λ[aˆ†bˆ†aˆbˆ− cˆ†aˆbˆ].
5. Salto do sı´tio 1 para o sı´tio 2:
Hˆ1→2 = Γ[(aˆ1)†aˆ1− (aˆ2)†aˆ1].
Obtemos pois, regras gerais para obtermos o Hamiltoniano de Doi para qualquer reac¸a˜o descrita por pro-
cessos do tipo k1X1+ k2X2+ · · ·+ knXn→ l1Y1+ l2Y2+ · · ·+ lmYm.
A.2.1. Comparac¸a˜o com o me´todo da segunda quantizac¸a˜o da mecaˆnica quaˆntica
Devemos ter em mente va´rios fatos enquanto trabalhamos com o me´todo da segunda quantizac¸a˜o para os
processos estoca´sticos cla´ssicos discutidos aqui. Existem algumas diferenc¸as imediatas com relac¸a˜o ao
modo como o me´todo e´ utilizado na mecaˆnica quaˆntica:
• Diferentemente da equac¸a˜o de Schrodinger, aqui na˜o existe o fator i na equac¸a˜o (A.14).
• O operador “quase” Hamiltoniano (Hˆ) geralmente na˜o e´ Hermitiano.
• A diferenc¸a mais importante esta´ no ca´lculo dos valores esperados de operadores. O valor esperado
do operador Aˆ quando o sistema esta´ no estado |φ(t)〉 na˜o e´ 〈φ|Aˆ|φ〉 como na teoria quaˆntica porque
〈φ|Aˆ|φ〉 e´ bilinear nas probabilidades P({ni}).
• Um observa´vel em um processo estoca´stico cla´ssico, digamos A({ni}), pode ser representado por
um operador Aˆ, que e´ diagonal na base de Fock, pela prescric¸a˜o de substituir a†i aˆi no lugar de ni.
• Para definir uma expressa˜o para o valor esperado em termos dos operadores, precisamos definir o
estado projec¸a˜o 〈P | com as seguintes propriedades:
〈P |0〉= 1, 〈P |aˆ†i = 〈P | (A.24)
As equac¸o˜es acima podem ser satisfeitas definindo 〈P | como
〈P |= 〈0|e∑i aˆi (A.25)
Demonstrac¸a˜o: Ver apeˆndice B.3.
• Para um processo estoca´stico cla´ssico, o valor esperado de um observa´vel A({ni}) e´ definido como
〈A〉= ∑
{ni}
P({ni}; t)A({ni}). (A.26)
Das propriedades do estado projec¸a˜o, temos que 〈P |∏
{ni}
a†nii |0〉 = 1. Usando este fato na equac¸a˜o
(A.26), vem
〈A〉 = ∑
{ni}
P({ni}; t)A({ni})〈P |∏
{ni}
a†nii |0〉
= ∑
{ni}
P({ni}; t)〈P |A({ni})∏
{ni}
a†nii |0〉
= ∑
{ni}
P({ni}; t)〈P |A({ni})|n1,n2, · · · 〉
= ∑
{ni}
P({ni}; t)〈P |Aˆ|n1,n2, · · · 〉
onde Aˆ e´ o operador obtido substituindo ni por a
†
i aˆi em A({ni}). E´ importante notar aqui que o
operador Aˆ e´ independente dos ı´ndices ni e assim
〈A〉= 〈P |Aˆ|φ(t)〉= 〈P |Aˆ e−Hˆt |φ(0)〉. (A.27)
131
Apeˆndice A. Mapeamento de Doi
• Colocando Aˆ = 1 na equac¸a˜o (A.27) obtemos
〈P |φ(t)〉= 〈P |e−Hˆt |φ(0)〉= 1. (A.28)
Esta equac¸a˜o significa que a probabilidade total e´ conservada durante a evoluc¸a˜o temporal.
• Colocando t = 0 na equac¸a˜o (A.28) obtemos
〈P |φ(0)〉= 1. (A.29)
Como a equac¸a˜o acima e a equac¸a˜o (A.28) sa˜o verdadeiras para qualquer |φ(0)〉, devemos ter
〈P |Hˆ = 0. (A.30)
• Se Hˆ for escrito na forma ordenada normal, isto e´, se todos os operadores de criac¸a˜o a† estiverem a`
esquerda de todos os operadores de aniquilac¸a˜o aˆ, enta˜o de (A.25) e de (A.30), temos que Hˆ deve
ser nulo se todos os operadores a† forem substituidos por 1.
A.2.2. O deslocamento de Doi
Usando a definic¸a˜o do operador projec¸a˜o 〈P |= 〈0|e∑i aˆi e a definic¸a˜o de experanc¸a de um observa´vel dada
pela equac¸a˜o (A.27), temos
〈A〉= 〈0|e∑i aˆi Aˆe−Hˆt |φ(0)〉. (A.31)
O fator e∑i aˆi pode ser comutado atrave´s dos operadores e retornar
〈A〉= 〈0|Aˆshiftede−Hˆshiftedte∑i aˆi |φ(0)〉 (A.32)
onde os operadores deslocados sa˜o obtidos dos na˜o deslocados pela substituic¸a˜o de 1+a†i no lugar de a
†
i .
Este passo e´ consequeˆncia do que foi feito no apeˆndice (B.3), de onde pode-se tirar a identidade eaˆ f (a†) =
f (a†+1)eaˆ. O Hamiltoniano com {a†j} → a†j +1 e´ conhecido como o Hamiltoniano deslocado de Doi e a
vantagem deste procedimento e´ que o Hamiltoniano fica ordenado normalmente (todos os a† a` direita dos
aˆ ) e como consequeˆncia o valor da esperanc¸a e´ zero.
132
Apeˆndice B
Formulac¸a˜o contı´nua em uma teoria de
campos
B.1. Representac¸a˜o em estados coerentes e integrais funcionais
Partindo da representac¸a˜o a la segunda quantizac¸a˜o, uma teoria de campos pode ser obtida pelas mesmas
te´cnicas desenvolvidas para os sistemas quaˆnticos de muitas partı´culas. Uma boa discussa˜o desses me´todos,
em particular sobre a representac¸a˜o em estados coerentes pode ser obtida em [62] assim como em [23].
Mas aqui procuraremos abordar o assunto da maneira mais autocontida possı´vel, onde alguns dos detalhes
a respeito dos estados coerentes esta˜o disponı´veis no apeˆndice B.2.
B.1.1. Representac¸a˜o em estados coerentes
A ide´ia ba´sica da construc¸a˜o de uma integral funcional para operadores de campo e´, como no caso da
mecaˆnica quaˆntica, segmentar a evoluc¸a˜o temporal do (quase) Hamiltoniano em intervalos de tempo infini-
tesimais e absorver o ma´ximo possı´vel da dinaˆmica durante um curto intervalo de tempo em um conjunto
adequadamente escolhido de auto-estados. Mas como estes auto-estados devem ser escolhidos? No con-
texto da mecaˆnica quaˆntica de uma u´nica partı´cula, a estrutura do Hamiltoniano sugere uma representac¸a˜o
em termos dos auto-estados da coordenada e do momento. Aqui, dados os Hamiltonianos conveniente-
mente descritos em termos de operadores de criac¸a˜o e aniquilac¸a˜o, uma ide´ia o´bvia seria procurar por auto-
estados destes operadores. Tais estados existem e sa˜o chamados estados coerentes. Como auto-estados do
operador criac¸a˜o na˜o sa˜o normaliza´veis [63], focaremos nos auto-estados do operador aniquilac¸a˜o.
Vamos denotar o estado coerente por |φ〉, enta˜o
aˆ|φ〉= φ|φ〉. (B.1)
Uma forma possı´vel para o estado coerente e´
|φ〉= N(φ) eφa† |0〉 (B.2)
onde N(φ) e´ uma func¸a˜o de φ a ser determinada. Podemos checar explicitamente que o estado dado acima
133
Apeˆndice B. Formulac¸a˜o contı´nua em uma teoria de campos
e´ um estado coerente:
aˆ|φ〉 = N(φ) aˆ eφa† |0〉
= N(φ)
[
aˆ,eφa
†
]
|0〉 (usamos aˆ|0〉= 0)
= N(φ)
[
∂
∂a†
,eφa
†
]
|0〉
= N(φ) φ eφa
† |0〉
= φ|φ〉
Como a fase global do vetor de estado na˜o tem significado, podemos escolher N(φ) real. A constante de
normalizac¸a˜o N(φ) pode ser determinada exigindo
1 = 〈φ|φ〉
= N2(φ)〈0|eφ∗aˆeφ a† |0〉
= N2(φ)
〈
0
∣∣∣∣∣ ∞∑m=0 φ
∗aˆm
m!
∞
∑
n=0
φn a†n
n!
∣∣∣∣∣0
〉
(B.3)
Para m 6= n,
〈0|aˆma†n|0〉= 0.
Para m = n, temos
〈0|aˆna†n|0〉 = 〈0|aˆn−1 [aˆ,a†n]|0〉 (usando aˆ|0〉= 0)
= 〈0|aˆn−1na†(n−1)|0〉 (usando (A.9))
= n〈0|aˆn−1a†(n−1)|0〉
= n(n−1)〈0|aˆn−2a†(n−2)|0〉
...
= n! (B.4)
Portanto, temos
〈0|aˆma†n|0〉= n!δm,n (B.5)
Substituindo este resultado em (B.3), vem
1 = N2(φ)
∞
∑
m,n=0
φ∗mφn
m!n!
n!δm,n
= N2(φ)
∞
∑
n=0
|φ|2n
n!
= N2(φ)e|φ|
2
⇒ N(φ) = e− 12 |φ|2 (B.6)
O estado coerente normalizado e´ enta˜o dado por
|φ〉= e− 12 |φ|2+φa† |0〉 (B.7)
134
B.1. Representac¸a˜o em estados coerentes e integrais funcionais
Estado projec¸a˜o
O estado projec¸a˜o 〈P | definido em (A.25) e´ tambe´m um estado coerente, ja´ que
〈P |= 〈0|eaˆ = e 12 〈1| (B.8)
onde usamos (B.7) fazendo φ→ 1, ou seja, |1〉= e− 12+a† |0〉.
Produto interno entre dois estados coerentes
Para uso subsequente vamos calcular uma expressa˜o para o produto interno entre dois estados coerentes
|φ1〉 e |φ2〉 :
〈φ1|φ2〉 =
〈
0
∣∣∣e− 12 |φ1|2+φ∗1aˆe− 12 |φ2|2+φ2a† ∣∣∣0〉
= e−
1
2 |φ1|2− 12 |φ2|2
〈
0
∣∣eφ∗1aˆeφ2a† ∣∣0〉
= e−
1
2 |φ1|2− 12 |φ2|2
〈
0
∣∣∣∣∣ ∞∑m=0 φ
∗m
1 aˆ
m
m!
∞
∑
n=0
φn2a
†n
n!
∣∣∣∣∣0
〉
= e−
1
2 |φ1|2− 12 |φ2|2
∞
∑
m,n=0
φ∗m1 φ
n
2
m! n!
〈0|aˆma†n|0〉
= e−
1
2 |φ1|2− 12 |φ2|2
∞
∑
m,n=0
φ∗m1 φ
n
2
m! n!
n! δm,n
= e−
1
2 |φ1|2− 12 |φ2|2
∞
∑
n=0
φ∗n1 φ
n
2
n!
= e−
1
2 |φ1|2− 12 |φ2|2eφ
∗
1φ2
⇒ 〈φ1|φ2〉 = e− 12 |φ1|2− 12 |φ2|2+φ∗1φ2 (B.9)
Resoluc¸a˜o da identidade em termos de estados coerentes
E´ possı´vel mostrar que a resoluc¸a˜o da identidade escrita em termos do estados φ e´ dada por
1 =
∫ d2φ
pi
|φ〉〈φ|. (B.10)
Veja apeˆndice (B.2.2) para maiores detalhes.
B.1.2. Formulac¸a˜o em integral funcional
Partiremos dos resultados relacionados aos estados coerentes discutidos acima a agora obteremos a formulac¸a˜o
em integral funcional para os processos estoca´sticos de interesse.
135
Apeˆndice B. Formulac¸a˜o contı´nua em uma teoria de campos
Processo estoca´stico com uma u´nica espe´cie em dimensa˜o 0
Por simplicidade derivaremos primeiro a formulac¸a˜o em integral de caminho de um processo estoca´stico
para uma u´nica espe´cie em um u´nico sı´tio da rede, evitando complicac¸o˜es desnecessa´rias na notac¸a˜o.
Seja Hˆ o quase-Hamiltoniano do processo. Sejam aˆ, a† e |0〉 o operador aniquilac¸a˜o, o de criac¸a˜o e o
estado de va´cuo respectivamente. O estado do sistema e´ dado por (A.11)∗ e a evoluc¸a˜o temporal e´ dada
pela equac¸a˜o (A.12).
Para derivar a formulac¸a˜o em integral funcional partimos de (A.12)
|ψ(t)〉= e−Hˆt |ψ(0)〉. (B.11)
Vamos discretizar o tempo t em N subintervalos de comprimento ∆t, i. e.
∆t =
t
N
. (B.12)
Estamos interessados no limite ∆t→ 0. Equac¸a˜o (B.11) pode ser escrita como
|ψ(t)〉= lim
∆t→0
e−HˆN∆t |ψ(0)〉 (B.13)
|ψ(t)〉 = lim
∆t→0
e−HˆN∆t |ψ(0)〉
= lim
∆t→0
e−Hˆ∆te−Hˆ∆t · · ·e−Hˆ∆t |ψ(0)〉
(B.14)
Inserindo a resoluc¸a˜o da identidade (B.10) entre cada subintervalo de tempo na equac¸a˜o acima, obteremos
|ψ(t)〉 = lim
∆t→0
∫ d2φt
pi
|φt〉〈φt |e−Hˆ∆t
∫ d2φt−∆t
pi
|φt−∆t〉〈φt−∆t |e−Hˆ∆t∫ d2φt−2∆t
pi
|φt−2∆t〉〈φt−2∆t |e−Hˆ∆t
... ∫ d2φ∆t
pi
|φ∆t〉〈φ∆t |e−Hˆ∆t
∫ d2φ0
pi
|φ0〉〈φ0|ψ(0)〉
= lim
∆t→0
N −1
∫ ( t
∏
τ=0
d2φτ
)
|φt〉〈φt |e−Hˆ∆t |φt−∆t〉〈φt−∆t |e−Hˆ∆t |φt−2∆t〉
· · · 〈φ∆t |e−Hˆ∆t |φ0〉〈φ0|ψ(0)〉
= lim
∆t→0
N −1
∫ ( t
∏
τ=0
d2φτ
)
|φt〉
[
t
∏
τ=∆t
〈φτ|e−Hˆ∆t |φτ−∆t〉
]
〈φ0|ψ(0)〉 (B.15)
onde N e´ a constante de normalizac¸a˜o.
Seja A(n) um observa´vel. O operador correspondente Aˆ e´ definido substituindo n por a†aˆ em A(n). O
valor esperado e´ dado por (A.27) como
∗Com ψ(t) no lugar de φ(t).
136
B.1. Representac¸a˜o em estados coerentes e integrais funcionais
〈A〉 = 〈P |Aˆ|ψ(t)〉
= e
1
2 〈1|Aˆ|ψ(t)〉
= N −1 lim
∆t→0
∫ ( t
∏
τ=0
d2φτ
)
〈1|Aˆ|φt〉
[
t
∏
τ=∆t
〈φτ|e−Hˆ∆t |φτ−∆t〉
]
〈φ0|ψ(0)〉. (B.16)
O fator e
1
2 foi absorvido na constante de normalizac¸a˜o.
Vamos agora calcular cada termo da equac¸a˜o (B.16) um a um. Consideremos o termo do meio:
〈φτ|e−Hˆ∆t |φt−∆t〉 = 〈φτ|[1− Hˆ∆t+O(∆t2)]|φτ−∆t〉
= 〈φτ|φτ−∆t〉−〈φτ|Hˆ|φτ−∆t〉∆t+O(∆t2)
= 〈φτ|φτ−∆t〉−〈φτ|Hˆ(φ∗τ ,φτ−∆t)|φτ−∆t〉∆t+O(∆t2)
= 〈φτ|φτ−∆t〉− Hˆ(φ∗τ ,φτ−∆t)〈φτ||φτ−∆t〉∆t+O(∆t2)
= 〈φτ|φτ−∆t〉[1− Hˆ(φ∗τ ,φτ−∆t)+O(∆t2)]
= 〈φτ|φτ−∆t〉e−Hˆ(φ∗τ ,φτ−∆t )∆t (B.17)
onde Hˆ(φ∗τ ,φτ−∆t) e´ obtido substituindo aˆ por φτ−∆t e a† por φ∗τ na forma ordenada normal. Usando agora
o produto interno de estados coerentes (B.9),
〈φτ|φτ−∆t〉 = e− 12 |φτ|2− 12 |φτ−∆t |2+φ∗τφτ−∆t
= e
1
2 (|φτ|2−|φτ−∆t |2)e−|φτ|
2+φ∗τφτ−∆t
= e
1
2 (|φτ|2−|φτ−∆t |2)e−φ
∗
τ (φτ−φτ−∆t ) (B.18)
Desta forma,
t
∏
τ=∆t
〈φτ|e−Hˆ∆t |φt−∆t〉 =
t
∏
τ=∆t
〈φτ|φτ−∆t〉e−Hˆ(φ∗τ ,φτ−∆t )∆t
=
t
∏
τ=∆t
e
1
2 (|φτ|2−|φτ−∆t |2)e−φ
∗
τ (φτ−φτ−∆t )e−Hˆ(φ
∗
τ ,φτ−∆t )∆t
=
[
t
∏
τ=∆t
e
1
2 (|φτ|2−|φτ−∆t |2)
][
t
∏
τ=∆t
e−φ
∗
τ (φτ−φτ−∆t )e−Hˆ(φ
∗
τ ,φτ−∆t )∆t
]
= e
1
2 ∑
t
τ=∆t (|φτ|2−|φτ−∆t |2)e−∑
t
τ=∆t [φ
∗
τ (φτ−φτ−∆t )+Hˆ(φ∗τ ,φτ−∆t )∆t]
= e
1
2 [|φt |2−|φt−∆t |2+|φt−∆t |2−|φt−2∆t |2+···+|φ∆t |2−|φ0|2]
× e−∑tτ=∆t [φ∗τ (φτ−φτ−∆t )+Hˆ(φ∗τ ,φτ−∆t )∆t]
= e
1
2 (|φt |2−|φ0|2)e−∑
t
τ=∆t [φ
∗
τ (φτ−φτ−∆t )+Hˆ(φ∗τ ,φτ−∆t )∆t] (B.19)
Vamos agora desenvolver um pouco o termo 〈1|Aˆ|φt〉 na equac¸a˜o (B.16):
137
Apeˆndice B. Formulac¸a˜o contı´nua em uma teoria de campos
〈1|Aˆ|φt〉 = A(φt)〈1|φt〉
= A(φt)e−
1
2 1
2− 12 |φt |2+φt
= A(φt)e−
1
2− 12 |φt |2+φt (B.20)
onde A(φt) e´ obtido substituindo 1 no lugar de a† e φt no lugar de aˆ no operador Aˆ escrito na forma ordenada
normal.
Vamos agora desenvolver 〈φ0|ψ(0)〉 em (B.16) utilizando como distribuic¸a˜o inicial a distribuic¸a˜o de
Poisson,
P(n,0) =
n¯n0
n!
e−n¯0 (B.21)
Primeiramente
|ψ(0)〉 = ∑
n
P(n,0)|n〉
= ∑
n
n¯n0
n!
e−n¯0a†n|0〉
= e−n¯0en¯0a
† |0〉
= e−n¯0 |n¯0〉 (B.22)
O pro´prio estado inicial |ψ(0)〉 e´ um estado coerente para distribuic¸o˜es iniciais de Poisson. Em sistemas
d−dimensionais com difusa˜o, e´ interessante notar que um processo de difusa˜o partindo de uma distribuic¸a˜o
de Poisson na auseˆncia de ruı´do permanece uma distribuic¸a˜o de Poisson nos tempos subsequentes. Por-
tanto, as flutuac¸o˜es estoca´sticas nestes sistemas descrevem apenas os desvios da distribuic¸a˜o Poissoniana.
Este fato esta´ relacionado com o surgimento de fatores negativos de origem complexa† nas correlac¸o˜es do
ruı´do que emergem naturalmente da teoria. Veremos um exemplo desta situac¸a˜o posteriormente.
Agora,
〈φ0|ψ(0)〉 = e−n¯0〈φ0|n¯0〉
= e−n¯0e−
1
2 |φ0|2− 12 |n¯0|2+φ∗0n¯0 . (B.23)
Substituindo as equac¸o˜es (B.19), (B.20) e (B.23) em (B.16) e absorvendo todas as constantes no fator de
normalizac¸a˜o, obteremos
†Complexo no sentido do conjunto dos nu´meros imagina´rios Z.
138
B.1. Representac¸a˜o em estados coerentes e integrais funcionais
〈A〉 = N −1 lim
∆t→0
∫ ( t
∏
τ=0
d2φτ
)
A(φt)e−
1
2 |φt |2+φt
× e 12 (|φt |2−|φ0|2)e−∑tτ=∆t [φ∗t (φt−φτ−∆t )+Hˆ(φ∗τ ,φτ−∆t )∆t]e− 12 |φ0|2+φ∗0n¯0
= N −1 lim
∆t→0
∫ ( t
∏
τ=0
d2φτ
)
A(φt)eφt e−|φ0|
2
eφ
∗
0n¯0
× exp
{
−
t
∑
τ=∆t
[φ∗τ(φτ−φτ−∆t)+ Hˆ(φ∗τ ,φτ−∆t)∆t]
}
= N −1 lim
∆t→0
∫ ( t
∏
τ=0
d2φτ
)
A(φt)eφt e−|φ0|
2
eφ
∗
0n¯0
× exp
{
−
t
∑
τ=∆t
[
φ∗τ
(
φτ−φτ−∆t
∆t
)
∆t+ Hˆ(φ∗τ ,φτ−∆t)∆t
]}
∆t→0
= N −1
∫
Dφ Dφ∗A[φ(t)]eφ(t)e−|φ(0)|
2
eφ
∗(0)n¯0
× exp
{
−
∫ t
0
dt ′
[
φ∗(t ′)
∂φ(t ′)
∂t ′
+ Hˆ[φ∗(t ′),φ(t ′)]
]}
= N −1
∫
Dφ Dφ∗ A[φ(t)]exp{−S[φ∗,φ]t0} (B.24)
onde
S[φ∗,φ]t0 = −[φ(t f )−|φ(0)|2+φ∗(0)n¯0]+
∫ t f
0
dt
{
φ∗(t)
∂φ(t)
∂t
+ Hˆ[φ∗(t),φ(t)]
}
= −[φ(t f )+φ∗(0){n¯0−φ(0)}]+
∫ t f
0
dt{φ∗(t) ∂tφ(t)+ Hˆ[φ∗(t),φ(t)]} (B.25)
Pontos importantes:
• Ao tomarmos o limite ∆t → 0, substituimos φ∗τ e φτ−∆t por φ∗(t ′) e φ(t ′). Por isso, sempre que um
termo contiver ambos φ∗(t ′) e φ(t ′), φ∗(t ′) devera´ ser considerado estar em um tempo posterior a
φ(t ′).
• A medida e´ definida como
Dφ Dφ∗ = lim
∆t→0
t
∏
τ=0
d2φτ. (B.26)
• A constante de normalizac¸a˜o e´ determinada impondo A(φ) = 1 na equac¸a˜o (B.24):
1 = N −1
∫
Dφ Dφ∗ 1 exp{−S[φ∗,φ]t0}
⇒N =
∫
Dφ Dφ∗ exp{−S[φ∗,φ]t0} (B.27)
Termo inicial
A integral com relac¸a˜o a φ∗0 na equac¸a˜o (B.20) pode ser feita:
∫
dφ∗0 e
−|φ0|2eφ
∗
0n¯0 =
∫
dφ∗0 exp{φ∗0(n¯0−φ0)}
= δ(n¯0−φ0) (B.28)
139
Apeˆndice B. Formulac¸a˜o contı´nua em uma teoria de campos
Agora a integral sobre φ0 tambe´m pode ser feita o que reforc¸a a condic¸a˜o inicial no integrando. O termo
[φ∗(0){n¯0−φ(0)}] pode ser tirado da ac¸a˜o (B.25), ja´ que a delta de Dirac impo˜e φ(0) = n¯0. Pore´m, uma
integral de caminho com o dado vı´nculo na˜o e´ diretamente susceptı´vel aos me´todos perturbativos.
Esquema para o ca´lculo perturbativo
A parte da ac¸a˜o S que e´ bilinear nos campos (∝ φ∗φ) e´ escolhida como “ac¸a˜o refereˆncia” e resolvida
exatamente. Os termos restantes sa˜o tratados perturbativamente. Nos ca´lculos perturbativos, um termo
sera´ zero se todo φ nele na˜o puder ser pareado com um φ∗ em um tempo anterior.
Na exponencial de φ∗(0)φ(0), todos os termos exceto o primeiro sa˜o nulos, ja´ que na˜o ha´ nenhum φ∗
anterior a` t = 0. Isto e´ equivalente a desprezar |φ|2 na ac¸a˜o. Desta forma, ficamos apenas com e−n¯0φ∗(0)
como termo inicial.
Termo final
O termo φ(t f ) na ac¸a˜o (B.25) pode ser tratado realizando o seguinte “deslocamento” no campo:
φ∗(t) = 1+ φ¯(t) (B.29)
Desta forma,
∫ t f
0
dt φ∗(t) ∂tφ(t) =
∫ t f
0
dt(1+ φ¯(t))∂tφ(t)
= φ(t f )−φ(0)+
∫ t f
0
dt φ¯(t)∂tφ(t) (B.30)
A ac¸a˜o em termos do campo deslocado se torna
S[φ¯(t),φ]t f0 =−[φ(0)+(1+ φ¯(0))n¯0]+
∫ t f
0
dt{φ¯(t)∂tφ(t)+ Hˆ[φ¯(t),φ(t)]}. (B.31)
Mais uma vez desprezamos φ(0) da ac¸a˜o por na˜o existir um φ¯ pareado anterior a t = 0. O termo n¯0 pode
ser absorvido na constante de normalizac¸a˜o. A ac¸a˜o se reduz a
S[φ¯(t),φ]t f0 =−φ¯(0)n¯0+
∫ t f
0
dt{φ¯(t)∂tφ(t)+ Hˆ[φ¯(t),φ(t)]} (B.32)
onde Hˆ[φ¯(t),φ(t)] e´ obtido substituindo 1+ φ¯(t) no lugar de a† e φ(t) no lugar de aˆ.
B.1.3. Processos estoca´sticos de uma u´nica espe´cie em d−dimenso˜es
De posse da formulac¸a˜o em integral de caminho para o caso de dimensa˜o 0, na˜o e´ difı´cil generalizar para
d dimenso˜es. No apeˆndice (B.4) executa-se esta generalizac¸a˜o bem como se obte´m o limite contı´nuo da
ac¸a˜o que fica determinada por
S[φ¯,φ]t f0 =−
∫
ddx φ¯(x,0) n¯0+
∫ t f
0
dt
∫
ddx
[
φ¯(x, t)(∂t −D∇2)φ(x, t)+ Hˆ ′(x)
]
(B.33)
onde Hˆ ′(x) e´ a parte do Hamiltoniano que na˜o corresponde a` difusa˜o.
140
B.1. Representac¸a˜o em estados coerentes e integrais funcionais
Processos estoca´sticos de mu´ltiplas espe´cies em dimensa˜o d
A ac¸a˜o obtida em (B.84) e reproduzida na sec¸a˜o anterior pode ser generalizada para o caso de mu´ltiplas
espe´cies. Escrevendo diretamente esta generalizac¸a˜o obteremos:
S[φ¯1,φ1, φ¯2,φ2, · · · ] = −
∫
ddx
[
φ¯1(x,0) n¯0,1+ φ¯2(x,0) n¯)0,2+ · · ·
]
= +
∫ t f
0
dt
∫
ddx[φ¯1(x, t)(∂t −D∇2)φ1(x, t)
+ φ¯2(x, t)(∂d−D∇2)φ2(x, t)
...
+ Hˆ ′(x)] (B.34)
Exemplos
Neste ponto, vamos obter H explı´citamente para o processo de aniquilac¸a˜o de pares limitado por difusa˜o,
A+A→ /0. Como o operador evoluc¸a˜o associado (A.13) ja´ esta´ ordenado normalmente, obtemos direta-
mente
H({φ∗},{φ}) = D
h2 ∑〈i j〉
(φ∗i −φ∗j)(φi−φ j)−λ∑
i
(1− (φ∗i )2)φ2i . (B.35)
Tomamos o limite contı´nuo, substituindo diferenc¸as finitas na rede por gradientes no espac¸o. A teoria de
campo resultante, antes de qualquer transformac¸a˜o de varia´veis sera´
S[φ˜,φ] =
∫
ddx
{
−φ(t f )+
∫ t f
0
dt
[
φ˜
(
∂t −D∇2
)
φ−λ0
(
1− φ˜2)φ2]−n0φ˜(0)} , (B.36)
onde λ0 ≡ λhd . Fazendo a mudanc¸a φ˜→ 1+ φ¯, obtemos
S[φ¯,φ] =
∫
ddx
{∫ t f
0
dt
[
φ¯
(
∂t −D∇2
)
φ−λ1φ¯φ2+λ2φ¯2φ2
]−n0φ¯(0)} (B.37)
com λ1 = 2λ0 e λ2 = λ0.
A descric¸a˜o para sistemas de mu´ltiplas espe´cies requer, no nı´vel da equac¸a˜o mestra, conjuntos de
nu´meros de ocupac¸a˜o adicionais. Por exemplo, a equac¸a˜o mestra para a reac¸a˜o de aniquilac¸a˜o de duas
espe´cies A+B→ /0 emprega a probabilidade P({m},{n}, t) onde {m} e {n} respectivamente denotam os
conjuntos de nu´meros de ocupac¸a˜o das partı´culas A e B. As partı´culas A e B se difundem na rede com cons-
tantes de difusa˜o DA e DB, possivelmente com DA 6= DB. Desenvolvendo todo o procedimento descrito
acima, obteremos a seguinte ac¸a˜o [27]:
S[a¯,a, b¯,b] =
∫
ddx
{∫ t f
0
dt
[
a¯
(
∂t −DA∇2
)
a+ b¯
(
∂t −DB∇2
)
b
+λ0(a¯+ b¯)ab+λ0a¯b¯ab
]
−a0a¯(0)−b0b¯(0)
}
. (B.38)
Generalizac¸a˜o para mu´ltiplas partı´culas e´ simples: para cada espe´cie de partı´cula nova, operadores da
segunda quantizac¸a˜o e campos correspondentes devem ser introduzidos. Os detalhes da reac¸a˜o ficam co-
dificados na equac¸a˜o mestra. Com alguma pra´tica, e´ simples partir diretamente da representac¸a˜o de Doi,
como exemplificado na sec¸a˜o A.2.
141
Apeˆndice B. Formulac¸a˜o contı´nua em uma teoria de campos
A ac¸a˜o (B.38) e´ linear em a¯ e b¯. O extremo da ac¸a˜o com b¯ = 1 e´:
δS
δa¯
= (∂t −DA∇2)a+2λ0ab−a0δ(t) = 0
e, com a¯ = 1 :
δS
δb¯
= (∂t −DB∇2)b+2λ0ab−b0δ(t) = 0.
Se λ0 = 0, as duas equac¸o˜es acima nada mais sa˜o que duas equac¸o˜es de difusa˜o independentes para as
densidades de partı´culas a e b com condic¸o˜es iniciais dadas por a(0) = a0 e b(0) = b0.
Relac¸a˜o com equac¸o˜es diferenciais parciais estoca´sticas (EDPE’s)
A teoria de campo desenvolvida acima a`s vezes pode ser colocada na forma de equac¸o˜es diferenciais par-
ciais estoca´sticas, via procedimento de Martin-Siggia-Rose-Janssen-de Dominicis (MSRJD) [64, 65, 66],
executado ao contra´rio. Ao inve´s de partirmos das EDPE’s e obtermos uma teoria de campo equivalente,
podemos partir da teoria de campo e obter as EDPE’s. Considere a ac¸a˜o (B.38) para o processo A+A→ /0.
Exceto o termo qua´rtico λ2φ¯2φ2, todos os termos sa˜o lineares em φ¯. Pore´m o termo qua´rtico pode ser
linearizado introduzindo o campo auxiliar η, onde
exp(−λ2φ¯2φ2) ∝
∫
dηexp
(
−η
2
2
)
exp(iη
√
2λ2φ¯φ). (B.39)
Substituindo esta relac¸a˜o na ac¸a˜o teremos treˆs campos que flutuam, φ¯, φ e η, mas com a vantagem de a
ac¸a˜o ser linear em φ¯. Ale´m disso, realizando a integrac¸a˜o funcional em Dφ¯
∫
Dηe
(
− η22
) ∫
Dφ
∫
Dφ¯e−
∫
φ¯(∂t−D∇2)φ+2λ0φ¯φ2+iη
√
2λ0φ¯φ
cria-se uma func¸a˜o δ que da´ origem a` equac¸a˜o
∂tφ= D∇2φ−2λ0φ2+ i
√
2λ0ηφ, (B.40)
onde η representa uma varia´vel estoca´stica Gaussiana com
〈η(x, t)〉= 0
e
〈η(x, t)η(x′, t ′)〉=−2λ0φ(x, t)2δ(x−x′)δ(t− t ′),
uma EDPE com ruido complexo! E´ razoa´vel que quando o nu´mero de partı´culas diminui, a intensidade do
ruı´do tambe´m diminua. Este fato esta´ expresso explı´citamente nesta u´ltima equac¸a˜o envolvendo o ruido η
na proporcionalidade direta entre 〈ηη′〉 e φ2. O aparecimento do sinal negativo para a correlac¸a˜o do ruı´do
e´ apenas consequeˆncia do fato de φ ser complexo e, portanto, na˜o poder ser diretamente interpretado como
a densidade das partı´culas, como se poderia supor de inı´cio. Existe uma raza˜o fı´sica do porqueˆ de na˜o
podermos obter um ruido branco real neste problema. Uma dada partı´cula, uma vez que na˜o aniquilada,
tera´ “varrido” uma a´rea ao seu redor sem qualquer partı´cula dentro. Desta forma, as partı´culas devem ser
anti-correlacionadas. O campo φ como uma quantidade flutuante na˜o e´ a mesma grandeza da densidade de
partı´culas. Sa˜o os valores esperados de φ e da densidade de partı´culas e´ que sa˜o os mesmos. Para vermos
isso, vamos calcular a densidade me´dia de partı´culas denotada por n≡ a†aˆ :
n = 〈0|eaˆa†aˆe−Hˆt |ψ(0)〉
= 〈0|(1+a†)eaˆaˆ|ψ(t)〉 (usando equac¸a˜o (B.11) e equac¸a˜o (B.65))
= 〈0|eaˆaˆ|ψ(t)〉+ 〈0|a†eaˆaˆ|ψ(t)〉
= 〈P |aˆ|ψ(t)〉 (usando 〈0|a† = 0 e equac¸a˜o (A.25)) (B.41)
142
B.2. Estados coerentes
Agora na formulac¸a˜o de integral de caminho
〈P |aˆ|ψ(t)〉=
∫
Dφ Dφ¯ φ e−S∫
Dφ Dφ¯ e−S
= 〈φ〉. (B.42)
Portanto, n = 〈φ〉. Apesar de o primeiro momento para a densidade de partı´culas e para o campo se-
rem iguais, isto na˜o e´ necessariamente va´lido para momentos maiores. Por exemplo, a densidade me´dia
quadra´tica n2 ≡ (a†aˆ)2 pode ser calculada de maneira ana´loga ao ca´lculo de n :
n2 = 〈0|eaˆ(a†aˆ)2|ψ(t)〉
= 〈0|eaˆa† aˆa†︸︷︷︸
= a†aˆ+1
aˆ|ψ(t)〉
= 〈0|eaˆa†(a†aˆ+1)aˆ|ψ(t)〉
= 〈0|eaˆ(a†)2aˆ2|ψ(t)〉+ 〈0|eaˆa†aˆ|ψ(t)〉
= 〈P |aˆ2|ψ(t)〉+ 〈P |aˆ|ψ(t)〉, (usando equac¸a˜o (B.11) (B.43)
ou seja,
n2 = 〈φ2〉+ 〈φ〉.
De modo geral, se φ possui distribuic¸a˜o Gaussiana, como deve ser o caso da difusa˜o pura, enta˜o a densi-
dade n deve possuir distribuic¸a˜o de Poisson. Isto e´ esperado, ja´ que um passeio aleato´rio simples possui
estatı´stica Poissoniana. O efeito das reac¸o˜es e´ modificar isso. Em certo sentido, o “ruı´do” η em (B.40) re-
presenta somente a parte fı´sica do ruı´do que se origina na “discreteza” dos processos de reac¸a˜o. Entretanto,
como esta parte do ruı´do na˜o pode ser verdadeiramente desmembrada do ruı´do fı´sico “total” que inclui o
associado a` difusa˜o, na˜o existe necessidade para suas correlac¸o˜es serem positivas.
Existe uma maneira de se obter uma EDPE para a densidade de partı´culas. Partindo da ac¸a˜o (B.36),
aplicamos a transformac¸a˜o na˜o linear de Cole-Hopf φ˜ = eρ˜, φ = ρe−ρ˜ tal que φ˜φ = ρ com Jacobiano
1. Esta transformac¸a˜o resulta em φ˜∂tφ= ∂t [ρ(1− ρ˜)]+ ρ˜∂tρ e, omitindo contribuic¸o˜es de fronteira, temos
−Dφ˜∇2φ=−Dφ∇2φ˜=−Dρ[∇2ρ˜+(∇ρ˜)2] para o termo de difusa˜o. O termo de interac¸a˜o fica representado
por −λ0(1− φ˜2)φ2 = λ0ρ2(1−e−2ρ˜) = 2λ0ρ˜ρ2−2λ0ρ˜2ρ2+ · · · . Vemos que o termo quadra´tico em ρ˜ tem
agora sinal oposto ao que tinha antes na representac¸a˜o em φ˜, e portanto corresponde a um ruido real, no
lugar do imagina´rio. Pore´m, truncar a se´rie em segunda ordem na˜o e´ justifica´vel e, ale´m disso, descrever a
dinaˆmica do sistema em termos de ρ e ρ˜ atrave´s da transformac¸a˜o de Cole-Hopf tem o custo adicional da
incorporac¸a˜o de um termo de “ruido de difusa˜o” associado ao acoplamento na˜o linear −Dρ(∇ρ˜)2.
Uma conclusa˜o importante que tiramos da discussa˜o acima e´ que simplesmente escrever as equac¸o˜es de
campo me´dio para a reac¸a˜o de aniquilac¸a˜o e adicionar ruido Gaussiano real, como comumente se faz, na˜o
necessariamente retorna as EDPE’s apropriadas. Outra observac¸a˜o e´ que a linearizac¸a˜o feita em φ¯ requer
que este campo aparec¸a quadraticamente na ac¸a˜o e, portanto, apenas reac¸o˜es de duas partı´culas podem ser
escritas em termos de EDPE’s. Se a reac¸a˜o for por exemplo da forma 3A→ /0, na˜o poderemos representar
o sistema como um conjunto de EDPE’s com ruido multiplicativo.
B.2. Estados coerentes
Neste apeˆndice faremos uma breve revisa˜o do conceito de estados coerentes e utilizaremos o oscilador
harmoˆnico como exemplo.
Primeiramente poderı´amos nos perguntar: o que e´ um estado coerente? Para responder a esta pergunta,
vamos primeiramente relembrar o conceito de pacote de onda de incerteza mı´nima. O estado de mı´nima
energia do oscilador harmoˆnico quaˆntico e´ representado por um pacote de onda de incerteza mı´nima. Para
143
Apeˆndice B. Formulac¸a˜o contı´nua em uma teoria de campos
vermos isso, utilizamos as seguintes representac¸o˜es para os operadores posic¸a˜o xˆ e momento pˆ em termos
dos operadores criac¸a˜o aˆ† e destruic¸a˜o aˆ :‡
x2 =
~
2mω
(a+a†)2 (B.44)
e
p2 =−mω~
2
(a−a†)2. (B.45)
Como
〈0|(a+a†)(a+a†)|0〉= 〈0|aa†|0〉= 1 (B.46)
e
〈0|(a−a†)(a−a†)|0〉=−〈0|aa†|0〉=−1, (B.47)
segue que
〈x2〉〈p2〉=−~
2
4
1(−1) = ~
2
4
e lembrando que 〈x〉= 〈p〉= 0, temos
〈(∆x)2〉〈(∆p)2〉= ~
2
4
. (B.48)
Poderı´amos agora perguntar em que circunstaˆncia um estado gene´rico |n〉 e´ tambe´m representado por um
pacote de onde de incerteza mı´nima. Correspondendo a (B.46) e (B.47) temos
〈n|(a+a†)(a+a†)|n〉= 〈n|aa†+a†a|n〉= 〈n|2a†a+[a,a†]|n〉= 2n+1
e similarmente
〈n|(a−a†)(a−a†)|n〉=−(2n+1),
o que implica
〈(∆x)2〉n〈(∆p)2〉n = ~
2
4
(2n+1)2 (B.49)
e enta˜o |n〉 na˜o e´ mı´nimo. Claramente o fato crucial para que |0〉 seja de mı´nima incerteza e´
a|0〉= 0 ⇒ 〈0|a†a|0〉= 0.
E´ natural esperar que outros pacotes de onda de incerteza mı´nima com valores esperados para x e p dife-
rentes de zero sejam ainda autofunc¸o˜es de a :
a|α〉= α|α〉 (B.50)
o que implica em 〈α|a†a|α〉= α〈α|a†|α〉= αα∗〈α|α〉= |α|2. E´ fa´cil checar que |α〉 define um pacote de
onde de incerteza mı´nima
〈α|(a+a†)|α〉= (α+α∗)
〈α|(a−a†)|α〉= (α−α∗)
‡Por simplicidade de notac¸a˜o omitiremos de agora em diante o chape´uˆdos operadores das equac¸o˜es.
144
B.2. Estados coerentes
〈α|(a+a†)(a+a†)|α〉= (α+α∗)2+1
〈α|(a−a†)(a−a†)|α〉= (α−α∗)2−1
e portanto
〈(∆x)2〉α = 〈x2〉α−〈x〉2α =
~
2mω
〈(∆p)2〉α = 〈p2〉α−〈p〉2α =
~mω
2
e, de acordo com as equac¸o˜es acima
〈(∆x)2〉α〈(∆p)2〉α = ~
2
4
. (B.51)
Assim, os estados |α〉 definidos por (B.48) satisfazem a condic¸a˜o de incerteza mı´nima. Eles sa˜o chamados
de estados coerentes.
B.2.1. Estados coerentes na representac¸a˜o n
Na base |n〉 o estado coerente se escreve
|α〉=∑
n
cn|n〉=∑
n
|n〉〈n|α〉. (B.52)
Como
|n〉= (a
†)n√
n!
|0〉 (B.53)
temos
〈n|α〉= α
n
√
n!
〈0|α〉 (B.54)
e enta˜o
|α〉= 〈0|α〉
∞
∑
n=0
αn√
n!
|n〉. (B.55)
A constante 〈0|α〉 e´ determinada por normalizac¸a˜o como segue:
1 =∑
n
〈α|n〉〈n|α〉= |〈0|α〉|2
∞
∑
m=0
|α|2m
m!
= |〈0|α〉|2e|α|2 .
Resolvendo esta equac¸a˜o para 〈0|α〉 teremos:
〈0|α〉= e− 12 |α|2 (B.56)
a menos de um fator de fase. Substituindo em (B.54) obtemos a forma final:
|α〉= e− 12 |α|2
∞
∑
n=0
αn√
n!
|n〉. (B.57)
145
Apeˆndice B. Formulac¸a˜o contı´nua em uma teoria de campos
Uma expressa˜o u´til pode ser obtida usando a forma explı´cita para |n〉, equac¸a˜o (B.53):
∞
∑
n=0
αn√
n!
|n〉=
∞
∑
n=0
αn
n!
(a†)n|0〉,
que implica
|α〉= e− 12 |α|2+αa† |0〉= eαa†−α∗a|0〉. (B.58)
B.2.2. Ortogonalidade e relac¸o˜es de completeza
Vamos calcular a sobreposic¸a˜o entre os estados coerentes usando a equac¸a˜o (B.57).
〈α|β〉=∑
n
〈α|n〉〈n|β〉= e− 12 |α|2− 12 |β|2∑
n
(α∗β)n
n!
= e−
1
2 |α|2− 12 |β|2+α∗β. (B.59)
Similarmente,
〈β|α〉= e− 12 |α|2− 12 |β|2+β∗α. (B.60)
Enta˜o
|〈α|β〉|2 = 〈α|β〉〈β|α〉= e−|α|2−|β|2+α∗β+αβ∗
ou
|〈α|β〉|2 = e−|α−β|2 . (B.61)
Como 〈α|β〉 6= 0 para α 6= β, dizemos que o conjunto {|α〉} e´ supercompleto§. Mas ainda e´ possı´vel
encontrar uma relac¸a˜o de completeza:
∫
d2α|α〉〈α|=
∫
d2αe−|α|
2∑
m,n
(a∗)nαm√
n!m!
|m〉〈n| (B.62)
onde d2α significa somar sobre todos os valores complexos de α, integrando sobre todo o plano complexo.
Escrevendo α na forma polar:
α= reiφ ⇒ d2α= dφdr r, (B.63)
obtemos
∫
d2αe−|α|
2
(α∗)nαm =
∫ ∞
0
dr r e−r
2
rm+n
∫ 2pi
0
dφei(m−n)φ = 2piδm,n
1
2
∫ ∞
0
dr2 (r2)m e−r
2
= pim!δm,n.
Usando este resultado obtemos finalmente∫
d2α|α〉〈α|= pi∑
n
|n〉〈n|= pi,
§Overcomplete
146
B.3. Prova propriedade de 〈P |
ou, equivalentemente,
∫ d2α
pi
|α〉〈α|= 1. (B.64)
B.3. Prova propriedade de 〈P |
Da definic¸a˜o do operador projec¸a˜o 〈P |, vem
〈P |a†i = 〈0|e∑i aˆia†i = 〈0|eaˆia†i e∑ j 6=i aˆ j .
Agora,
eaˆa† = [eaˆ,a†]+a†eaˆ
=
[
eaˆ,− ∂
∂aˆ
]
+a†eaˆ
=
[
∂
∂aˆ
,eaˆ
]
+a†eaˆ
=
∂
∂aˆ
eaˆ− eaˆ ∂
∂aˆ
+a†eaˆ
= eaˆ
∂
∂aˆ
+ eaˆ− eaˆ ∂
∂aˆ
+a†eaˆ
= eaˆ+a†eaˆ
⇒ eaˆa† = (1+a†)eaˆ (B.65)
Usando este resultado, temos
〈P |e†i = 〈0|(1+a†i )eaˆie∑ j 6=i aˆ j
= 〈0|e∑i aˆi (usando 〈0|a†i = 0)
= 〈P |. (B.66)
Ale´m disso,
〈P |0〉 = 〈0|
(
1+∑
i
aˆi+ · · ·
)
|0〉= 1, (B.67)
onde usamos aˆi|0〉= 0.
147
Apeˆndice B. Formulac¸a˜o contı´nua em uma teoria de campos
B.4. Generalizac¸a˜o da formulac¸a˜o em integral funcional para d
dimenso˜es
Seja Hˆ o “quase” Hamiltoniano para o processo. Sejam aˆi, a
†
i e |0〉 o operador de aniquilac¸a˜o, de criac¸a˜o
e o estado de va´cuo respectivamente. O subscrito i e´ um marcador que denota o sı´tio da rede.
A resoluc¸a˜o da identidade em termos de estados coerentes para cada rede sera´
1 =
∫ (
∏
i
d2φi
pi
)
|{φ}〉〈{φ}| (B.68)
onde |{φ}〉 = ⊗i|φi〉. A expressa˜o em integral de caminho ana´loga a` equac¸a˜o (B.15) para o caso de di-
mensa˜o 0 sera´
|ψ(t)〉= lim
∆t→0
N −1
∫ ( t
∏
τ=0
∏
i
d2φi,τ
)
|{φ}t〉
[
t
∏
τ=∆t
∏
i
〈φi,τ|e−Hˆ ∆t |φi,τ−∆t〉
]
〈{φ}0|ψ(0)〉 (B.69)
O operador Aˆ correspondente a um observa´vel A({n}) e´ obtido substituindo ni por a†i aˆi. O valor esperado
ana´logo a (B.16) sera´ dado por
〈A〉 = lim
∆t→0
N −1
∫ ( t
∏
τ=0
∏
i
d2φi,τ
)
〈{1}|Aˆ|{φ}t〉
×
[
t
∏
τ=∆t
∏
i
〈φi,τ|e−Hˆ ∆t |φi,τ−∆t〉
]
〈{φ}0|ψ(0)〉. (B.70)
O termo do meio na equac¸a˜o de cima e´ dado por
t
∏
τ=∆t
∏
i
〈φi,τ|e−Hˆ ∆t |φi,τ−∆t〉 = exp
{
1
2∑i
(|φi,t |2−|φi,0|2)
}
× exp
{
−
t
∑
τ=∆t
[
∑
i
φ∗i,τ(φi,τ−φi,τ−∆t)+ Hˆ({φ∗τ},{φτ−∆t})∆t
]}
. (B.71)
O primeiro termo em (B.70) sera´
〈{1}|Aˆ|{φ}t〉 = 〈{1}|A({φ}t)|{φ}t〉
= A({φ}t)〈{1}|{φ}t〉
= A({φ}t)∏
i
exp
{
−1
2
− 1
2
|φi,t |2+φi,t
}
∝ A({φ}t)exp
{
∑
i
[
−1
2
|φi,t |2+φi,t
]}
(B.72)
Assumindo distribuic¸a˜o inicial como sendo Poisson, obtemos para o terceiro termo de (B.70):
148
B.4. Generalizac¸a˜o da formulac¸a˜o em integral funcional para d dimenso˜es
〈{φ}0|ψ(0)〉 ∝ ∏
i
exp
{
−1
2
|φi,0|2+φ∗i,0n¯0
}
∝ exp
{
∑
i
[
−1
2
|φi,0|2+φ∗i,0 n¯0
]}
(B.73)
Substituindo as equac¸o˜es (B.71), (B.72) e (B.73) em (B.70) e absorvendo todas as constantes no fator de
normalizac¸a˜o, teremos
〈A〉 = N −1 lim
∆t→0
∫ ( t
∏
τ=0
∏
i
d2φi,τ
)
A({φ}t)exp
{
∑
i
[φi,t −|φi,0|2+φ∗i,0 n¯0]
}
× exp
{
−
t
∑
τ=∆t
[
∑
i
φ∗i,τ(φi,τ−φi,τ−∆t)+ Hˆ({φ∗τ},{φτ−∆t})∆t
]}
= N −1
∫ (
∏
i
Dφi Dφ∗i
)
A[{φ(t)}]exp
{
∑
i
[φi(t)−|φi(0)|2+φ∗i (0)n¯0]
}
× exp
{
−
∫ t
0
dt ′
[
∑
i
φ∗i (t
′)
∂φi(t ′)
∂t ′
+ Hˆ[{φ∗(t ′)},{φ(t ′)}]
]}
= N −1
∫ (
∏
i
Dφi Dφ∗i
)
A[{φ(t)}]exp{−S[{φ∗},{φ}]t0} (B.74)
onde
S[{φ∗},{φ}]t f0 = −∑
i
[φi(t f )−|φi(0)|2+φ∗i (0)n¯0]
+
∫ t f
0
dt
[
∑
i
φ∗i (t)
∂φi(t)
∂t
+ Hˆ[{φ∗(t)},{φ(t)}]
]
(B.75)
O termo contendo φ(0) pode ser desprezado da ac¸a˜o ja´ que na˜o temos φ∗ para t < 0. Assim,
S[{φ∗},{φ}]t f0 = −∑
i
[φi(t f )+φ∗i (0)n¯0]
+
∫ t f
0
dt
[
∑
i
φ∗i (t)
∂φi(t)
∂t
+ Hˆ[{φ∗(t)},{φ(t)}]
]
(B.76)
Deslocando o campo φ∗ para 1+ φ¯, obtemos
S[{φ∗},{φ}]t f0 = −∑
i
[φi(0)+{1+ φ¯i(0)}n¯0]
+
∫ t f
0
dt
[
∑
i
φ¯i(t)
∂φi(t)
∂t
+ Hˆ[{φ¯(t)},{φ(t)}]
]
(B.77)
149
Apeˆndice B. Formulac¸a˜o contı´nua em uma teoria de campos
Novamente podemos descprezar o termo φ(0) e a constante e obtermos
S[{φ¯},{φ}]t f0 =−∑
i
φ¯i(0) n¯0+
∫ t f
0
dt
[
∑
i
φ¯i(t)
∂φi(t)
∂t
+ Hˆ[{φ¯(t)},{φ(t)}]
]
(B.78)
Limite contı´nuo
Obtemos o limite contı´nuo seguindo o seguinte esquema:
∑
i
→
∫ ddx
hd
φi(t)→ hd φ(x, t)
φ¯i(t)→ φ¯(x, t)
(B.79)
onde h e´ a constante da rede.
Vamos assumir que cada processo estoca´stico elementar que ocorre em cada sı´tio ocorra em toda a rede e
que ainda haja difusa˜o entre os sı´tios vizinhos. Vamos assumir ainda que as taxas de transic¸a˜o e de difusa˜o
sa˜o independentes dos sı´tios. Desta forma, o Hamiltoniano completo pode ser dividido em duas partes
como segue:
Hˆ = Hˆdifusa˜o+ Hˆreac¸a˜o. (B.80)
Para um sistema de uma espe´cie e constante de difusa˜o D
Hˆdifusa˜o = D∑
〈i, j〉
(φ¯i− φ¯ j)(φi−φ j)
= D
∫ ddx
hd
{h ∇ ¯φ(x)} · {h ∇φ(x) hd}} (limite contı´nuo)
= Dh2
∫
ddx ∇φ¯(x) ·∇φ(x)
= −Dh2
∫
ddx φ¯(x) ∇2φ(x) (B.81)
onde no u´ltimo passo realizamos uma integrac¸a˜o por partes e desprezamos os termos de fronteira.
A parte de reac¸a˜o do Hamiltoniano pode ser escrita como
Hˆreac¸a˜o = ∑
i
Hˆ ′i
=
∫ ddx
hd
Hˆ ′(x) (B.82)
onde Hˆ ′i e´ a parte de reac¸a˜o do Hamiltoniano em cada sı´tio i. Substituindo na expressa˜o para a ac¸a˜o,
obteremos a ac¸a˜o no limite contı´nuo
150
B.4. Generalizac¸a˜o da formulac¸a˜o em integral funcional para d dimenso˜es
S[φ¯,φ] = −
∫ ddx
hd
φ¯(x,0) n¯0
+
∫ t f
0
dt
∫ ddx
hd
[
φ¯(x, t)
∂φ(x, t)
∂t
hd−Dhd+2 φ¯(x) ∇2φ(x)+ Hˆ ′(x)
]
= −
∫ ddx
hd
φ¯(x,0) n¯0
+
∫ t f
0
dt
∫
ddx
[
φ¯(x, t)
∂φ(x, t)
∂t
−Dh2 φ¯(x) ∇2φ(x)+ 1
hd
Hˆ ′(x)
]
(B.83)
Podemos redefinir todas as constantes de modo a absorver todos os fatores de h, e assim, obteremos final-
mente a ac¸a˜o
S[φ¯,φ]t f0 =−
∫
ddx φ¯(x,0) n¯0+
∫ t f
0
dt
∫
ddx
[
φ¯(x, t)(∂t −D∇2)φ(x, t)+ Hˆ ′(x)
]
. (B.84)
151
152
Refereˆncias Bibliogra´ficas
[1] Santos, R. V. and Silva, L. M. 08 2013 PLoS ONE 8(8), e69131.
(Cited on pages i, iii, and 83.)
[2] Santos, R. V. and Silva, L. M. (2013) Journal of Theoretical Biology 335, 79 – 87.
(Cited on pages i, iii, and 101.)
[3] deOliveira, M. M., Santos, R. V., and Dickman, R. (2012) Phys. Rev. E 86, 011121.
(Cited on pages i, iii, 19, and 20.)
[4] Santos, R. V. and Dickman, R. (2013) Journal of Statistical Mechanics: Theory and Experiment
2013(07), P07004.
(Cited on pages i, iii, and 31.)
[5] Santos, R. V. The importance of being discrete in sex. Submitted to PLOS ONE (2013).
(Cited on pages i, iii, 53, and 54.)
[6] Santos, R. V. Discreteness induced coexistence. To appear in Physica A (2013).
(Cited on pages i, iii, and 111.)
[7] Plischke, M. and Bergersen, B. (2006) Equilibrium Statistical Physics, World Scientific Publishing
Company, Incorporated, .
(Cited on page 1.)
[8] Goldenfeld, N. July 1992 Lectures On Phase Transitions And The Renormalization Group (Frontiers
in Physics, 85), Westview Press, illustrated edition edition.
(Cited on page 1.)
[9] Amit, D. (1984) Field theory, the renormalization group, and critical phenomena, World Scientific, .
(Cited on page 1.)
[10] Binney, J. J., Dowrick, N. J., Fisher, A. J., and Newman, M. E. J. (1992) The theory of critical
phenomena, Oxford University Press, Oxford.
(Cited on page 1.)
[11] Cardy, J. (1996) Scaling and Renormalization in Statistical Physics, Cambridge University Press, .
(Cited on page 1.)
[12] Parisi, G. (1988) Statistical Field Theory, Frontiers in PhysicsAddison-Wesley, .
(Cited on pages 1 and 2.)
[13] Mussardo, G. (2010) Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical
Physics, Oxford Graduate TextsOUP Oxford, .
(Cited on page 1.)
[14] vanSaarloos, W. (2003) Physics reports 386(2), 29–222.
(Cited on page 1.)
[15] McKane, A. and Newman, T. (2005) Physical review letters 94(21), 218102.
(Cited on page 1.)
[16] Butler, T. and Goldenfeld, N. (2011) Physical Review E 84(1), 011112.
(Cited on page 1.)
[17] Lande, R. (1998) Oikos pp. 353–358.
153
Refereˆncias Bibliogra´ficas
(Cited on page 1.)
[18] Dennis, B. (2002) Oikos 96(3), 389–401.
(Cited on page 1.)
[19] Allee, W. and Bowen, E. S. (1932) Journal of Experimental Zoology 61(2), 185–207.
(Cited on page 1.)
[20] Allee, W., Park, O., Emerson, A., Park, T., and Schmidt, K. (1949) Principles of animal ecology, WB
Saunders London, .
(Cited on page 1.)
[21] Paulsson, J. (2004) Nature 427(6973), 415–418.
(Cited on page 1.)
[22] Thattai, M. and Van Oudenaarden, A. (2001) Proceedings of the National Academy of Sciences
98(15), 8614–8619.
(Cited on page 2.)
[23] Negele, J. W. and Orland, H. November 1998 Quantum Many-particle Systems (Advanced Book
Classics), Westview Press, .
(Cited on pages 2, 129, and 133.)
[24] Murray, J. (2003) Mathematical Biology II: Spatial Models and Biomedical Applications, Interdisci-
plinary Applied MathematicsSpringer, .
(Cited on page 3.)
[25] Cantrell, R. and Cosner, C. (2003) Spatial Ecology via Reaction-Diffusion Equations, Wiley Series
in Mathematical and Computational BiologyJohn Wiley & Sons, .
(Cited on page 3.)
[26] Toussaint, D. and Wilczek, F. (1983) The Journal of Chemical Physics 78, 2642.
(Cited on page 3.)
[27] Tauber, U. C., Howard, M., and Vollmayr-Lee, B. P. (2005) J Phys A-Math Gen 38(17), 79.
(Cited on pages 3, 8, 9, 16, 31, and 141.)
[28] Peskin, M. and Schroeder, D. (1995) An Introduction To Quantum Field Theory, Advanced Book
ProgramWestview Press, .
(Cited on pages 5, 7, and 10.)
[29] Mattu¨ck, R. (1967) A guide to Feynman diagrams in the many-body problem, European physics
seriesMcGraw-Hill Pub. Co., .
(Cited on page 7.)
[30] Zinn-Justin, J. (2002) Quantum Field Theory and Critical Phenomena (International Series of Mono-
graphs on Physics), Clarendon Press, 4 edition.
(Cited on page 12.)
[31] Gustafson, K. E. July 1997 Introduction to Partial Differential Equations and Hilbert Space Methods,
Dover Publications, 3 edition.
(Cited on page 16.)
[32] Cardy, J. (2006) A+ A 100, 26.
(Cited on page 17.)
[33] Peliti, L. Path integral approach to birth-death processes on a lattice Technical report March 1985.
(Cited on pages 20 and 125.)
[34] Gabel, A., Meerson, B., and Redner, S. (2013) Phys. Rev. E 87, 010101.
(Cited on pages 31 and 32.)
[35] Smith, J. and Dawkins, R. (1993) The Theory of Evolution, Canto SeriesCambridge University Press,
.
(Cited on page 53.)
[36] Jarne, P. and Auld, J. (2006) Evolution 60(9), 1816–1824.
(Cited on page 53.)
[37] Eppley, S. and Jesson, L. (2008) Journal of evolutionary biology 21(3), 727–736.
(Cited on page 53.)
[38] Lehtonen, J., Jennions, M. D., and Kokko, H. (2012) Trends in ecology & evolution 27(3), 172–178.
(Cited on page 53.)
[39] Meirmans, S., Meirmans, P. G., and Kirkendall, L. R. (2012) The Quarterly Review of Biology 87(1),
154
Refereˆncias Bibliogra´ficas
19–40.
(Cited on page 53.)
[40] Otto, S. P. and Lenormand, T. (2002) Nature Reviews Genetics 3(4), 252–261.
(Cited on page 53.)
[41] Driessens, G., Beck, B., Caauwe, A., Simons, B. D., and Blanpain, C. August 2012 Nature 488(7412),
527–530.
(Cited on page 83.)
[42] Chen, J., Li, Y., Yu, T.-S., McKay, R. M., Burns, D. K., Kernie, S. G., and Parada, L. F. August 2012
Nature 488(7412), 522–526.
(Cited on page 83.)
[43] Ishizawa, K., Rasheed, Z., Karisch, R., Wang, Q., Kowalski, J., Susky, E., Pereira, K., Karamboulas,
C., Moghal, N., Rajeshkumar, N., et al. (2010) Cell stem cell 7(3), 279–282.
(Cited on page 83.)
[44] Stewart, J., Shaw, P., Gedye, C., Bernardini, M., Neel, B., and Ailles, L. (2011) Proceedings of the
National Academy of Sciences 108(16), 6468.
(Cited on page 83.)
[45] Vargaftig, J., Taussig, D., Griessinger, E., Anjos-Afonso, F., Lister, T., Cavenagh, J., Oakervee, H.,
Gribben, J., and Bonnet, D. (2011) Leukemia.
(Cited on page 83.)
[46] Sarry, J., Murphy, K., Perry, R., Sanchez, P., Secreto, A., Keefer, C., Swider, C., Strzelecki, A.,
Cavelier, C., Re´cher, C., et al. (2011) The Journal of Clinical Investigation 121(1), 384.
(Cited on page 83.)
[47] Zhong, Y., Guan, K., Zhou, C., Ma, W., Wang, D., Zhang, Y., and Zhang, S. (2010) Cancer letters
292(1), 17–23.
(Cited on page 83.)
[48] Baker, M. (2008) Nature 456(7222), 553.
(Cited on page 83.)
[49] Johnston, M., Maini, P., Jonathan Chapman, S., Edwards, C., and Bodmer, W. (2010) Journal of
theoretical biology 266(4), 708–711.
(Cited on page 83.)
[50] Baker, M. (2008) Nature Reports Stem Cells.
(Cited on page 83.)
[51] Schatton, T., Murphy, G., Frank, N., Yamaura, K., Waaga-Gasser, A., Gasser, M., Zhan, Q., Jordan,
S., Duncan, L., Weishaupt, C., et al. (2008) Nature 451(7176), 345–349.
(Cited on page 83.)
[52] Boiko, A., Razorenova, O., van deRijn, M., Swetter, S., Johnson, D., Ly, D., Butler, P., Yang, G.,
Joshua, B., Kaplan, M., et al. (2010) Nature 466(7302), 133–137.
(Cited on page 83.)
[53] (2011) Nature Medicine 17, 278 – 279.
(Cited on page 83.)
[54] Quintana, E., Shackleton, M., Sabel, M., Fullen, D., Johnson, T., and Morrison, S. (2008) Nature
456(7222), 593–598.
(Cited on page 83.)
[55] Morrison, S. and Kimble, J. (2006) Nature 441(7097), 1068–1074.
(Cited on page 83.)
[56] Rapp, U., Ceteci, F., Schreck, R., et al. (2008) Cell Cycle 7(1), 45.
(Cited on page 83.)
[57] Chaffer, C., Brueckmann, I., Scheel, C., Kaestli, A., Wiggins, P., Rodrigues, L., Brooks, M., Rei-
nhardt, F., Su, Y., Polyak, K., et al. (2011) Proceedings of the National Academy of Sciences
108(19), 7950.
(Cited on page 83.)
[58] Strauss, R., Hamerlik, P., Lieber, A., and Bartek, J. (2012) Molecular Therapy.
(Cited on page 83.)
[59] Hardin, G. et al. (1960) Science 131(3409), 1292–1297.
155
Refereˆncias Bibliogra´ficas
(Cited on page 111.)
[60] Gardiner, C. (2009) Stochastic methods: a handbook for the natural and social sciences, Springer
series in synergeticsSpringer, .
(Cited on page 111.)
[61] Doi, M. (1976) J Phys A-Math Gen 9(9), 1479.
(Cited on pages 125 and 128.)
[62] Schulman, L. S. December 2005 Techniques and Applications of Path Integration, Dover Publicati-
ons, .
(Cited on page 133.)
[63] Fan, H. and Wu¨nsche, A. (2001) The European Physical Journal D-Atomic, Molecular, Optical and
Plasma Physics 15(3), 405–412.
(Cited on page 133.)
[64] Martin, P. C., Siggia, E. D., and Rose, H. A. July 1973 Physical Review A 8(1), 423+.
(Cited on page 142.)
[65] Janssen, H. (1976) Zeitschrift fu¨r Physik B Condensed Matter 23(4), 377–380.
(Cited on page 142.)
[66] deDominicis, C. (1976) Le Journal de Physique Colloques 37(C1), 1–1.
(Cited on page 142.)
156