Métodos estocásticos e de teoria de campos aplicados a problemas motivados na ecologia e oncologia

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 nonequilibrium 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 renormalization 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 subsequent 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.