Caminhos da evolução: aplicações de teoria dos jogos, probabilidade e cálculo estocástico

Abstract

The famous quote “Nothing in biology makes sense, except in the light of evolution”, by Theodosius Dobzhansky, perfectly encapsulates the tremendous explanatory power of the theory of evolution in the face of the countless biological phenomena we observe today, or find preserved in the fossil record. Since it was independently proposed by Darwin and Wallace, the theory has been considered the best way to explain the diversity of life on Earth. The evidence that all living beings share a common origin only continues to accumulate over time. Today, research in this area is more interdisciplinary than ever, involving biologists, physicists, mathematicians, and computer scientists. The approach we take is to try to describe evolution through mathematical models, involving deterministic rate equations, but primarily through stochastic models, where the parameters of interest follow probability distributions. In Chapter 2, we study the replicator dynamics and introduce game theory into the study of evolution, addressing an interesting question about the escalation of aggression in animal conflicts. In Chapter 3, we extend the famous replicator equation to the case where a population of prey with two distinct types evolves in the presence of predators. In Chapter 4, we delve deeply into the Moran process so that we can use it in Chapter 5 to study an original group selection model, aiming to explain the emergence of cooperation. It is possible to show that, even losing to defectors within the same group, cooperators may have an overall advantage and have a probability close to 1 of dominating the population. Finally, in Chapter 6, we propose a simple mathematical model that encompasses the three basic principles of evolution: heredity, mutation, and selection. We demonstrate how it is possible to obtain demographic variations of a trait in the population in terms of variations at the reproductive level, and also how to predict the evolutionary path of a given trait in the long term, given the initial evolutionary path. In other words, we propose a model that aims to predict evolution.