Probabilidades de fixação assintóticas para o processo de Moran com duas estratégias e o processo de Moran para três estratégias

Abstract

In this work we present a study on the Moran process [14] for two and three strategies. We begin by presenting a brief review on Evolutionary Game Theory and Birth-Death Processes, including the main results in the literature on the Moran process for two strategies. We also present some results of our authorship on birth-death processes. We continue with a study of the population dynamics in the Moran process with two strategies, when the size N of the population tends to infinity, seeking to make an analogy with the deterministic case. We present original results that show that when N is large enough just 5 out of the 8 evolutionary scenarios classified in [25] can occur. We also study the Moran process for three strategies. We will show that in the Moran process with three strategies, the fixation probabilities are given as a solution of a system of linear equations and not by an exact formula, as in the case of two strategies. We present general results, based on Markov chain coupling,that provide us with upper and lower bounds for the fixation probabilities. We also present some original particular results, considering the behavior of the fitnesses in some specific regions. Finally, we use the results presented on the Moran process for three strategies to try to understand, from the stochastic point of view, the problem of the evolution of cooperation treated in [21].