Probabilidades de fixação na aproximação de baixa migração do processo de Moran em metapopulações estruturadas por grafos

Abstract

In this thesis we present some original results based on the work of Yagoobi and Traulsen [44] on the low migration regime for the Moran process in network structured meta-populations with two types of individuals [26]. We investigate the fixation probability of both types in the case of frequency independent fitnesses and under two update rules: birth-death (Bd) and death-birth (dB). Our analysis includes two particular cases of meta-populations: two-patch meta-population and meta-star. In both cases, we use the low migration approximation to obtain exact formulas for the fixation probability with initial condition in an arbitrary homogeneous state of the meta-population. After that, we deduce an asymptotic formula for the fixation probabilities obtained in the low migration approximation; the precise description of the asymptotic limit will be given in due course. Furthermore, we apply the asymptotic approximation to quantify the size of the advantage or disadvantage that population structure imposes on the fixation of a mutation introduced uniformly into the meta-population. Based on our asymptotic results, we also establish conditions that permit us to infer whether the metapopulation structure under consideration acts as, for example, a suppressor or amplifier of natural selection. We do that analysis both in the Bd rule and in the dB rule. In particular, our results show that, contrary to what Yagoobi and Traulsen found, the meta-star under Bd rule may act as a suppressor of natural selection or even as reducer. In addtion, depending on the ratio between the size of the population at centre and the total size of population at leaves, the meta-star under dB rule can exhibit behaviour that is neither amplification nor suppression.