The Stochastic Evolutionary Models Based On Single And Grouping Populations

Evolutionary game theory is derived from the thoughts of evolutionary biology, which involves mathematics, physics, biology, game theory and other subjects. Evolutionary game theory develops rapidly in recent years and has been used widely in many fields such as evolution of ecological system, neural network, swarm intelligence, cognitive science, self-organization emerging behavior, dynamics of economy, etc. Replicator dynamics equation provides effective approaches to the study of the deterministic dynamics of infinite populations. However, the real populations are finite, so it is very meaningful to study the dynamics of finite populations by using the stochastic methods.This paper mainly studies the stochastic evolutionary model based on single and grouping populations. Because of the importance of fixation time in finite populations, from the integral formulas deduced from the backward Kolmogorow equation chapter 2 derives the analytical expressions of the conditional fixation times of the Wright-Fisher process and Moran process based on single population under weak selection. We also make a comparison of the conditional fixation times of the two frequency-dependent processes. In chapter 3, we studies the asynchronous update processes, i.e., Moran process, pairwise comparison process, and local update process, based on grouping population. We get the evolutionary conditions of strategies for the three asynchronous update processes under weak selection from the expressions for transition probabilities. Then we use the approximate formulas of within-group invariance and between-groups invariance to derive the conditions for the evolution of cooperation in Prisoner’s dilemma game based on pairwise comparison process. We also introduce mutation to the grouping population, and discuss the evolutionary dynamics under weak mutation. In chapter 4 we incorporate the more realistic punishment mechanism and revenge mechanism into the grouping population, and then we obtain the conditions for the evolution of cooperation.