Evolutionary Dynamics For Strategies Based On Birth–death Processes

Evolutionary game dynamics is one of active fields of multiple subjects, which involves mathematics, evolutionary biology, game theory and other subjects. It has been developed to a mature theoretical system in the past three decades. The theory of replicator dynamics provided powerful mathematical foundations to the study of evolutionary game dynamics for infinite populations. Because any real populations are finite, the methods of stochastic processes were introduced to the area for studying evolutionary dynamics of finite populations in recent years, and made this field develop rapidly.In Chapter 2, we study evolutionary dynamics of cooperation based on Moran process by two-level selection under active linking, and give the analytical approximation of fixation probability for each strategy under weak selection. We also derive the conditions for the evolution of cooperation in Prisoner's dilemma game and snowdrift game. Some numerical simulations are presented. Since fixation time is an important measurement for evolution of strategies, Chapter 3 mainly discusses the fixation times for local update processes under weak selection. We give the approximate presentations of fixation times of A and B strategies. Our results show that the conditional fixation times of a single A and a single B mutant are approximately identical, they depends only on the density dependent term of the payoff difference of two strategies, and the unconditional mean fixation time depends only on the constant term of the payoff difference. Finally, the relationship between population size, the entries of payoff matrix and the intensity of selection is illustrated by figures. In Chapter 4, we study the evolutionary dynamics of pairwise comparison processes under mutation, and obtain the fixation probabilities of stratigies in this process without mutation and the stochastic dynamics equation with mutation. When the system is under mutation, the processes are ergodic, and then the limit distributions of the stochastic processes are discussed.