Stochastic Co-Evolution Models Of Two Populations

Evolutionary game theory originated from the field of biomathematics, which is used by many scholars to describe the behavior dynamics of individuals in a single population and the evolutionary dynamics of a single species. But, in many cases in the real world, the population is not isolated, and at least interacts with one of the other populations. Therefore, it is significant to study the co-evolution of the two populations with mutual interaction.In the first chapter, we introduce the research background for the co-evolution of two populations,the preliminary knowledge for the games between the two populations with two payoff matrices, and the main research work of this paper. In the second chapter, we consider the stochastic evolutionary dynamics of the two populations with both internal and external games. The individuals of the games can not only come from different populations, but also from the same population. The game between different populations is asymmetric, and the game between the same species is symmetric. We first introduce the game models for two populations which are described by four payoff matrices, and then analyze the stochastic co-evolution of the games by selection-mutation Moran process. At last we derive the expressions for equilibrium frequencies of strategies under neutral selection and weak selection. In the third chapter, we first introduce the model of volunteer's dilemma under the co-evolution mechanism of the two populations, study the evolutionary dynamics for the model in the infinite populations, and analyze the stability of the equilibrium points of the replication dynamic equations. We then study the stochastic evolutionary dynamics for the model in finite populations by selection-mutation Moran process and derive the expressions for joint frequencies at equilibrium under weak selection. In the fourth chapter, we mainly study the games of two infinite populations in continuous strategy space. We construct two models for the evolutional games of continuous strategies. The one only considers the games by individuals from different populations, while the other considers the games by individuals both from different populations and in the same populations.We derive the dynamic equations of continuous strategies for the two models, and discuss the existence and stability of the internal equilibrium points of the continuous dynamic equations for the two game models. Finally the two models are compared.