Study On Dynamics Of Replicator Equation In Evolutionary Games

In evolutionary game theory,the study of dynamic behavior of replicator equation is an important subject.The idea of equivalent transformation between replicator equation and Lotka-Volterra equation is established in [2,3],but it is only discussed in the case of constant coefficients.Following their equivalent transformation idea,this paper generalizes and studies the corresponding transformation problem in the case of time-varying coefficients,based on the previous research results on Lotka-Volterra system,some new results on the dynamic behavior of replicator equation are obtained.The content and structure of this paper are as follows:The first chapter is the preface and the preparatory knowledge.This part briefly introduces the replicator equation and Lotka-Volterra equation,as well as some results of previous studies on these two equations respectively.In addition,the related concepts and results to be used in this paper are also given.Chapter 2 is one of the main parts of this paper.We mainly study the dynamical properties of the replicator equation with constant coefficients payoff matrix when9)= 2,3,4.Firstly,the replicator equation is transformed into the corresponding LotkaVolterra equation.The dynamic properties of the corresponding Lotka-Volterra equation are studied in detail,the corresponding classification of phase diagrams is obtained,then the relevant conclusions about the replication equation are derived.Chapter 3 mainly studies the replicator equation with time-varying payoff matrix.Firstly,the idea of equivalent transformation between Lotka-Volterra equation and replicator equation under constant coefficients in [2,3] is generalized.The transformation theorem of replicator equation with time-varying payoff matrix and Lotka-Volterra equation with time-varying coefficient is obtained.By applying this theorem,and combining some existing results on periodic solution,quasi-periodic solution and almost periodic solution of Lotka-Volterra equation with time-varying coefficient,some new sufficient conditions for the existence of Reciprocating oscillation function solutions for the corresponding time-varying coefficient replicator equation are obtained.Chapter 4 To illustrates the applicability of the theoretical results obtained in this paper,some theoretical analysis and computational simulation for a few of examples are given.