Complete Classification And Dynamical Behavior Of Alternating Prisoner’s Dilemma Cycle Dominant

In this paper,based on the alternating prisoner’s dilemma payoff matrix,based on the second-order game matrix of any two-strategy game,by reducing its diagonal elements to zero,using the dominance theorem of the replication equation,we give a complete classification of the dominance cycle of the two-two game under the basic condition of sixteen strategies T>R>P>S,2 R>T+S.By introducing parameters a,b,c so that P=S+a,R=P+b,T=R+c.And then specifically discussing the polynomial with undetermined positive and negative,thus fully classifying the dominant cycles of two-two games with different payoff parameters T,R,P,S,proving that the maximum number of cycles of the sixteen-strategy game under the alternating prisoner’s dilemma is fourteen.By choosing appropriate parameters to ensure that the three-dimensional Lotka-Volterra system is stable at the positive equilibrium point,and then transforming the coefficient matrix of its linear system into a diagonal type,and then using the central manifold theorem to transform it into a two-dimensional system,and finally calculating the first-order focal quantities of the system,and then proving the existence of the limit rings of the four-loop system[S1,S15,S10,S8].