Almost Sure Exponential Stability in the Stochastic Delay Replicator Dynamics for Evolutionary Snowdrift Games

We investigate stochastic snowdrift games in which the payoff of players are disturbed by stochastic noise and time delay in this paper. The stochastic replicator dynamic model is proposed so as to investigate stochastic stability of evolutionary games. Our most interesting results concern the impact of multiplicative noise and time delay on the cooperation behavior in large well-mixed populations. The theory of stochastic delay differential equation is used as the main research tool. A sufficient condition on time delay and noise is proposed that it is corrected to guarantee the stable equilibrium point is indeed the almost sure exponential stablility (ASES). Moreover, we get a generalization of the cooperation evolution in two-strategy games. Stochastic Lyapunov framework is used to prove the stochastic stability and finding the value of delay is the key and challenging in our case. Examples are illustrated our results.