Delayed Evolutionary game dynamics with non-uniform interactions in two communities

In this paper we study the evolution of a population composed of several communities where the interaction between them is non-uniform. The main objective is to study the effect of slow time scale delays on the convergence of the replicator dynamics to the Evolutionary Stable Strategy (ESS). In particular, we investigate the effect of time delays on the stability of the replicator dynamics. We can distinguish two types of the time delay: strategic delay which is the delay associated with the strategies of players, and spatial delay, which is the delay associated with the communities of the players. We prove that for large strategic delays, the mixed intermediate ESS may become an unstable state for the replicator dynamics, while for any value of the spatial delay, the replicator dynamics converge to the intermediate ESS. We also show that when both types of the time delay coexist, the instability property is emphasized.