Discrete and Continuous Distributed Delays in Replicator Dynamics

In this paper, we study evolutionary games and we examine the stability of the evolutionarily stable strategy (ESS) in the continuous-time replicator dynamics with distributed time delays. In many examples, the interactions between individuals take place instantaneously, but their impacts are not immediate and take a certain amount of time which is usually random. In this paper, we study the consequences of distributed delays on the stability of the replicator dynamics. The main results are that (i) under the exponential delay distribution, the ESS is asymptotically stable for any value of the rate parameter; (ii) the necessary and sufficient conditions for the asymptotic stability of the ESS under uniform and Erlang distributions; and (iii) for the discrete distributed delays, we derive a necessary and sufficient delay-independent stability conditions. We illustrate our results with numerical examples from the Hawk–Dove game.