Robust Stability of Uncertain Replicator Population Dynamics with Time Delay.

In this paper, we generalize the replicator dynamics with time delay of Tao [13] to the time-delayed uncertain population replicator dynamics. Considering the uncertainties of payoff functions, we propose three perturbed models of the traditional time-delayed replicator dynamics. The stability of the evolutionary stable strategy (ESS) is then studied. We prove the ESS of traditional replicator dynamics is also asymptotically stable under some constraints on the time delay in our first model and the result of Tao [13] is a special case of our first model. Then We further study the uniformly perturbed time-delayed replicator dynamics with two delays. In our third model, we show the stability of the ESS is irrelevant to the rate parameter. Some numerical examples are illustrated.