Hopf Bifurcations in Replicator Dynamics with Distributed Delays

In this paper, we study the existence and the property of the Hopf bifurcation in the two-strategy replicator dynamics with distributed delays. In evolutionary games, we assume that a strategy would take an uncertain time delay to have a consequence on the fitness (or utility) of the players. As the mean delay increases, a change in the stability of the equilibrium (Hopf bifurcation) may occur at which a periodic oscillation appears. We consider Dirac, uniform, Gamma, and discrete delay distributions, and we use the PoincaréLindstedt’s perturbation method to analyze the Hopf bifurcation. Our theoretical results are corroborated with numerical simulations.