  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
Poincar\'e- Lindstedt's perturbation method to analyze the Hopf bifurcation.
Our theoretical results are corroborated with numerical simulations.
