Hopf Bifurcations in Replicator Dynamics with Distributed Delays
For researchers in evolutionary game theory, this work provides a theoretical analysis of how distributed delays induce oscillations, but it is incremental as it extends known bifurcation analysis to specific delay distributions.
This paper studies Hopf bifurcations in two-strategy replicator dynamics with distributed delays, showing that as mean delay increases, periodic oscillations emerge via Hopf bifurcation for various delay distributions.
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 Poincaré- Lindstedt's perturbation method to analyze the Hopf bifurcation. Our theoretical results are corroborated with numerical simulations.