Hopf Bifurcations in Replicator Dynamics with Distributed Delays

TL;DR

This paper investigates how distributed delays in two-strategy replicator dynamics can lead to Hopf bifurcations, causing stability changes and periodic oscillations, with analysis across various delay distributions supported by numerical simulations.

Contribution

It introduces a comprehensive analysis of Hopf bifurcations in replicator dynamics with multiple delay distributions using perturbation methods.

Findings

Hopf bifurcations occur as mean delay increases.

Different delay distributions affect stability and oscillation patterns.

Numerical simulations confirm theoretical predictions.

Abstract

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.