Bifurcation Analysis of Replicator Dynamics with Logistic Growth and Strategy-Dependent Time Delays in Snowdrift Game

This study examines the stability effects of strategy-dependent time delays on replicator dynamics with logistic population growth. Using a two-player snowdrift game, we incorporate social forms of delay where individuals' payoffs are delayed, impacting offspring production. While a stable mixed-strategy equilibrium exists without delays, introducing time delays can destabilize this equilibrium through Hopf bifurcations, inducing oscillations. Employing Hopf bifurcation theory, we explore how critical delay values depend on whether delays are equal across strategies, unequal with one strategy experiencing no delay, or fully asymmetric. The analysis reveals rich phenomena, including regions where increasing a delay can transition the system between stability and instability. Our work elucidates the complex interplay between time delays, strategic dynamics, and finite population constraints.