Evolutionary dynamics of the delayed replicator-mutator equation: Limit cycle and cooperation

TL;DR

This paper investigates how delay and mutation influence the dynamics of cooperation and defection in evolutionary game theory, revealing conditions for stable oscillations and coexistence in classic symmetric games.

Contribution

It analytically and numerically characterizes the conditions under which delay induces Hopf bifurcations and stable limit cycles in the delayed replicator-mutator equation.

Findings

Delay can induce stable oscillations in cooperation and defection.

Mutation alone does not cause oscillatory behavior.

Delay can trigger Hopf bifurcation even without mutation.

Abstract

Game theory deals with strategic interactions among players and evolutionary game dynamics tracks the fate of the players' populations under selection. In this paper, we consider the replicator equation for two-player-two-strategy games involving cooperators and defectors. We modify the equation to include the effect of mutation and also delay that corresponds either to the delayed information about the population state or in realizing the effect of interaction among players. By focusing on the four exhaustive classes of symmetrical games -- the Stag Hunt game, the Snowdrift game, the Prisoners' Dilemma game, and the Harmony game -- we analytically and numerically analyze delayed replicator-mutator equation to find the explicit condition for the Hopf bifurcation bringing forth stable limit cycle. The existence of the asymptotically stable limit cycle imply the coexistence of the cooperators and the defectors; and in the games, where defection is a stable Nash strategy, a stable limit cycle does provide a mechanism for evolution of cooperation. We find that while mutation alone can never lead to oscillatory cooperation state in two-player-two-strategy games, the delay can change the scenario. On the other hand, there are situations when delay alone cannot lead to the Hopf bifurcation in the absence of mutation in the selection dynamics.