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

Area of Science:

• Evolutionary Game Theory
• Mathematical Biology
• Population Dynamics

Background:

• Game theory analyzes strategic interactions, while evolutionary game dynamics models population changes under selection.
• The replicator equation is a key model for these dynamics in two-player, two-strategy games.

Purpose of the Study:

• To investigate the impact of mutation and time delays on the replicator equation in symmetrical two-player, two-strategy games.
• To identify conditions for Hopf bifurcation and the emergence of stable limit cycles, indicating sustained oscillations.

Main Methods:

• Analytical and numerical analysis of the delayed replicator-mutator equation.
• Focus on four symmetrical games: Stag Hunt, Snowdrift, Prisoner's Dilemma, and Harmony game.
• Investigation of Hopf bifurcation conditions leading to stable limit cycles.

Main Results:

• Mutation alone does not induce oscillatory cooperation in these games.
• Time delays, in conjunction with mutation, can lead to oscillatory cooperation and stable limit cycles.
• Delay alone may not cause Hopf bifurcation without mutation.

Conclusions:

• Stable limit cycles, facilitated by delay and mutation, can promote the evolution of cooperation.
• Time delays play a crucial role in generating cooperative dynamics previously unattainable with mutation alone.