Finite-size effects and switching times for Moran process with mutation

Area of Science:

• Evolutionary game theory
• Mathematical biology
• Population dynamics

Background:

• The Moran process models population dynamics and evolution.
• The iterated Prisoner's Dilemma is a key model for cooperation and competition.
• Understanding evolutionarily stable strategies is crucial for predicting population behavior.

Purpose of the Study:

• To analyze the Moran process with two competing populations under an iterated Prisoner's Dilemma with mutation.
• To investigate the impact of multiple evolutionarily stable strategies.
• To explore asymmetries in stochastic models compared to deterministic limits.

Main Methods:

• Bifurcation analysis of the deterministic system for infinite population size.
• Study of the Master equation for the stochastic process.
• Asymptotic analysis for invariant distribution and metastable switching times in large, finite populations.

Main Results:

• A complete bifurcation analysis of the deterministic system was performed.
• Asymptotics for the invariant distribution and switching times were obtained for the stochastic process.
• The stochastic system exhibits asymmetries (skew) for parameter values where the deterministic limit is symmetric.

Conclusions:

• The study highlights the importance of considering stochastic effects in evolutionary dynamics.
• Asymmetries in stochastic models can arise even when deterministic models predict symmetry.
• These findings have implications for understanding the evolution of cooperation and competition in structured populations.