Mixing times in evolutionary game dynamics

Abstract

Without mutation and migration, evolutionary dynamics ultimately leads to the extinction of all but one species. Such fixation processes are well understood and can be characterized analytically with methods from statistical physics. However, many biological arguments focus on stationary distributions in a mutation-selection equilibrium. Here, we address the equilibration time required to reach stationarity in the presence of mutation, this is known as the mixing time in the theory of Markov processes. We show that mixing times in evolutionary games have the opposite behaviour from fixation times when the intensity of selection increases: In coordination games with bistabilities, the fixation time decreases, but the mixing time increases. In coexistence games with metastable states, the fixation time increases, but the mixing time decreases. Our results are based on simulations and the WKB approximation of the master equation.

5 pages, 3 figures

References in corpus (3)

• Statistical physics of social dynamics
• Coevolutionary Dynamics: From Finite to Infinite Populations
• Fixation of strategies for an evolutionary game in finite populations

Cited by in corpus (10)

• WKB theory of large deviations in stochastic populations
• Master equations and the theory of stochastic path integrals
• Fixation in finite populations evolving in fluctuating environments
• When the mean is not enough: Calculating fixation time distributions in birth-death processes
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• Stochastic tunneling and metastable states during the somatic evolution of cancer
• A Finite Population Destroys a Traveling Wave in Spatial Replicator Dynamics
• Evolutionary Games on Networks: Phase Transition, Quasi-equilibrium, and Mathematical Principles
• Applications of WKB and Fokker-Planck methods in analyzing population extinction driven by weak demographic fluctuations
• Fixation and Fluctuations in Two-Species Cooperation