Stochasticity and evolutionary stability

Abstract

In stochastic dynamical systems, different concepts of stability can be obtained in different limits. A particularly interesting example is evolutionary game theory, which is traditionally based on infinite populations, where strict Nash equilibria correspond to stable fixed points that are always evolutionarily stable. However, in finite populations stochastic effects can drive the system away from strict Nash equilibria, which gives rise to a new concept for evolutionary stability. The conventional and the new stability concepts may apparently contradict each other leading to conflicting predictions in large yet finite populations. We show that the two concepts can be derived from the frequency dependent Moran process in different limits. Our results help to determine the appropriate stability concept in large finite populations. The general validity of our findings is demonstrated showing that the same results are valid employing vastly different co-evolutionary processes.