Oscillatory evolutionarily stable state and limit cycle in replicator dynamics

TL;DR

This paper introduces the concept of oscillatory ESS (OESS) in replicator dynamics, demonstrating that it corresponds to stable limit cycles and analyzing their uniqueness and phase space locations.

Contribution

It defines and proves the stability of oscillatory ESS (OESS) as a new type of stable limit cycle in evolutionary game dynamics.

Findings

OESS corresponds to a stable limit cycle in replicator dynamics.

Conditions for the uniqueness of OESS are established.

Locations of multiple OESSes in phase space are characterized.

Abstract

The idea of evolutionarily stable state (ESS) of a population is a cornerstone of evolutionary game theory; moreover, it coincides with the game-theoretic concept of Nash equilibrium. Such a state corresponds to a strategy adopted by the population such that a rare mutant strategy cannot invade the population. In parallel, the dynamical formulation of evolutionary game theory -- particularly through replicator dynamics embodying the tenet of survival of the fittest -- provides a framework for modelling frequency-dependent selection over time. While it is well known that an ESS corresponds to stable fixed point in replicator dynamics, the evolutionary game-theoretic characterization of limit cycles is unknown. Here we fill this lacuna by defining oscillatory ESS (OESS) which we prove to be a stable limit cycle. We also show when an OESS is unique and if there are multiple OESSes, then what their locations are in the phase space.