Oscillatory evolutionarily stable state and limit cycle in replicator dynamics
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.
Keywords
Cite
@article{arxiv.2601.01996,
title = {Oscillatory evolutionarily stable state and limit cycle in replicator dynamics},
author = {Suman Chakraborty and Vikash Kumar Dubey and Vaibhav Madhok and Sagar Chakraborty},
journal= {arXiv preprint arXiv:2601.01996},
year = {2026}
}