Chaos and unpredictability in evolutionary dynamics in discrete time

TL;DR

This paper investigates how discrete-time replicator dynamics in two-strategy games can exhibit chaotic and unpredictable behavior, contrasting with continuous models, and explores the implications of such dynamics for population stability.

Contribution

It introduces a discrete-time replicator model showing chaotic dynamics and complex attractor structures, expanding understanding of evolutionary game behavior beyond continuous models.

Findings

Chaotic and periodic behaviors emerge in discrete-time replicator dynamics.

Bimodality leads to more complex attractor variations.

Unphysical stationary solutions imply population unpredictability.

Abstract

A discrete-time version of the replicator equation for two-strategy games is studied. The stationary properties differ from that of continuous time for sufficiently large values of the parameters, where periodic and chaotic behavior replace the usual fixed-point population solutions. We observe the familiar period-doubling and chaotic-band-splitting attractor cascades of unimodal maps but in some cases more elaborate variations appear due to bimodality. Also unphysical stationary solutions have unusual physical implications, such as uncertainty of final population caused by sensitivity to initial conditions and fractality of attractor preimage manifolds.