Characterization of spiraling patterns in spatial rock-paper-scissors games

TL;DR

This paper investigates the formation and stability of spiral wave patterns in spatial rock-paper-scissors models, combining analytical and numerical methods to understand how parameters influence pattern emergence and stability.

Contribution

It provides a detailed phase diagram and analytical characterization of spiral patterns in a generic cyclic competition model near Hopf bifurcation.

Findings

Spiral patterns are stable near the Hopf bifurcation.

Nonlinear mobility influences spiral wave stability.

The complex Ginzburg-Landau equation describes pattern dynamics.

Abstract

The spatio-temporal arrangement of interacting populations often influences the maintenance of species diversity and is a subject of intense research. Here, we study the spatio-temporal patterns arising from the cyclic competition between three species in two dimensions. Inspired by recent experiments, we consider a generic metapopulation model comprising "rock-paper-scissors" interactions via dominance removal and replacement, reproduction, mutations, pair-exchange and hopping of individuals. By combining analytical and numerical methods, we obtain the model's phase diagram near its Hopf bifurcation and quantitatively characterize the properties of the spiraling patterns arising in each phase. The phases characterizing the cyclic competition away far from the Hopf bifurcation (at low mutation rate) are also investigated. Our analytical approach relies on the careful analysis of the properties of the complex Ginzburg-Landau equation derived through a controlled (perturbative) multiscale expansion around the model's Hopf bifurcation. Our results allows us to clarify when spatial "rock-paper-scissors" competition leads to stable spiral waves and under which circumstances they are influenced by nonlinear mobility.