Phys. Rev. E 90 (2014) 032704. doi:10.1103/PhysRevE.90.032704

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

Bartosz Szczesny, Mauro Mobilia and Alastair M. Rucklidge

Department of Applied Mathematics,
University of Leeds, Leeds, LS2 9JT, UK

Abstract. The spatiotemporal arrangement of interacting populations often influences the maintenance of species diversity and is a subject of intense research. Here, we study the spatiotemporal 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.

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