Phys. Rev. E 90 (2014) 032704.
Characterization of spiraling patterns in spatial rock-paper-scissors games
Alastair M. Rucklidge
Department of Applied Mathematics,
University of Leeds, Leeds, LS2 9JT, UK
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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