  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.
