Beyond rock–paper–scissors systems — Deterministic models of cyclic ecological systems with more than three species

We consider cyclic ecological systems consisting of more than three species, extending the well-studied three-species rock–paper–scissors (RPS) system. For such systems alliance and partial alliance states consisting of non-competing species can form, a phenomenon not present in classical RPS systems. We develop, analyze and simulate a May–Leonard type model accounting for cyclic dynamics and spatial diffusion. We analyze in detail four-species and five-species systems, determine dynamics for both spatially homogeneous initial conditions (i.e., ode models) and the propagation and interaction of patches of different species driven by diffusion and the cyclic dynamics (pde models). We show markedly different behavior between the four-and five-species models, with alliance states as stable steady states for the former whereas both coexistence of all species and heteroclinic cycles are possible stable outcomes for the latter. Finally, we discuss the extension of this even/odd split to systems with more than five species.