Effect of Parity in Cyclically Competing Communities

Communities exhibiting cyclic (i.e., ``winnerless’') competition occur throughout nature. These systems can exhibit qualitatively different long-term behavior depending on the strength of inter-species competition and a specific property of the number of interacting species: parity. We consider how ecological communities can adapt to and transition between an odd and an even number of cyclically competing species. Previous studies have shown that, for strong inter-species competition, odd parity communities are dynamically unstable, while species in even parity communities form stable alliances of maximal noncompeting sets. We trace a homotopy between two May-Leonard type models: the odd parity N=3 model and the even parity N=4 system. We identify all physical steady states and analyze their stabilities. We show the existence of stable transitional states that are unique to our symmetric model. We then use a piecewise linear approximation technique to characterize the stability of heteroclinic cycles. We use numerical simulations to map distinct regimes in parameter space. Our computations confirm our analytical results and also reveal a window in parameter space where limit cycles occur. We illustrate that parity transitions can be complex and characterized by diverse parameter-dependent pathways.