Parity law for population dynamics of N-species with cyclic advantage competitions

Abstract

We study population dynamics on lattice space with N species ( N=3, 4, 5, and 6) with cyclic competitive advantage, having both a weaker species and a stronger one. The steady state with equal population densities is the globally stable fixed point. For the biased-rate cyclic advantage population, we can observe the different features of the steady states and their stabilities for even N and for odd N, and we call it as “parity law”. We obtain the following results on the parity law: (i) for odd number N all the species can coexist following the biased rate and we have the counterintuition for the change of equilibrium densities; (ii) for even N all the species cannot always coexist and it coincides with our intuition. We also analyzed mean-field approximation. It is revealed that most of the results of Monte Carlo simulation can be understood by mean-field approximation dynamics. However, equilibrium densities cannot be always predicted by mean-field approximation because of the spatial fixation.