Segregating partition problem in competition-diffusion systems

Abstract

We consider a reaction-diffusion system to describe the interaction of three competing species which move by diffusion in ${\mathbf{R}}^2$, under the situation where all of the diffusion rates are small and all of the inter-specific competition rates are large. The resulting system possesses three locally stable spatially constant equilibria, each of which implies that only one of the competing species survive and the other two are extinct. Since the diffusion rates are small, internal layer regions appear as sharp interfaces with triple junctions, which generally divide the whole plane into three different regions occupied by only one of the species. The dynamics of interfaces as well as triple junctions are numerically studied. More specifically, assuming that three competing species are almost equal in competitive strength, we derive an angle condition between any neighboring interfaces at triple junctions by formal asymptotic analysis. Furthermore, for more general cases, we numerically study the dynamics of segregating patterns of three competing species from interfacial view points.