On the limit configuration of four species strongly competing systems

Abstract

We analysed some qualitative properties of the limit configuration of the solutions of a reaction-diffusion system of four competing species as the competition rate tends to infinity. Large interaction induces the spatial segregation of the species and only two limit configurations are possible: either there is a point where four species concur, a 4-point, or there are two points where only three species concur. We characterized, for a given datum, the possible 4-point configuration by means of the solution of a Dirichlet problem for the Laplace equation.Mathematics Subject Classification (2010). 35J65; 35Bxx; 92D25.

Some remarks on segregation of $k$ species in strongly competing systems

“…In Section 3 we study the geometry of the limiting configuration in any number of species. If k D 2s, starting from the argument used in [19] for k D 4 species, we prove that some limiting configurations are strictly related to the solution of a Dirichlet problem for the Laplace equation. Our results rely on the construction of a harmonic function which assumes the value P 2s j D1 .…”

“…Propositions 4.4 and 4.5 show that the limiting configurations, which elements of Z 3 .U / have only even multiplicity, are closely connected to harmonic solutions of (3.2) with alternate boundary datum. Since such solutions have to satisfy some integral conditions (see (3.13)), it follows that the most probable segregation configurations have only points in Z 3 .U / with odd multiplicity (see also Remark 3.13 in [19]).…”

“…An open problem we wish to handle is to compare results by reaction-diffusion systems in populations competition, as in [3,4,9,11,12,19,23], and results obtained by differential games theory contained in [16,20] (see also the references therein), in order to reach a better understanding of the territoriality of the competing species or groups (see also [5]).…”

“…The description of the qualitative properties of the limiting configurations in the planar case (i.e., n D 2) was considered in [11] for k D 3 and in [19] for k D 4. The aim of this paper is to describe the geometry of the limiting configurations in the planar case, for any number of species.…”

Some remarks on segregation of k species in strongly competing system

“…The description of the qualitative properties of the limiting configurations in the planar case (i.e. n = 2) was considered in [11] for k = 3 and in [19] for k = 4. The aim of this paper is to describe the geometry of the limiting configurations in the planar case, for any number of species.…”