Some remarks on segregation of k species in strongly competing system

TL;DR

This paper investigates the spatial segregation patterns of multiple competing species modeled by a system of differential equations, focusing on the geometric structure of the limiting configurations as competition intensifies.

Contribution

It provides new insights into the geometric structure of the limiting configurations for any number of species in a strongly competitive regime, especially relating to even numbers of species.

Findings

Limiting configurations are connected to solutions of Laplace's equation for even numbers of species.

The study characterizes the geometry of population densities as competition parameter tends to infinity.

Results apply to planar domains with general boundary conditions.

Abstract

Spatial segregation occurs in population dynamics when $k$ species interact in a highly competitive way. As a model for the study of this phenomenon, we consider the competition-diffusion system of $k$ differential equations \[ -\Delta u_i(x)=-\mu u_i (x)\displaystyle{\sum_{j\neq i}} u_j (x) \quad i=1,...,k \] in a domain $D$ with appropriate boundary conditions. Any $u_i$ represents a population density and the parameter $\mu$ determines the interaction strength between the populations. The purpose of this paper is to study the geometry of the limiting configuration as $\mu\longrightarrow+\infty$ on a planar domain for any number of species. If $k$ is even we show that some limiting configurations are strictly connected to the solution of a Dirichlet problem for the Laplace equation.