Some remarks on segregation of k species in strongly competing system

Flavia Lanzara; Eugenio Montefusco

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arXiv · 2103.06808

Some remarks on segregation of k species in strongly competing system

Flavia Lanzara·Eugenio Montefusco

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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.

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Flavia Lanzara, Eugenio Montefusco. 2021-03-11. Some remarks on segregation of k species in strongly competing system. https://arxiv.org/abs/2103.06808

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