Some remarks on segregation of k species in strongly competing system

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 $$-\mathrm{\Delta }{u}_i(x)=-\mu u_i(x){\displaystyle \sum _{j\mathrm{\ne }i}{u}_j(x)i=1,\mathrm{.}\mathrm{.}\mathrm{.},k}$$ 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 $\mu ⟶+\mathrm{\infty }$ 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.