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

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 competitiondiffusion system of k differential equations −∆ui(x) = −μui(x) ∑ j 6=i uj(x) i = 1, ..., k in a domain D with appropriate boundary conditions. Any ui represents a population density and the parameter μ determines the interaction strength between the populations. The purpose of this paper is to study the geometry of the limiting configuration as μ −→ +∞ 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. 2010 Mathematics Subject Classification: Primary 35Bxx, 35J47; Secondary 92D25.