  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.
