Instability of non-constant equilibrium solutions of a system of competition-diffusion equations

The system of interaction-diffusion equations describing competition between two species is investigated. By using a version of the Perron-Frobenius theorem of positive matrices generalized to function spaces, it is proved that any non-constant equilibrium solution of the system is unstable both under Neumann boundary conditions (for the rectangular parallelepiped domain) and under periodic conditions. It is conjectured that this result extends to convex domains, and that the simple interaction-diffusion model cannot explain spatially segregated distributions of two competing species in such domains.