Instability of Stationary Solutions for a Lotka]Volterra Competition Model with Diffusion

We consider a Lotka]Volterra competition model with diffusion on R which describes the dynamics of the population of two competing species, and study the stability of positive stationary solutions of the model relative to the space X of bounded uniformly continuous functions with the supremum norm. In consideration of earlier results, we shall arrive at the study of the distribution of the real eigenvalues of the linearized operator around the stationary solution, and prove the existence of only one positive eigenvalue whose eigenfunction exponentially decays to 0 as jª"`. This suggests that the instability of the stationary solution is not due to the space X.