Instability of Stationary Solutions for Lotka-Volterra Competition Model with Diffusion
We consider the Lotka-Volterra competition model with diffusion on R, and study the stability of axial symmetric positive stationary solutions relative to the space X of bounded uniformly continuous functions with the supremum norm. In consideration of the result of Alexander et al.  and Derndinger , we shall arrive at studying 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 ε --> ±∞. This suggests that the instability of the stationary solution is not due to the space X.