Effects of Small Spatial Variation of the Reproduction Rate in a Two Species Competition Model

Of concern is the effect of a small spatially inhomogeneous perturbation of the reproduction rate of the first species in a two-species LotkaVolterra competition-diffusion problem with spatially homogeneous reaction terms. Apart from this perturbation and the diffusion rates, the two species are assumed to be identical. Our main result shows that the first species can always invade, whereas the second species can only invade under certain conditions which yield uniform persistence of both species. The proof relies on comparison techniques and properties of the principal eigenvalue of reactiondiffusion equations.