Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model

The asymptotic behavior, as the coefficient of the advection term approaches infinity, of the principal eigenvalue of an elliptic operator is determined. The limiting profiles of the corresponding eigenfunctions are also given. As an application a Lotka-Volterra reaction-diffusion-advection model for two competing species in a heterogeneous environment is investigated. The two species are assumed to be identical except for their dispersal strategies: one disperses by random diffusion only, and the other by both random diffusion and advection along an environmental gradient. When the advection is strong relative to random dispersal, both species can coexist. In some situations, it is further shown that the density of the species with large advection in the direction of resources is concentrated at the spatial location with maximum resources; detailed asymptotic profiles of all coexistence states are also given.