Existence and Asymptotic Stability of Traveling Waves of Two-species Lotka-Volterra Competition-Diffusion Systems via Geometric Singular Perturbations

Abstract

We consider the two-species Lotka-Volterra competition-diffusion system where one species has a small diffusion rate relative to the other species and has a small competition coefficient. By the geometric singular perturbation theory, we prove the existence of the wavefront connecting the coexistence state to the trivial state, with traveling speed greater than the minimal speed. For such wavefronts, we show that the profiles of both species are monotone. Finally, we use the geometric singular perturbation theory to estimate the associated Evans function, and give the related stability results.