Uniqueness and stability of traveling waves for a three-species competition system with nonlocal dispersal

This paper is concerned with the traveling waves for a three-species competitive system with nonlocal dispersal. It has been shown by Dong, Li and Wang (DCDS 37 (2017) 6291-6318) that there exists a minimal wave speed such that a traveling wave exists if and only if the wave speed is above this minimal wave speed. In this paper, we first investigate the asymptotic behavior of traveling waves at negative infinity by a modified version of Ikehara's Theorem. Then we prove the uniqueness of traveling waves by applying the stronger comparison principle and the sliding method. Finally, we establish the exponential stability of traveling waves with large speeds by the weighted-energy method and the comparison principle, when the initial perturbation around the traveling wavefront decays exponentially as x → -∞, but can be arbitrarily large in other locations.