The dynamics of traveling wavefronts for a nonlocal delay competition system with local vs. nonlocal diffusions

Abstract

In this paper, we investigate the existence and exponential stability of traveling wavefronts for a two-component nonlocal delay Lotka–Volterra competition system with local vs. nonlocal diffusions. We first obtain the existence of monostable traveling wavefronts connecting two boundary equilibria by Schauder’s fixed point theorem with an explicit construction of a pair of super- and subsolutions. Furthermore, applying the weighted energy method together with the comparison principle, we prove that all the solutions of the corresponding Cauchy problem converge exponentially to the traveling wavefronts provided that the initial perturbations around the traveling wavefronts belong to a certain weighted Sobolev space.