GLOBAL ASYMPTOTIC STABILITY OF CONSTANT EQUILIBRIUM IN A NONLOCAL DIFFUSION COMPETITION MODEL WITH FREE BOUNDARIES

In this paper we give a classification of the global asymptotic stability for a nonlocal diffusion competition model with free boundaries consisting of an invasive species with density u and a native species with density v. We not only prove that such nonlocal diffusion problem has a unique global solution and also determine the long-time asymptotic behavior of the solution for three competition cases : (I) u is an inferior competitor, (II) u is a superior competitor and (III) the weak competition case. Especially, in case (II), under some additional conditions, we determine the long-time asymptotic behavior of the solution when vanishing happens. Moreover, the criteria for spreading and vanishing are obtained.