Long-time dynamics of a competition model with nonlocal diffusion and free boundaries: Vanishing and spreading of the invader

In this work, we investigate the long-time dynamics of a two species competition model of Lotka-Volterra type with nonlocal diffusions. One of the species, with density v (t, x), is assumed to be a native in the environment (represented by the real line), while the other species, with density u (t, x), is an invading species which invades the territory of v with two fronts, x=g (t) on the left and x=h (t) on the right. So the population range of u is the evolving interval g (t), h (t) and the reaction-diffusion equation for u has two free boundaries, with g (t) decreasing in t and h (t) increasing in t, and the limits h_: =h () and g_: =g () - thus always exist. We obtain detailed descriptions of the long-time dynamics of the model according to whether h_-g_ is or finite. In the latter case, we reveal in what sense the invader u vanishes in the long run and v survives the invasion, while in the former case, we obtain a rather satisfactory description of the long-time asymptotic limit for both u (t, x) and v (t, x) when a certain parameter k in the model is less than 1. This research is continued in a separate work, where sharp criteria are obtained to distinguish the case h_-g_= from the case h_-g_ is finite, and new phenomena are revealed for the case k 1. The techniques developed in this paper should have applications to other models with nonlocal diffusion and free boundaries.