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.