Long-time dynamics of a competition model with nonlocal diffusion and free boundaries: Chances of successful invasion

This is a continuation of our work dns-part1 to investigate the long-time dynamics of a two species competition model of Lotka-Volterra type with nonlocal diffusions, where the territory (represented by the real line) of a native species with density v (t, x), is invaded by a competitor with density u (t, x), via 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. Let h_: =h () and g_: =g () -. In dns-part1, we obtained detailed descriptions of the long-time dynamics of the model according to whether h_-g_ is or finite. In the latter case, we demonstrated in what sense the invader u vanishes in the long run and v survives the invasion, while in the former case, we obtained a rather satisfactory description of the long-time asymptotic limits of u (t, x) and v (t, x) when the parameter k in the model is less than 1. In the current paper, we obtain sharp criteria to distinguish the case h_-g_= from the case h_-g_ is finite. Moreover, for the case k 1 and u is a weak competitor, we obtain biologically meaningful conditions that guarantee the vanishing of the invader u, and reveal chances for u to invade successfully. In particular, we demonstrate that both h_==-g_ and h_= but g_ is finite are possible; the latter seems to be the first example for this kind of population models, with either local or nonlocal diffusion.