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.