2024/03/28 by Yihong Du, Du, Yihong, Wenjie Ni +3
Computer Science · Medicine · Physics and Astronomy · #35K57 #35R20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Dynamics and Pattern Formation #Opinion Dynamics and Social Influence
paper · pdf · doi:10.48550/arxiv.2403.19134
openalex publication_date 2024/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This is a continuation of our work \citedns-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 \R) 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 \citedns-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.