{
  "id": "2403.19134",
  "version": "v1",
  "published": "2024-03-28T04:05:08.000Z",
  "updated": "2024-03-28T04:05:08.000Z",
  "title": "Long-time dynamics of a competition model with nonlocal diffusion and free boundaries: Chances of successful invasion",
  "authors": [
    "Yihong Du",
    "Wenjie Ni",
    "Linfei Shi"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This is a continuation of our work \\cite{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 $\\R$) 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 $h_\\infty:=h(\\infty)\\leq \\infty$ and $g_\\infty:=g(\\infty)\\geq -\\infty$. In \\cite{dns-part1}, we obtained detailed descriptions of the long-time dynamics of the model according to whether $h_\\infty-g_\\infty$ is $\\infty$ 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 $h_\\infty-g_\\infty=\\infty$ from the case $h_\\infty-g_\\infty$ is finite. Moreover, for the case $k\\geq 1$ 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 $h_\\infty=\\infty=-g_\\infty$ and $h_\\infty=\\infty$ but $g_\\infty$ is finite are possible; the latter seems to be the first example for this kind of population models, with either local or nonlocal diffusion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-28T04:05:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K57",
      "35R20"
    ],
    "keywords": [
      "long-time dynamics",
      "nonlocal diffusion",
      "free boundaries",
      "successful invasion",
      "species competition model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}