{
  "id": "2403.19131",
  "version": "v1",
  "published": "2024-03-28T03:59:59.000Z",
  "updated": "2024-03-28T03:59:59.000Z",
  "title": "Long-time dynamics of a competition model with nonlocal diffusion and free boundaries: Vanishing and spreading of the invader",
  "authors": [
    "Yihong Du",
    "Wenjie Ni",
    "Linfei Shi"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "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 $\\R$), 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 $h_\\infty:=h(\\infty)\\leq \\infty$ and $g_\\infty:=g(\\infty)\\geq -\\infty$ thus always exist. We obtain 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 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 $h_\\infty-g_\\infty=\\infty$ from the case $h_\\infty-g_\\infty$ is finite, and new phenomena are revealed for the case $k\\geq 1$. The techniques developed in this paper should have applications to other models with nonlocal diffusion and free boundaries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-28T03:59:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K57",
      "35R20"
    ],
    "keywords": [
      "nonlocal diffusion",
      "long-time dynamics",
      "free boundaries",
      "long-time asymptotic limit",
      "species competition model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}