{
  "id": "2103.02812",
  "version": "v1",
  "published": "2021-03-04T03:25:08.000Z",
  "updated": "2021-03-04T03:25:08.000Z",
  "title": "Travelling waves, blow-up and extinction in the Fisher-Stefan model",
  "authors": [
    "Scott W. McCue",
    "Maud El-Hachem",
    "Matthew J. Simpson"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "While there is a long history of employing moving boundary problems in physics, in particular via Stefan problems for heat conduction accompanied by a change of phase, more recently such approaches have been adapted to study biological invasion. For example, when a logistic growth term is added to the governing partial differential equation in a Stefan problem, one arrives at the Fisher-Stefan model, a generalisation of the well-known Fisher-KPP model, characterised by a leakage coefficient $\\kappa$ which relates the speed of the moving boundary to the flux of population there. This Fisher-Stefan model overcomes one of the well-known limitations of the Fisher-KPP model, since time-dependent solutions of the Fisher-Stefan model involve a well-defined front with compact support which is more natural in terms of mathematical modelling. Almost all of the existing analysis of the standard Fisher-Stefan model involves setting $\\kappa > 0$, which can lead to either invading travelling wave solutions or complete extinction of the population. Here, we demonstrate how setting $\\kappa < 0$ leads to retreating travelling waves and an interesting transition to finite-time blow-up. For certain initial conditions, population extinction is also observed. Our approach involves studying time-dependent solutions of the governing equations, phase plane and scaling analysis, leading to new insight into the possibilities of travelling waves, blow-up and extinction for this moving boundary problem. Matlab software used to generate the results in this work are available on Github.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-04T03:25:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "travelling wave",
      "extinction",
      "moving boundary problem",
      "time-dependent solutions",
      "stefan problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}