{
  "id": "2412.07398",
  "version": "v1",
  "published": "2024-12-10T10:43:16.000Z",
  "updated": "2024-12-10T10:43:16.000Z",
  "title": "Quasistationarity and extinction for population processes",
  "authors": [
    "Damian Clancy"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider stochastic population processes that are almost surely absorbed at the origin within finite time. Our interest is in the quasistationary distribution, $\\boldsymbol{u}$, and the expected time, $\\tau$, from quasistationarity to extinction, both of which we study via WKB approximation. This approach involves solving a Hamilton-Jacobi partial differential equation specific to the model. We provide conditions under which analytical solution of the Hamilton-Jacobi equation is possible, and give the solution. This provides a first approximation to both $\\boldsymbol{u}$ and $\\tau$. We provide further conditions under which a corresponding `transport equation' may be solved, leading to an improved approximation of $\\boldsymbol{u}$. For multitype birth and death processes, we then consider an alternative approximation for $\\boldsymbol{u}$ that is valid close to the origin, provide conditions under which the elements of this alternative approximation may be found explicitly, and hence derive an improved approximation for $\\tau$. We illustrate our results in a number of applications.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-10T10:43:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extinction",
      "hamilton-jacobi partial differential equation specific",
      "quasistationarity",
      "alternative approximation",
      "stochastic population processes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}