{
  "id": "0904.3468",
  "version": "v1",
  "published": "2009-04-22T14:30:58.000Z",
  "updated": "2009-04-22T14:30:58.000Z",
  "title": "Quasi-stationary distributions for structured birth and death processes with mutations",
  "authors": [
    "Pierre Collet",
    "Servet Martinez",
    "Sylvie Méléard",
    "Jaime San Martin"
  ],
  "comment": "39 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the probabilistic evolution of a birth and death continuous time measure-valued process with mutations and ecological interactions. The individuals are characterized by (phenotypic) traits that take values in a compact metric space. Each individual can die or generate a new individual. The birth and death rates may depend on the environment through the action of the whole population. The offspring can have the same trait or can mutate to a randomly distributed trait. We assume that the population will be extinct almost surely. Our goal is the study, in this infinite dimensional framework, of quasi-stationary distributions when the process is conditioned on non-extinction. We firstly show in this general setting, the existence of quasi-stationary distributions. This result is based on an abstract theorem proving the existence of finite eigenmeasures for some positive operators. We then consider a population with constant birth and death rates per individual and prove that there exists a unique quasi-stationary distribution with maximal exponential decay rate. The proof of uniqueness is based on an absolute continuity property with respect to a reference measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-04-22T14:30:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "92D25",
      "60K35",
      "60J70",
      "60J80"
    ],
    "keywords": [
      "quasi-stationary distribution",
      "death processes",
      "structured birth",
      "individual",
      "death rates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0904.3468C"
    }
  }
}