{
  "id": "1507.00397",
  "version": "v1",
  "published": "2015-07-02T00:08:34.000Z",
  "updated": "2015-07-02T00:08:34.000Z",
  "title": "Scaling limits of a model for selection at two scales",
  "authors": [
    "Shishi Luo",
    "Jonathan C. Mattingly"
  ],
  "comment": "23 pages, 1 figure",
  "categories": [
    "math.PR",
    "math.DS"
  ],
  "abstract": "The dynamics of a population undergoing selection is a central topic in evolutionary biology. This question is particularly intriguing in the case where selective forces act in opposing directions at two population scales. For example, a fast-replicating virus strain outcompetes slower-replicating strains at the within-host scale. However, if the fast-replicating strain causes host morbidity and is less frequently transmitted, it can be outcompeted by slower-replicating strains at the between-host scale. Here we consider a stochastic ball-and-urn process which models this type of phenomenon. We prove the weak convergence of this process under two natural scalings. The first scaling leads to a deterministic nonlinear integro-partial differential equation on the interval $[0,1]$ with dependence on a single parameter, $\\lambda$. We show that the fixed points of this differential equation are Beta distributions and that their stability depends on $\\lambda$ and the behavior of the initial data around $1$. The second scaling leads to a measure-valued Fleming-Viot process, an infinite dimensional stochastic process that is frequently associated with a population genetics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-02T00:08:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "scaling limits",
      "deterministic nonlinear integro-partial differential equation",
      "virus strain outcompetes slower-replicating strains",
      "infinite dimensional stochastic process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}