{
  "id": "1406.5884",
  "version": "v2",
  "published": "2014-06-23T12:38:50.000Z",
  "updated": "2018-07-11T21:28:38.000Z",
  "title": "Rescaling limits of the spatial Lambda-Fleming-Viot process with selection",
  "authors": [
    "Alison Etheridge",
    "Amandine Veber",
    "Feng Yu"
  ],
  "comment": "77 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the spatial Lambda-Fleming-Viot process model [BEV10] for frequencies of genetic types in a population living in R^d, with two types of individuals (0 and 1) and natural selection favouring individuals of type 1. We consider two cases, one in which the dynamics of the process are driven by purely `local' (fixed radius) events and one incorporating large-scale extinction-recolonisation events whose radii have a polynomial tail distribution. In both cases, we consider a sequence of spatial Lambda-Fleming-Viot processes indexed by n, and we assume that the fraction of individuals replaced during a reproduction event and the relative frequency of events during which natural selection acts tend to 0 as n tends to infinity. We choose the decay of these parameters in such a way that when reproduction is only local, the measure-valued process describing the local frequencies of the less favoured type converges in distribution to a (measure-valued) solution to the stochastic Fisher-KPP equation in one dimension, and to a (measure-valued) solution to the deterministic Fisher-KPP equation in more than one dimensions. When large-scale extinction-recolonisation events occur, the sequence of processes converges instead to the solution to the analogous equation in which the Laplacian is replaced by a fractional Laplacian (again, noise can be retained in the limit only in one spatial dimension). We also define the process of `potential ancestors' of a sample of individuals taken from these populations, which takes the form of a system of branching and coalescing symmetric jump processes. We show their convergence in distribution towards a system of Brownian or stable motions which branch at some finite rate. In one dimension, in the limit, pairs of particles also coalesce at a rate proportional to the local time at zero of their separation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-23T12:38:50.000Z",
      "abstract": "We consider the spatial Lambda-Fleming-Viot process model for frequencies of genetic types in a population living in Rd, with two types of individuals (0 and 1) and natural selection favouring individuals of type 1. We first prove that the model with selection is well-defined. Next, we consider two cases, one in which the dynamics of the process are driven by purely local events and one incorporating large-scale extinction-recolonisation events. In both cases, we consider a sequence of spatial Lambda-Fleming-Viot processes indexed by n, and we assume that the fraction of individuals replaced during a reproduction event and the relative frequency of events during which natural selection acts tend to 0 as n tends to infinity. We choose the decay of these parameters in such a way that the frequency of the less favoured type converges in distribution to the solution to the Fisher KPP-equation (with noise in one dimension) when reproduction is only local, or to the solution to an analogous equation in which the Laplacian is replaced by a fractional Laplacian when large-scale extinction-recolonisation events occur. We also define the process of potential ancestors of a sample of individuals taken from these populations, and show their convergence in distribution towards a system of Brownian or stable motions which branch at some finite rate. In one dimension, in the limit, pairs of particles also coalesce at a rate proportional to their local time together. In contrast to previous proofs of scaling limits for the spatial Lambda-Fleming-Viot process here the convergence of the more complex forwards-in-time processes is used to prove the convergence of the potential ancestries.",
      "comment": "59 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2018-07-11T21:28:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G57",
      "60J25",
      "92D10",
      "60J75",
      "60G52"
    ],
    "keywords": [
      "rescaling limits",
      "natural selection acts tend",
      "spatial lambda-fleming-viot process model",
      "large-scale extinction-recolonisation events occur",
      "incorporating large-scale extinction-recolonisation events"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 77,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.5884E"
    }
  }
}