{
  "id": "1903.06731",
  "version": "v1",
  "published": "2019-03-15T18:22:51.000Z",
  "updated": "2019-03-15T18:22:51.000Z",
  "title": "General selection models: Bernstein duality and minimal ancestral structures",
  "authors": [
    "Fernando Cordero",
    "Sebastian Hummel",
    "Emmanuel Schertzer"
  ],
  "comment": "47 pages, 12 figures",
  "categories": [
    "math.PR",
    "q-bio.PE"
  ],
  "abstract": "The $\\Lambda$-Wright--Fisher process describes the type-frequency evolution of an infinite population. We model frequency-dependent selection pressure with a general polynomial drift vanishing at the boundary. An appropriate decomposition of the drift allows us to construct a series of Moran-type models that converge under suitable conditions to the solution of the associated stochastic differential equation. The genealogical structure inherent in the graphical representation of these finite population models can be seen in the large population limit as a generalisation of the ancestral selection graph of Krone and Neuhauser. We introduce an ancestral process that keeps track of the sampling distribution along the ancestral structures and that satisfies a duality relation with the type-frequency process. We refer to it as Bernstein coefficient process and to the relation as Bernstein duality. The latter is a generalisation of the classic moment duality. Many classic results in the restricted setting of a moment duality generalise into our framework. In particular, we derive criteria for the accessibility of the boundary and determine the time to absorption. It turns out that multiple ancestral processes are associated to the same forward dynamics. We characterise the set of optimal ancestral structures and provide a recipe to construct them from the drift. In particular, this allows us to recover well-known ancestral structures of the literature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-15T18:22:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82C22",
      "92D15",
      "60J25",
      "60J27"
    ],
    "keywords": [
      "general selection models",
      "minimal ancestral structures",
      "bernstein duality",
      "model frequency-dependent selection pressure",
      "classic moment duality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}