{
  "id": "2004.14137",
  "version": "v1",
  "published": "2020-04-29T12:39:44.000Z",
  "updated": "2020-04-29T12:39:44.000Z",
  "title": "Spatial populations with seed-bank: well-posedness, duality and equilibrium",
  "authors": [
    "Andreas Greven",
    "Frank den Hollander",
    "Margriet Oomen"
  ],
  "comment": "74 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a system of interacting Fisher-Wright diffusions with seed-bank. Individuals live in colonies and are subject to resampling and migration as long as they are active. Each colony has a structured seed-bank into which individuals can retreat to become dormant, suspending their resampling and migration until they become active again. As geographic space labelling the colonies we consider a countable Abelian group $\\mathbb{G}$ endowed with the discrete topology. The key example of interest is the Euclidean lattice $\\mathbb{G}=\\mathbb{Z}^d$. Our goal is to classify the long-time behaviour of the system in terms of the underlying model parameters. In particular, we want to understand in what way the seed-bank enhances genetic diversity. We introduce three models of increasing generality, namely, individuals become dormant: (1) in the seed-bank of their colony; (2) in the seed-bank of their colony while adopting a random colour that determines their wake-up time; (3) in the seed-bank of a random colony while adopting a random colour. The extension in (2) allows us to model wake-up times with fat tails while preserving the Markov property of the evolution. For each of the three models we show that the system converges to a unique equilibrium depending on a single density parameter that is determined by the initial state, and exhibits a dichotomy of coexistence (= locally multi-type equilibrium) versus clustering (= locally mono-type equilibrium) depending on the parameters controlling the migration and the seed-bank. The dichotomy between clustering and coexistence in model 1 is determined by migration only. In models (2) and (3), when the wake-up time has infinite mean, the dichotomy is determined by both the exchange with the seed-bank and migration. It turns out that the seed-bank affects the long-time behaviour both quantitatively and qualitatively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-29T12:39:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J70",
      "60K35",
      "92D25"
    ],
    "keywords": [
      "spatial populations",
      "equilibrium",
      "random colour",
      "seed-bank enhances genetic diversity",
      "well-posedness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 74,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}