{
  "id": "2210.01681",
  "version": "v1",
  "published": "2022-10-04T15:27:42.000Z",
  "updated": "2022-10-04T15:27:42.000Z",
  "title": "Adaptation in a heterogeneous environment II: To be three or not to be",
  "authors": [
    "M. Alfaro",
    "F. Hamel",
    "F. Patout",
    "L. Roques"
  ],
  "categories": [
    "math.AP",
    "q-bio.PE"
  ],
  "abstract": "We propose a model to describe the adaptation of a phenotypically structured population in a $H$-patch environment connected by migration, with each patch associated with a different phenotypic optimum, and we perform a rigorous mathematical analysis of this model. We show that the large-time behaviour of the solution (persistence or extinction) depends on the sign of a principal eigenvalue, $\\lambda_H$, and we study the dependency of $\\lambda_H$ with respect to $H$. This analysis sheds new light on the effect of increasing the number of patches on the persistence of a population, which has implications in agroecology and for understanding zoonoses; in such cases we consider a pathogenic population and the patches correspond to different host species. The occurrence of a springboard effect, where the addition of a patch contributes to persistence, or on the contrary the emergence of a detrimental effect by increasing the number of patches on the persistence, depends in a rather complex way on the respective positions in the phenotypic space of the optimal phenotypes associated with each patch. From a mathematical point of view, an important part of the difficulty in dealing with $H\\ge 3$, compared to $H=1$ or $H=2$, comes from the lack of symmetry. Our results, which are based on a fixed point theorem, comparison principles, integral estimates, variational arguments, rearrangement techniques, and numerical simulations, provide a better understanding of these dependencies. In particular, we propose a precise characterisation of the situations where the addition of a third patch increases or decreases the chances of persistence, compared to a situation with only two patches.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-04T15:27:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B30",
      "35B40",
      "35K40",
      "35Q92",
      "92D25"
    ],
    "keywords": [
      "heterogeneous environment",
      "adaptation",
      "persistence",
      "third patch increases",
      "population"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}