{
  "id": "1605.02027",
  "version": "v1",
  "published": "2016-05-06T18:39:25.000Z",
  "updated": "2016-05-06T18:39:25.000Z",
  "title": "Stochastic population growth in spatially heterogeneous environments: The density-dependent case",
  "authors": [
    "Alexandru Hening",
    "Dang H. Nguyen",
    "George Yin"
  ],
  "comment": "40 pages, 10 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "This work is devoted to studying the dynamics of a population subject to the combined effects of stochastic environments, competition for resources, and spatio-temporal heterogeneity and dispersal. The population is spread throughout $n$ patches whose population abundances are modeled as the solutions of a system of nonlinear stochastic differential equations living on $[0,\\infty)^n$. We prove that $\\lambda$, the stochastic growth rate of the system in the absence of competition, determines the long-term behaviour of the population. The parameter $\\lambda$ can be expressed as the Lyapunov exponent of an associated linearized system of stochastic differential equations. Detailed analysis shows that if $\\lambda>0$, the population abundances converge polynomially fast to a unique invariant probability measure on $(0,\\infty)^n$, while when $\\lambda<0$, the abundances of the patches converge almost surely to $0$ exponentially fast. Compared to recent developments, our model incorporates very general density-dependent growth rates and competition terms. Another significant generalization of our work is allowing the environmental noise driving our system to be degenerate. This is more natural from a biological point of view since, for example, the environments of the different patches can be perfectly correlated.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-06T18:39:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "92D25",
      "37H15",
      "60H10"
    ],
    "keywords": [
      "stochastic population growth",
      "spatially heterogeneous environments",
      "density-dependent case",
      "stochastic differential equations living",
      "unique invariant probability measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}