Persistence criteria for populations with non-local dispersion

Abstract

In this article, we analyse the non-local model:

$$\begin{aligned} \partial _t u(t,x)=J\star u(t,x) -u(t,x) + f(x,u(t,x)) \quad \text {for }\;t>0,\;\text { and }\; x \in {\mathbb {R}}^N, \end{aligned}$$

where J is a positive continuous dispersal kernel and f(xu) is a heterogeneous KPP type non-linearity describing the growth rate of the population. The ecological niche of the population is assumed to be bounded (i.e. outside a compact set, the environment is assumed to be lethal for the population). For compactly supported dispersal kernels J, we derive an optimal persistence criteria. We prove that a positive stationary solution exists if and only if the generalised principal eigenvalue \(\lambda _p\) of the linear problem

$$\begin{aligned} J\star \varphi -\varphi + \partial _sf(x,0)\varphi +\lambda _p\varphi =0 \quad \text { in }\; {\mathbb {R}}^N, \end{aligned}$$

is negative. \(\lambda _p\) is a spectral quantity that we defined in the spirit of the generalised first eigenvalue of an elliptic operator. In addition, for any continuous non-negative initial data that is bounded or integrable, we establish the long time behaviour of the solution u(tx). We also analyse the impact of the size of the support of the dispersal kernel on the persistence criteria. We exhibit situations where the dispersal strategy has “no impact” on the persistence of the species and other ones where the slowest dispersal strategy is not any more an “Ecological Stable Strategy”. We also discuss persistence criteria for fat-tailed kernels.

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Acknowledgments

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n321186: “Reaction-Diffusion Equations, Propagation and Modelling” held by Henri Berestycki. J. Coville acknowledges support from the ANR JCJC project MODEVOL: ANR-13-JS01-0009 and the ANR project NONLOCAL (ANR-14-CE25-0013). The authors would also thank the anonymous referees for their judicious comments and advices.

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Correspondence to Jérôme Coville.

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Berestycki, H., Coville, J. & Vo, H. Persistence criteria for populations with non-local dispersion. J. Math. Biol. 72, 1693–1745 (2016). https://doi.org/10.1007/s00285-015-0911-2

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Keywords

  • Heterogeneous KPP nonlocal equation
  • Persistence criteria
  • Dispersal budget
  • Asymptotic behaviours
  • ESS

Mathematics Subject Classification

  • Primary 35R09
  • 45K05
  • 92D25
  • Secondary 45C05
  • 45M20
  • 47B65