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Persistence criteria for populations with non-local dispersion

H. Berestycki 1 Jerome Coville 2 H.-H. Vo 1
1 CAMS
CAMS - Centre d'Analyse et de Mathématique sociales
Abstract : n this article, we analyse the non-local model: ∂tu(t,x)=J⋆u(t,x)−u(t,x)+f(x,u(t,x))for t>0, and x∈RN, where J is a positive continuous dispersal kernel and f(x, u) 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 λp of the linear problem J⋆φ−φ+∂sf(x,0)φ+λpφ=0 in RN, is negative. λp 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(t, x). 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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Submitted on : Wednesday, May 27, 2020 - 6:13:32 PM
Last modification on : Tuesday, December 1, 2020 - 3:04:55 AM

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H. Berestycki, Jerome Coville, H.-H. Vo. Persistence criteria for populations with non-local dispersion. Journal of Mathematical Biology, Springer Verlag (Germany), 2016, 72 (7), pp.1693-1745. ⟨10.1007/s00285-015-0911-2⟩. ⟨hal-02635497⟩

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