Journal of Mathematical Biology

, Volume 72, Issue 7, pp 1693–1745 | Cite as

Persistence criteria for populations with non-local dispersion

  • Henri Berestycki
  • Jérôme CovilleEmail author
  • Hoang-Hung Vo


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.


Heterogeneous KPP nonlocal equation Persistence criteria Dispersal budget Asymptotic behaviours ESS 

Mathematics Subject Classification

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



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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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Henri Berestycki
    • 1
  • Jérôme Coville
    • 2
    Email author
  • Hoang-Hung Vo
    • 3
  1. 1.CAMS - École des Hautes Études en Sciences SocialesParisFrance
  2. 2.UR 546 Biostatistique et Processus Spatiaux, INRAAvignonFrance
  3. 3.Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityViet Nam

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