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- W2803050518 abstract "In this article, we analyse the non-local model : $$ frac{partial u}{partial t}=Jstar u -u + f(x,u) quad text{ with }quad x in R^N, $$ 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 $lambda_p$ of the linear problem $$ Jstar varphi(x) -varphi(x) + partial_sf(x,0)varphi(x)+lambda_pvarphi(x)=0 quad text{ in }quad R^N,$$ 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 $u(t,x)$. We also analyse the 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." @default.
- W2803050518 created "2018-05-17" @default.
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- W2803050518 date "2014-06-24" @default.
- W2803050518 modified "2023-10-06" @default.
- W2803050518 title "Nonlocal heterogeneous KPP equations in $R^N$" @default.
- W2803050518 hasPublicationYear "2014" @default.
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