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NONLOCAL REFUGE MODEL WITH A PARTIAL CONTROL

Abstract : In this paper, we analyse the structure of the set of positive solutions of an heterogeneous nonlocal equation of the form: $$ \int_{\Omega} K(x, y)u(y)\,dy -\int_ {\Omega}K(y, x)u(x)\, dy + a_0u+\lambda a_1(x)u -\beta(x)u^p=0 \quad \text{in}\quad \times \O$$ where $\Omega\subset \R^n$ is a bounded open set, $K\in C(\R^n\times \R^n) $ is nonnegative, $a_i,\beta \in C(\Omega)$ and $\lambda\in\R$. Such type of equation appears in some studies of population dynamics where the above solutions are the stationary states of the dynamic of a spatially structured population evolving in a heterogeneous partially controlled landscape and submitted to a long range dispersal. Under some fairly general assumptions on $K,a_i$ and $\beta$ we first establish a necessary and sufficient criterium for the existence of a unique positive solution. Then we analyse the structure of the set of positive solution $(\lambda,u_\lambda)$ with respect to the presence or absence of a refuge zone (i.e $\omega$ so that $\beta_{|\omega}\equiv 0$).
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https://hal.archives-ouvertes.fr/hal-00828256
Contributor : Jerome Coville <>
Submitted on : Thursday, May 30, 2013 - 3:55:13 PM
Last modification on : Wednesday, July 7, 2021 - 3:36:10 AM
Long-term archiving on: : Saturday, August 31, 2013 - 6:45:08 AM

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  • HAL Id : hal-00828256, version 1
  • ARXIV : 1305.7122

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Jerome Coville. NONLOCAL REFUGE MODEL WITH A PARTIAL CONTROL. Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2015, 35 (4), pp.1421 - 1446. ⟨hal-00828256⟩

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