# NONLOCAL REFUGE MODEL WITH A PARTIAL CONTROL

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1 BIOSP - Biostatistique et Processus Spatiaux

Abstract : In this paper, we analyse the structure of the set of positive solutions of an heterogeneous nonlocal equation of the form: $$ \int {\Omega} Kx, yuy\,dy -\int {\Omega}Ky, xux\, dy + a 0u+\lambda a 1xu -\betaxu^p=0 \quad \text{in}\quad \times \O$$ 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 $\beta {|\omega}\equiv 0$.

Keywords : Nonlocal diffusion operators principal eigenvalue non trivial solution asymptotic behaviour partially controlled refuge model

Author: ** Jerome Coville - **

Source: https://hal.archives-ouvertes.fr/