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Nonlocal refuge model with a partial control

In this paper, we analyse the structure of the set of positive solutions of an heterogeneous nonlocal equation of the form: $$ \int_Ω K(x, y)u(y)\,dy -\int_ ΩK(y, x)u(x)\, dy + a_0u+λa_1(x)u -β(x)u^p=0 \quad \text{in}\quad \times Ø$$ where $Ω\subset \R^n$ is a bounded open set, $K\in C(\R^n\times \R^n) $ is nonnegative, $a_i,β\in C(Ω)$ and $λ\in\R$. 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 $β$ 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 $(λ,u_λ)$ with respect to the presence or absence of a refuge zone (i.e $ω$ so that $β_{|ω}\equiv 0$).