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: $$\int_{\Omega} K(x, y)u(y)\,dy -\int_ {\Omega}K(y, x)u(x)\, dy + a_0u+\lambda a_1(x)u -\beta(x)u^p=0 \quad \text{in}\quad \times \O$$ 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 $\beta_{|\omega}\equiv 0$).