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: $${\int }_{\Omega }K(x,y)u(y)dy-{\int }_{\Omega }K(y,x)u(x)dy+a_{0}u+\lambda a_1(x)u-\beta (x)u^p=0\text{in}\times Ø$$ where $\Omega \subset {\mathbb{R}}^n$ is a bounded open set, $K\in C({\mathbb{R}}^n\times {\mathbb{R}}^n)$ is nonnegative, $a_i,\beta \in C(\Omega )$ and $\lambda \in \mathbb{R}$. 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 ${\beta }_{|\omega }\equiv 0$).