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: ∫ Ω K(x, y)u(y) dy − ∫ Ω K(y, x)u(x) dy + a0u+ λa1(x)u− β(x)u p = 0 in ×Ω where Ω ⊂ R is a bounded open set, K ∈ C(R × R) is nonnegative, ai, β ∈ C(Ω) and λ ∈ 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,ai 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, uλ) with respect to the presence or absence of a refuge zone (i.e ω so that β|ω ≡ 0).