Nonlocal refuge model with a partial control

Original · EN

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 -∫ ΩK(y, x)u(x) dy + a₀u+λa₁(x)u -β(x)uᵖ=0 in × Ø where Ω⊂ ⁿ is a bounded open set, K∈ C(ⁿ× ⁿ) is nonnegative, aᵢ,β∈ C(Ω) and λ∈. 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).