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: ∫Ω K(x, y)u(y) dy - ∫Ω K(y, x)u(x) dy + a0u + λa1(x)u - β(x)up = 0 in Ω where Ω ⊂ ℝn is a bounded domain, K ∈ C(ℝn × ℝn) is non-negative, ai; β ∈ C(Ω) and λ ∈ ℝ. Such type of equation appears in some studies of population dynamics where the population evolves in a partially controlled heterogeneous landscape and disperses on long ranges. Under some fairly general assumptions on K; ai andβ, we first establish a necessary and sufficient condition for the existence of a unique positive solution. Then, we analyse the structure of the set of positive solutions (λ; uλ) depending on the presence or absence of a refuge zone (i.e ω so that β|ω ≡ 0).