A heterogeneous diffusive logistic model of a single species population dynamics with predation and harvesting terms. (June 2017) - Rokn-e-vafa, Saeed Shabani; Tehrani, Hossein Torabi
Abstract: We study existence and multiplicity of positive solutions of a heterogeneous diffusive logistic equation with predation and harvesting terms, − Δ u = a u − b ( x ) u 2 − c u 1 + m u − d h ( x ) in Ω, ∂ u ∂ ν = 0 on ∂ Ω, where a, c, m and d are positive constants, Ω a bounded smooth domain in R N, and b ( x ) is a nonnegative function on Ω ¯, with Ω 0 a region such that Ω ¯ 0 ⊂ Ω and Ω ¯ 0 = x ∈ Ω : b ( x ) = 0 . Under the strong growth rate assumption, that is, when a is greater than the first eigenvalue of − Δ in Ω 0 with Dirichlet boundary condition, we show that the equation has at least one positive solution for 0 ≤ d < d a, for some d a > 0 . In addition, in case c < a we prove some multiplicity results when m < m 0, with m 0 = ∫ Ω b ( x ) d x / ( a | Ω | ) .