Diffusive Holling type-II predator–prey system with harvesting of prey

We study diffusive heterogeneous predator–prey system with Holling type-II functional response in the presence of harvesting in prey equation i.e. { − Δ u = λ u − a ( x ) u 2 − b u v 1 + m u − c h ( x ) in Ω , − Δ v = μ v − v 2 + d u v 1 + m u in Ω , ∂ ν u = ∂ ν v = 0 on ∂ Ω , where Ω is a smooth bounded region in R N and a ( x ) is a nonnegative continuous function on Ω ¯ , with Ω 0 a region such that Ω ¯ 0 ⊂ Ω and a ( x ) > 0 on Ω ¯ ∖ Ω ¯ 0 . We show that in the case of weak prey growth rate, i.e. λ less than the first eigenvalue of −Δ in Ω 0 , the system has a positive solution for the values of μ in a bounded interval and c > 0 sufficiently small. We also investigate the multiplicity of positive solutions in this case. When the prey growth rate is strong, we will show that the system has a positive solution for large positive value of μ, again provided that c is sufficiently small.