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.{−Δu=λu−a(x)u2−buv1+mu−ch(x)in Ω,−Δv=μv−v2+duv1+muin Ω,∂νu=∂νv=0on ∂Ω, where Ω is a smooth bounded region in RN and a(x) is a nonnegative continuous function on Ω¯, with Ω0 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 Ω0, 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.