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. {-Delta u = lambda u - a(x)u(2) - buv/1+mu - ch(x) in Omega, -Delta v = mu v - v(2) + duv/1+mu in Omega, partial derivative(v)u = partial derivative(v)v = 0 on partial derivative Omega, where Omega is a smooth bounded region in R-N and a(x) is a nonnegative continuous function on (Omega) over bar, with Omega(0) a region such that (Omega) over bar (0) subset of Omega and a(x) > 0 on (Omega) over bar\(Omega) over bar (0). We show that in the case of weak prey growth rate, i.e. lambda less than the first eigenvalue of -Delta in Omega(0), the system has a positive solution for the values of mu 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 mu, again provided that c is sufficiently small. (C) 2013 Elsevier Inc. All rights reserved.