Gilpin-Ayala model with spatial diffusion and its optimal harvesting policy

In this paper we consider spatially nonhomogeneous Gilpin-Ayala diffusive equation suffered from exploitation with spatially variable harvesting effort E(x):∂u∂t=DΔu+r(x)u1-(uK(x))θ-E(x)u,(t,x)∈(0,∞)×Ω,u(0,x)=ϕ(x),x∈Ω,∂u∂n=0,t∈(0,∞),x∈∂Ω,which describes the growth of the single-species with Neumann boundary condition and initial value condition. We investigate the global stability of equilibrium solution and optimal harvesting policy. It is found that in the case of ordinary differential equation, our results generalize the spatially homogeneous equations discussed in [C.W. Clark, Mathematical Bio-economics: The Optimal Management of renewable Resources, Wiley, New York, 1976; C.W. Clark, Mathematical Bio-economics: The Optimal Management of Renewable Resources, second ed., Wiley, New York, 1990; M. Fan, K. Wang, Optimal harvesting policy for single population with periodic coefficients, Math. Biosci. 152 (1998) 165; H. Li, A class of single-species models with periodic coefficients and their optimal harvesting policy, J. Biomath. 14(4) (1999) 293], our brief results also generalize the corresponding results in [H. Li, Logistic model for single-species with spatial diffusion and its optimal harvesting policy, J. Biomath. 14(3) (1999) 293] when θ = 1.