Effect of harvesting quota and protection zone in a nonlocal dispersal reaction-diffusion equation

In this article, we study the nonlocal dispersal reaction-diffusion equation with spatially non-homogeneous harvesting ut=∫RNJ(x−y)u(y,t)dy−u(x,t)+au(1−u)−ch(x)p(u),inΩ×(0,∞),u(x,t)=0inRN∖Ω×(0,∞),u(x,0)=u0(x),inΩ̄,where Ω⊂RN is a bounded smooth domain, a>0 and c>0 are constants, J is a continuous and nonnegative dispersal kernel, p(u) is a harvesting response function which satisfies Holling type II growth condition, and h(x) is the harvesting distribution function which may be zero in some subdomain of Ω. We first establish the existence and uniqueness of positive stationary solutions. Then we show that when the intrinsic growth rate a is larger than the principal eigenvalue of the protection zone, then the population is always sustainable; while in the opposite case, there exists a maximum allowable catch to avoid the population extinction. The existence of an optimal harvesting pattern is also shown.