Effect of harvesting quota and protection zone in a nonlocal dispersal reaction–diffusion equation. (February 2019) - Chen, Yujuan; Zhu, Yueping
Abstract: In this article, we study the nonlocal dispersal reaction–diffusion equation with spatially non-homogeneous harvesting u t = ∫ R N J ( x − y ) u ( y, t ) d y − u ( x, t ) + a u ( 1 − u ) − c h ( x ) p ( u ), i n Ω × ( 0, ∞ ), u ( x, t ) = 0 i n R N ∖ Ω × ( 0, ∞ ), u ( x, 0 ) = u 0 ( x ), i n Ω ̄, where Ω ⊂ R N 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.