Effect of harvesting quota and protection zone in a nonlocal dispersal reaction–diffusion equation. (February 2019)

• Record Type: Journal Article
• Authors: Chen, Yujuan; Zhu, Yueping
• Is Part Of: Nonlinear analysis. Volume 45(2019)
• Page Start: 854
• Page End: 865
• Publication Date: 2019-02
• Subjects: Nonlocal dispersal -- Stationary solution -- Existence and uniqueness -- Principal eigenvalue -- Harvesting -- Protection zone; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2018.08.007
• Languages: English
• ISSNs: 1468-1218

Abstract

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.