Diffusive logistic equations with harvesting and heterogeneity under strong growth rate

Advances in Nonlinear Analysis

article

Saeed Shabani Rokn-e-vafa¹  Hossein T. Tehrani² 

[1] Department of Mathematical Sciences, Sharif University of Technology;Department of Mathematical Sciences, University of Nevada

关键词: Logistic equation;    harvesting;    heterogeneity;    strong growth rate;    comparison principles;    stable solutions;   

DOI  :  10.1515/anona-2016-0208

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

We consider the equation - Δ ⁢ u = a ⁢ u - b ⁢ ( x ) ⁢ u 2 - c ⁢ h ⁢ ( x )   in  ⁢ Ω , u = 0   on  ⁢ ∂ ⁡ Ω , -\Delta u=au-b(x)u^{2}-ch(x)\quad\text{in }\Omega,\qquad u=0\quad\text{on }% \partial\Omega, where Ω is a smooth bounded domain in ℝ N {\mathbb{R}^{N}} , b ⁢ ( x ) {b(x)} and h ⁢ ( x ) {h(x)} are nonnegative functions, and there exists Ω 0 ⊂ ⊂ Ω {\Omega_{0}\subset\subset\Omega} such that { x : b ⁢ ( x ) = 0 } = Ω ¯ 0 {\{x:b(x)=0\}=\overline{\Omega}_{0}} . We investigate the existence of positive solutions of this equation for c large under the strong growth rate assumption a ≥ λ 1 ⁢ ( Ω 0 ) {a\geq\lambda_{1}(\Omega_{0})} , where λ 1 ⁢ ( Ω 0 ) {\lambda_{1}(\Omega_{0})} is the first eigenvalue of the - Δ {-\Delta} in Ω 0 {\Omega_{0}} with Dirichlet boundary condition. We show that if h ≡ 0 {h\equiv 0} in Ω ∖ Ω ¯ 0 {\Omega\setminus\overline{\Omega}_{0}} , then our equation has a unique positive solution for all c large, provided that a is in a right neighborhood of λ 1 ⁢ ( Ω 0 ) {\lambda_{1}(\Omega_{0})} . For this purpose, we prove and utilize some new results on the positive solution set of this equation in the weak growth rate case.

【 授权许可】

CC BY