Effect of harvesting quota and protection zone in a reaction-diffusion model arising from fishery management

Renhao Cui; Haomiao Li; Linfeng Mei; Junping Shi

doi:10.3934/dcdsb.2017039

Article Contents

2017, Volume 22, Issue 3: 791-807. Doi: 10.3934/dcdsb.2017039

This issue Previous Article A priori estimates for positive solutions to subcritical elliptic problems in a class of non-convex regions Next Article Effects of superinfection and cost of immunity on host-parasite co-evolution

Effect of harvesting quota and protection zone in a reaction-diffusion model arising from fishery management

1.
Institute for Mathematical Sciences, Renmin University of China, Beijing, 100872, China
2.
Y.Y.Tseng Functional Analysis Research Center and School of Mathematical Sciences, Harbin Normal University, Harbin, Heilongjiang, 150025, China
3.
Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA
4.
Department of Mathematics, Henan Normal University, Xinxiang, Henan, 453007, China

* Corresponding author: Junping Shi
* Corresponding author: Junping Shi

Dedicated to Professor Steve Cantrell on the occasion of his 60th birthday

Received: September 30, 2015

Revised: November 30, 2015

Published: September 2017

R.-H. Cui is partially supported by the National Natural Science Foundation of China (No. 11401144,11471091 and 11571364), Project Funded by China Postdoctoral Science Foundation (2015M581235), Natural Science Foundation of Heilongjiang Province (JJ2016ZR0019); L.-F. Mei is partially supported by the National Natural Science Foundation of China (No. 11371117); H.-M. Li and J.-P. Shi are partially supported by US-NSF grants DMS-1313243 and DMS-1331021.

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Abstract

A reaction-diffusion logistic population model with spatially nonhomogeneous harvesting is considered. It is shown that when the intrinsic growth rate 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 steady state solutions is also studied for both cases. The existence of an optimal harvesting pattern is also shown, and theoretical results are complemented by some numerical simulations for one-dimensional domains.

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Mathematics Subject Classification: Primary:35J55, 35B32, 92D25, 92D40.

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Figure 1. Simulation of solutions of (33) when $c = C_1^*(h)$ (left) and $c = C_2^*(h)$ (right). Here $\Omega =(0,10)$, $h(x)=0.2\chi_{[2.5,7.5]}$, and initial value $u_0(x)=1+0.1\sin(2\pi x/10)$

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Figure 2. Plot of $C_1^*(h)$ (left) and $C_2^*(h)$ (right) versus parameter $a$ for (33) with protection zone in the interior of $\Omega $. Here the harvesting functions are $h_0(x)$ and $h(x)=h_i^*(x)$ for $1\le i\le 4$, $L=10$ and $b=1$. The values of $C_1^*(h)$ and $C_2^*(h)$ are numerically estimated with each increment of $a$ by $0.025$

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Figure 3. Plot of $C_1^*(h)$ (left) and $C_2^*(h)$ (right) versus parameter $a$ for (33) with harvesting zone in the interior of $\Omega $. Here the harvesting functions are $h_0(x)$ and $h(x)=h_i(x)$ for $1\le i\le 4$, $L=10$ and $b=1$. The values of $C_1^*(h)$ and $C_2^*(h)$ are numerically estimated with each increment of $a$ by $0.025$

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