# American Institute of Mathematical Sciences

May  2017, 22(3): 791-807. doi: 10.3934/dcdsb.2017039

## 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

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

Received  October 2015 Revised  December 2015 Published  January 2017

Fund Project: 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.

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.

Citation: Renhao Cui, Haomiao Li, Linfeng Mei, Junping Shi. Effect of harvesting quota and protection zone in a reaction-diffusion model arising from fishery management. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 791-807. doi: 10.3934/dcdsb.2017039
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##### References:
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 $u_0(x)=1+0.1\sin(2\pi x/10)$
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 $0.025$
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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