# 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
##### References:

show all references

##### 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 $0.025$
 [1] Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342 [2] Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118 [3] Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136 [4] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [5] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [6] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [7] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 [8] Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 [9] Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 [10] Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454 [11] Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448 [12] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [13] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [14] Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074 [15] Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 [16] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [17] S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435 [18] Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276 [19] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445 [20] Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

2019 Impact Factor: 1.27