# 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 and Continuous Dynamical Systems - B, 2017, 22 (3) : 791-807. doi: 10.3934/dcdsb.2017039
##### References:
 [1] A. Avasthi, California tries to connect its scattered marine reserves, Science, 308 (2005), 487-488.  doi: 10.1126/science.308.5721.487. [2] B. A. Block and H. Dewar, Migratory movements, depth preferences, and thermal biology of {A}tlantic bluefin tuna, Science, 293 (2001), 1310-1314.  doi: 10.1126/science.1061197. [3] K. Brown, W. N. Adger, E. Tompkins, P. Bacon, D. Shim and K. Young, Trade-off analysis for marine protected area management, Ecological Economics, 37 (2001), 417-434.  doi: 10.1016/S0921-8009(00)00293-7. [4] R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments, Proc. Roy. Soc. Edinburgh Sect. A, 112 (1989), 293-318.  doi: 10.1017/S030821050001876X. [5] R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations Wiley Series in Mathematical and Computational Biology. John Wiley & Sons Ltd. , Chichester, 2003. [6] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2. [7] R.-H. Cui, J.-P. Shi and B.-Y. Wu, Strong Allee effect in a diffusive predator-prey system with a protection zone, J. Differential Equations, 256 (2014), 108-129.  doi: 10.1016/j.jde.2013.08.015. [8] E. N. Dancer and Y.-H. Du, Effects of certain degeneracies in the predator-prey model, SIAM J. Math. Anal., 34 (2002), 292-314.  doi: 10.1137/S0036141001387598. [9] Y. -H. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations. Vol. 1 volume 2 of {Series in Partial Differential Equations and Applications}, World Scientific Publishing Co. Pte. Ltd. , Hackensack, NJ, 2006. [10] Y. -H. Du, Change of environment in model ecosystems: Effect of a protection zone in diffusive population models, In Recent progress on reaction-diffusion systems and viscosity solutions, World Sci. Publ. , Hackensack, NJ, (2009), 49–73. [11] Y.-H. Du and X. Liang, A diffusive competition model with a protection zone, J. Differential Equations, 244 (2008), 61-86.  doi: 10.1016/j.jde.2007.10.005. [12] Y.-H. Du, R. Peng and M.-X. Wang, Effect of a protection zone in the diffusive {L}eslie predator-prey model, J. Differential Equations, 246 (2009), 3932-3956.  doi: 10.1016/j.jde.2008.11.007. [13] Y.-H. Du and J.-P. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91.  doi: 10.1016/j.jde.2006.01.013. [14] Y. -H. Du and J. -P. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment, In Nonlinear Dynamics and Evolution Equations, of Fields Inst. Commun. . Amer. Math. Soc. , Providence, RI, 48 (2006), 95–135. [15] Y.-H. Du and J.-P. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593.  doi: 10.1090/S0002-9947-07-04262-6. [16] S. D. Gaines, C. White, M. H. Carr and S. R. Palumbi, Designing marine reserve networks for both conservation and fisheries management, Proc. Nati. Acad. Scie., 107 (2010), 18286-18293.  doi: 10.1073/pnas.0906473107. [17] S. Gubbay, Marine protected areas --past, present and future, Springer, 1995. Ecological Economics, 37 (2001), 417-434.  doi: 10.1007/978-94-011-0527-9_1. [18] J. B. C. Jackson and M. X. Kirby, Historical overfishing and the recent collapse of coastal ecosystems, Science, 293 (2001), 629-637.  doi: 10.1126/science.1059199. [19] P. J. S. Jones, Marine protected area strategies: issues, divergences and the search for middle ground, Reviews in Fish Biology and Fisheries, 11 (2001), 197-216.  doi: 10.1023/A:1020327007975. [20] H. M. Joshi, G. E. Herrera, S. Lenhart and M. G. Neubert, Optimal dynamic harvest of a mobile renewable resource, Nat. Resour. Model, 22 (2009), 322-343.  doi: 10.1111/j.1939-7445.2008.00038.x. [21] K. Kurata and J.-P. Shi, Optimal spatial harvesting strategy and symmetry-breaking, Appl. Math. Optim., 58 (2008), 89-110.  doi: 10.1007/s00245-007-9032-7. [22] Y. Lou and E. Yanagida, Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics, Japan J. Indust. Appl. Math., 23 (2006), 275-292.  doi: 10.1007/BF03167595. [23] R. M. May, Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477.  doi: 10.1038/269471a0. [24] I. Noy-Meir, Stability of grazing systems: An application of predator-prey graphs, Jour. Ecology, 63 (1975), 459-481.  doi: 10.2307/2258730. [25] M. G. Neubert, Marine reserves and optimal harvesting, Ecol. Lett., 6 (2003), 843-849.  doi: 10.1046/j.1461-0248.2003.00493.x. [26] K. Oeda, Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone, J. Differential Equations, 250 (2011), 3988-4009.  doi: 10.1016/j.jde.2011.01.026. [27] S. Oruganti, J.-P. Shi and R. Shivaji, Diffusive logistic equation with constant yield harvesting. Ⅰ. Steady states, Trans. Amer. Math. Soc., 354 (2002), 3601-3619.  doi: 10.1090/S0002-9947-02-03005-2. [28] D. Pauly and V. Christensen, Towards sustainability in world fisheries, Nature, 148 (2002), 689-695.  doi: 10.1038/nature01017. [29] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9. [30] J.-P. Shi and R. Shivaji, Global bifurcations of concave semipositone problems, J. Differential Equations, 246 (2009), 2788-2812. [31] J.-P. Shi and X.-F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.  doi: 10.1016/j.jde.2008.09.009.

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##### References:
 [1] A. Avasthi, California tries to connect its scattered marine reserves, Science, 308 (2005), 487-488.  doi: 10.1126/science.308.5721.487. [2] B. A. Block and H. Dewar, Migratory movements, depth preferences, and thermal biology of {A}tlantic bluefin tuna, Science, 293 (2001), 1310-1314.  doi: 10.1126/science.1061197. [3] K. Brown, W. N. Adger, E. Tompkins, P. Bacon, D. Shim and K. Young, Trade-off analysis for marine protected area management, Ecological Economics, 37 (2001), 417-434.  doi: 10.1016/S0921-8009(00)00293-7. [4] R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments, Proc. Roy. Soc. Edinburgh Sect. A, 112 (1989), 293-318.  doi: 10.1017/S030821050001876X. [5] R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations Wiley Series in Mathematical and Computational Biology. John Wiley & Sons Ltd. , Chichester, 2003. [6] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2. [7] R.-H. Cui, J.-P. Shi and B.-Y. Wu, Strong Allee effect in a diffusive predator-prey system with a protection zone, J. Differential Equations, 256 (2014), 108-129.  doi: 10.1016/j.jde.2013.08.015. [8] E. N. Dancer and Y.-H. Du, Effects of certain degeneracies in the predator-prey model, SIAM J. Math. Anal., 34 (2002), 292-314.  doi: 10.1137/S0036141001387598. [9] Y. -H. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations. Vol. 1 volume 2 of {Series in Partial Differential Equations and Applications}, World Scientific Publishing Co. Pte. Ltd. , Hackensack, NJ, 2006. [10] Y. -H. Du, Change of environment in model ecosystems: Effect of a protection zone in diffusive population models, In Recent progress on reaction-diffusion systems and viscosity solutions, World Sci. Publ. , Hackensack, NJ, (2009), 49–73. [11] Y.-H. Du and X. Liang, A diffusive competition model with a protection zone, J. Differential Equations, 244 (2008), 61-86.  doi: 10.1016/j.jde.2007.10.005. [12] Y.-H. Du, R. Peng and M.-X. Wang, Effect of a protection zone in the diffusive {L}eslie predator-prey model, J. Differential Equations, 246 (2009), 3932-3956.  doi: 10.1016/j.jde.2008.11.007. [13] Y.-H. Du and J.-P. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91.  doi: 10.1016/j.jde.2006.01.013. [14] Y. -H. Du and J. -P. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment, In Nonlinear Dynamics and Evolution Equations, of Fields Inst. Commun. . Amer. Math. Soc. , Providence, RI, 48 (2006), 95–135. [15] Y.-H. Du and J.-P. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593.  doi: 10.1090/S0002-9947-07-04262-6. [16] S. D. Gaines, C. White, M. H. Carr and S. R. Palumbi, Designing marine reserve networks for both conservation and fisheries management, Proc. Nati. Acad. Scie., 107 (2010), 18286-18293.  doi: 10.1073/pnas.0906473107. [17] S. Gubbay, Marine protected areas --past, present and future, Springer, 1995. Ecological Economics, 37 (2001), 417-434.  doi: 10.1007/978-94-011-0527-9_1. [18] J. B. C. Jackson and M. X. Kirby, Historical overfishing and the recent collapse of coastal ecosystems, Science, 293 (2001), 629-637.  doi: 10.1126/science.1059199. [19] P. J. S. Jones, Marine protected area strategies: issues, divergences and the search for middle ground, Reviews in Fish Biology and Fisheries, 11 (2001), 197-216.  doi: 10.1023/A:1020327007975. [20] H. M. Joshi, G. E. Herrera, S. Lenhart and M. G. Neubert, Optimal dynamic harvest of a mobile renewable resource, Nat. Resour. Model, 22 (2009), 322-343.  doi: 10.1111/j.1939-7445.2008.00038.x. [21] K. Kurata and J.-P. Shi, Optimal spatial harvesting strategy and symmetry-breaking, Appl. Math. Optim., 58 (2008), 89-110.  doi: 10.1007/s00245-007-9032-7. [22] Y. Lou and E. Yanagida, Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics, Japan J. Indust. Appl. Math., 23 (2006), 275-292.  doi: 10.1007/BF03167595. [23] R. M. May, Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477.  doi: 10.1038/269471a0. [24] I. Noy-Meir, Stability of grazing systems: An application of predator-prey graphs, Jour. Ecology, 63 (1975), 459-481.  doi: 10.2307/2258730. [25] M. G. Neubert, Marine reserves and optimal harvesting, Ecol. Lett., 6 (2003), 843-849.  doi: 10.1046/j.1461-0248.2003.00493.x. [26] K. Oeda, Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone, J. Differential Equations, 250 (2011), 3988-4009.  doi: 10.1016/j.jde.2011.01.026. [27] S. Oruganti, J.-P. Shi and R. Shivaji, Diffusive logistic equation with constant yield harvesting. Ⅰ. Steady states, Trans. Amer. Math. Soc., 354 (2002), 3601-3619.  doi: 10.1090/S0002-9947-02-03005-2. [28] D. Pauly and V. Christensen, Towards sustainability in world fisheries, Nature, 148 (2002), 689-695.  doi: 10.1038/nature01017. [29] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9. [30] J.-P. Shi and R. Shivaji, Global bifurcations of concave semipositone problems, J. Differential Equations, 246 (2009), 2788-2812. [31] J.-P. Shi and X.-F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.  doi: 10.1016/j.jde.2008.09.009.
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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