Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Effect of harvesting quota and protection zone in a reaction-diffusion model arising from fishery management
Pages: 791 - 807, Issue 3, May 2017

doi:10.3934/dcdsb.2017039      Abstract        References        Full text (527.5K)           Related Articles

Renhao Cui - Institute for Mathematical Sciences, Renmin University of China, Beijing, 100872, China (email)
Haomiao Li - Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, United States (email)
Linfeng Mei - Department of Mathematics, Henan Normal University, Xinxiang, Henan 453007, China (email)
Junping Shi - Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795, United States (email)

1 A. Avasthi, California tries to connect its scattered marine reserves, Science, 308 (2005), 487-488.
2 B. A. Block and H. Dewar, et al, Migratory movements, depth preferences, and thermal biology of Atlantic bluefin tuna, Science, 293 (2001), 1310-1314.
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.
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.       
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.       
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.       
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.       
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.       
12 Y.-H. Du, R. Peng and M.-X. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differential Equations, 246 (2009), 3932-3956.       
13 Y.-H. Du and J.-P. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91.       
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.       
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.
17 S. Gubbay, Marine protected areas - past, present and future, Springer, 1995. Ecological Economics, 37 (2001), 417-434.
18 J. B. C. Jackson and M. X. Kirby, et al, Historical overfishing and the recent collapse of coastal ecosystems, Science, 293 (2001), 629-637.
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.
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.       
21 K. Kurata and J.-P. Shi, Optimal spatial harvesting strategy and symmetry-breaking, Appl. Math. Optim., 58 (2008), 89-110.       
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.       
23 R. M. May, Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477.
24 I. Noy-Meir, Stability of grazing systems: An application of predator-prey graphs, Jour. Ecology, 63 (1975), 459-481.
25 M. G. Neubert, Marine reserves and optimal harvesting, Ecol. Lett. 6 (2003), 843-849.
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.       
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.       
28 D. Pauly and V. Christensen, et al, Towards sustainability in world fisheries, Nature, 148 (2002), 689-695.
29 P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.       
30 J.-P. Shi and R. Shivaji, Global bifurcations of concave semipositone problems, In Evolution equations, volume 234 of Lecture Notes in Pure and Appl. Math., pages 385-398. Dekker, New York, 2003.       
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.       

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