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Global dynamics and bifurcation in a stage structured prey-predator fishery model with harvesting. (English) Zbl 1246.92026

Summary: This paper describes a prey-predator model with stage structure for predator and selective harvesting effort on the predator population. The Holling type II functional response function is taken into consideration. All the equilibria of the proposed system are determined and the behavior of the system is investigated near them. Local stability of the system is analyzed. A geometric approach is used to derive sufficient conditions for global stability of the system. The occurrence of Hopf bifurcations of the model system in the neighborhood of the co-existing equilibrium point is shown through considering the maximal relative increase of predation as a bifurcation parameter. Fishing effort used to harvest the predator population is considered as a control to develop a dynamic framework to investigate the optimal utilization of the resource, sustainability properties of the stock and the resource rent earned from the resource. Pontryagin’s maximum principle is used to characterize the optimal control. The optimal system is derived and then solved numerically using an iterative method with a Runge-Kutta fourth-order scheme. Simulation results show that the optimal control scheme can achieve a sustainable ecosystem.

MSC:

92D40 Ecology
49N90 Applications of optimal control and differential games
91B76 Environmental economics (natural resource models, harvesting, pollution, etc.)
37N25 Dynamical systems in biology
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[1] Chen, C.C.; Hsui, C.Y., Fishery policy when considering the future opportunity of harvesting, Mathematical biosciences, 207, 138-160, (2007) · Zbl 1114.92065
[2] Kar, T.K.; Chakraborty, K., Effort dynamics in a prey – predator model with harvesting, International journal of information systems science, 6, 3, 318-332, (2010) · Zbl 1201.91148
[3] Sebestyén, Z.; Varga, Z.; Garay, J.; Cimmaruta, R., Dynamic model and simulation analysis of the genetic impact of population harvesting, Applied mathematics and computation, 216, 2, 565-575, (2010) · Zbl 1184.92037
[4] Xiao, D.; Li, W.; Han, W., Dynamics in a ratio dependent predator prey model with predator harvesting, Mathematical analysis and applications, 324, 1, 4-29, (2006) · Zbl 1122.34035
[5] Garza-Gill, M.D.; Varela-Lafuente, M.M.; Suris-Regueiro, J.C., European hake fishery bioeconomic management (southern stock) applying effort tax, Fisheries science, 60, 199-206, (2003)
[6] Kar, T.K.; Chakraborty, K., A bioeconomic assessment of the bangladesh shrimp fishery, World journal of modelling and simulation, 7, 1, 58-69, (2011)
[7] Matsuda, H.; Nishimori, K., A size-structured model for a stock-recovery program for an exploited endemic fisheries resource, Fisheries science, 1468, 1-14, (2002)
[8] Wang, W.; Mulone, G.; Salemi, F.; Salone, V., Permanence and stability of a stage-structured predator prey model, Journal of mathematical analysis and applications, 262, 499-528, (2001) · Zbl 0997.34069
[9] Leard, B.; Rebaza, J., Analysis of predator – prey models with continuous threshold harvesting, Applied mathematics and computation, 217, 12, 5265-5278, (2011) · Zbl 1207.92046
[10] Chakraborty, K.; Das, S.; Kar, T.K., Optimal control of effort of a stage structured prey – predator fishery model with harvesting, Nonlinear analysis: real world applications, 12, 6, 3452-3467, (2011) · Zbl 1231.49017
[11] Clark, C.W., Mathematical bioeconomics: the optimal management of renewable resources, (1990), Wiley Series New York · Zbl 0712.90018
[12] Clark, C.W., Mathematical models in the economics of renewable resources, SIAM review, 21, 1, 81-99, (1979) · Zbl 0401.90028
[13] Hannesson, R., Optimal harvesting of ecologically interdependent fish species, Journal of environmental economics and management, 10, 329-345, (1983)
[14] Ragozin, D.L.; Brown, G.J., Harvest policies and nonmarket valuation in a predator prey system, Journal of environmental economics and management, 12, 55-168, (1985)
[15] Ströbele, W.J.; Wacker, H., The economics of harvesting predator – prey systems, Journal of economics, 61, 65-81, (1995) · Zbl 0829.90038
[16] Kar, T.K.; Matsuda, H., Controllability of a harvested prey – predator system with time delay, Journal of biological systems, 14, 2, 243-254, (2006) · Zbl 1105.92040
[17] Kar, T.K.; Matsuda, H., Global dynamics and controllability of a harvested prey – predator system with Holling type III functional response, Nonlinear analysis: hybrid systems, 1, 59-67, (2007) · Zbl 1117.93311
[18] Saito, Y.; Jung, C.; Lim, Y., Effects of cannibalism on a basic stage structure, Applied mathematics and computation, 217, 5, 2133-2141, (2010) · Zbl 1200.92034
[19] Kar, T.K.; Chaudhuri, K.S., Regulation of a prey – predator fishery by taxation: a dynamics reaction model, Journal of biological systems, 11, 2, 173-187, (2003) · Zbl 1074.91035
[20] Chaudhuri, K.S.; Johnson, T., Bioeconomic dynamics of a fishery modeled as an S-system, Mathematical biosciences, 99, 231-249, (1990) · Zbl 0699.92024
[21] Ganguly, S.; Chaudhuri, K.S., Regulations of a single species fishery by taxation, Ecological modelling, 82, 51-60, (1995)
[22] Anderson, L.G.; Lee, D.R., Optimal governing instrument, operation level and enforcement in natural resource regulation: the case of the fishery, American journal of agricultural economics, 68, 678-690, (1986)
[23] Krishna, S.V.; Srinivasu, P.D.N.; Kaymakcalan, B., Conservation of an ecosystem through optimal taxation, Bulletin of mathematical biology, 60, 569-584, (1998) · Zbl 1053.92521
[24] Pradhan, T.; Chaudhuri, K.S., A dynamic reaction model of two species fishery with taxation as a control instrument: a capital theoretic analysis, Ecological modelling, 121, 1-16, (1999)
[25] Jang, S.R.-J.; Diamond, S.L., Population-level effects of density dependence in a size-structured fishery model, Applied mathematics and computation, 139, 1, 133-155, (2003) · Zbl 1018.92027
[26] Berryman, A.A., The origins and evolutions of predator – prey theory, Ecology, 73, 1530-1535, (1992)
[27] Xu, R.; Ma, Z., Stability and Hopf bifurcation in a ratio-dependent predator prey system with stage-structure, Chaos, solitons and fractals, 38, 669-684, (2008) · Zbl 1146.34323
[28] Gao, S.J.; Chen, L.S.; Teng, Z.D., Hopf bifurcation and global stability for a delayed predator – prey system with stage structure for predator, Applied mathematics and computation, 202, 721-729, (2008) · Zbl 1151.34067
[29] Kuang, Y.; Takeuchi, Y., Predator – prey dynamics in models of prey dispersal in two-patch environments, Mathematical biosciences, 120, 77-98, (1994) · Zbl 0793.92014
[30] Song, X.; Guo, H., Global stability of a stage-structured predator – prey system, International journal of biomathematics, 1, 3, 313-326, (2008) · Zbl 1173.34043
[31] Chakraborty, K.; Chakraborty, M.; Kar, T.K., Optimal control of harvest and bifurcation of a prey – predator model with stage structure, Applied mathematics and computation, 217, 21, 8778-8792, (2011) · Zbl 1215.92059
[32] Arditi, R.; Ginzburg, L.R., Coupling in predator – prey dynamics: ratio-dependence, Journal of theoretical biology, 139, 311-326, (1989)
[33] Venkatsubramanian, V.; Schattler, H.; Zaborszky, J., Local bifurcation and feasibility regions in differential – algebraic systems, IEEE transactions on automatic control, 40, 12, 1992-2013, (1995) · Zbl 0843.34045
[34] Li, M.Y.; Muldowney, J.S., A geometric approach to global stability problems, SIAM journal on mathematical analysis, 27, 4, 1070-1083, (1996) · Zbl 0873.34041
[35] Haque, M.; Zhen, J.; Venturino, E., An ecoepidemiological predator prey model with standard disease incidence, Mathematical methods in the applied sciences, 32, 7, 875-898, (2008) · Zbl 1158.92035
[36] Bunomo, B.; Onofrio, A.; Lacitignola, D., Global stability of an SIR epidemic model with information dependent vaccination, Mathematical biosciences, 216, 1, 9-16, (2008) · Zbl 1152.92019
[37] Kar, T.K.; Mondal, P.K., Global dynamics and bifurcation in a delayed SIR epidemic model, Nonlinear analysis: real world applications, 12, 2058-2068, (2011) · Zbl 1235.34216
[38] Martin, R.H., Logarithmic norms and projections applied to linear differential systems, Journal of mathematical analysis and applications, 45, 432-454, (1974) · Zbl 0293.34018
[39] Workman, J.T.; Lenhart, S., Optimal control applied to biological models, (2007), Chapman and Hall/CRC · Zbl 1291.92010
[40] Hackbush, W., A numerical method for solving parabolic equations with opposite orientations, Computing, 20, 3, 229-240, (1978) · Zbl 0391.65044
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