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Bifurcation and control of a bioeconomic model of a prey-predator system with a time delay. (English) Zbl 1245.92060

The dynamical behaviour of a bioeconomic model system of differential algebraic equations is analysed. The system describes a prey-predator fishery with two zones (free fishing and protected). It is shown that a singularity-induced bifurcation occurs when a variation of economic interest of harvesting is considered, thus the interior equilibrium changes from being stable to being unstable. A sufficient condition is obtained for the interior equilibrium to be stable when a state feedback controller is introduced. Further, when the time delay is incorporated as a parameter, Hopf bifurcation also occurs. Then these results are verified by some numerical simulations.

MSC:

92D40 Ecology
91B76 Environmental economics (natural resource models, harvesting, pollution, etc.)
34K18 Bifurcation theory of functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
65C20 Probabilistic models, generic numerical methods in probability and statistics
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References:

[1] Marszalek, W. G.; Trzaska, Z. W., Singularity induced bifurcations in electrical power system, IEEE Transactions on Power Systems, 20, 302-310 (2005)
[2] Ayasun, S.; Nwankpa, C. O.; Kwatny, H. G., Computation of singular and singularity induced bifurcation points of differential-algebraic power system mode, IEEE Transactions on Circuits and Systems, 51, 1525-1537 (2004) · Zbl 1374.34027
[3] Yue, M.; Schlueter, R., Bifurcation subsystem and its application in power system analysis, IEEE Transactions on Power Systems, 19, 1885-1893 (2004)
[4] 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
[5] Kar, T. K.; Pahari, U. K., Modelling and analysis of a prey-predator system with stage-structure and harvesting, Nonlinear analysis: Real World Applications, 8, 601-609 (2007) · Zbl 1152.34374
[6] Feng, W., Dynamics in 3-species predator-prey models with time delays, Discrete and Continuous Dynamical Systems, Supplement, 364-372 (2007) · Zbl 1163.35323
[7] Dai, G.; Tang, M., Coexistence region and global dynamics of a harvested predator-prey system, SIAM Journal on Applied Mathematics, 58, 193-210 (1998) · Zbl 0916.34034
[8] Myerscough, M. R.; Gray, B. F.; Hogarth, W. L.; Norbury, J., An analysis of an ordinary differential equation model for a two-species predator-prey system with harvesting and stocking, Journal of Mathematical Biology, 30, 389-411 (1992) · Zbl 0749.92022
[9] Xiao, D.; Ruan, S., Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Institute Communications, 21, 493-506 (1999) · Zbl 0917.34029
[10] Berryman, A. A., The origin and evolution of predator-prey theory, Ecology, 75, 1530-1535 (1992)
[11] Kar, T. K., Selective harvesting in a prey-predator fishery with time delay, Mathematical and Computer Modelling, 38, 449-458 (2003) · Zbl 1045.92046
[12] Martin, A.; Ruan, S., Predator-prey models with delay and prey harvesting, Journal of Mathematical Biology, 43, 247-267 (2001) · Zbl 1008.34066
[13] Toaha, S.; Hassan, M. A., Stability analysis of predator-prey population model with time delay and constant rate of harvesting, Journal of Mathematics, 40, 37-48 (2008) · Zbl 1226.37057
[14] Ruan, S., On nonlinear dynamics of predator models with discrete delay, Mathematical Modelling of Natural Phenomena, 4, 2, 140-188 (2009) · Zbl 1172.34046
[15] Kar, T. K.; Chakraborty, K., Bioeconomic modelling of a prey predator system using differential algebraic equations, International Journal of Engineering, Science and Technology, 2, 1, 13-34 (2010)
[16] Zhang, X.; Zhang, Q.; Zhang, Y., Bifurcations of a class of singular biological economic models, Chaos, Solitons and Fractals, 40, 3, 1309-1318 (2009) · Zbl 1197.37129
[17] Zhang, G.; Zhu, L.; Chen, B., Hopf bifurcation and stability for a differential-algebraic biological economic system, Applied Mathematics and Computation, 217, 1, 330-338 (2010) · Zbl 1197.92051
[18] Liu, C.; Duan, X.; Yang, C., Dynamic analysis in A differential-algebraic harmful phytoplankton blooms model, International Journal of Information and Systems Sciences, 5, 3-4, 340-350 (2009) · Zbl 1498.92306
[19] Liu, C.; Zhang, Q.; Huang, J.; Tang, W., Dynamical behavior of a harvested prey-predator model with stage structure and discrete time delay, Journal of Biological Systems, 17, 4, 759-777 (2009) · Zbl 1342.92184
[20] Liu, C.; Zhang, Q.; Zhang, X., Dynamic analysis in a harvested differential-algebraic prey-predator model, Journal of Mechanics in Medicine and Biology, 9, 1, 123-140 (2009)
[21] Liu, C.; Duan, X.; Zhang, Q.; Wang, C., The dynamics of a differential-algebraic food web with harvesting, International Journal of Information and Systems Sciences, 5, 3-4, 457-466 (2009) · Zbl 1498.92307
[22] Liu, C.; Zhang, Q.; Duan, X., Dynamical behavior in a harvested differential algebraic prey-predator model with discrete time delay and stage structure, Journal of the Franklin Institute, 346, 10, 1038-1059 (2009) · Zbl 1185.49043
[23] Liu, C.; Zhang, Q.; Zhang, X.; Duan, X., Dynamical behavior in a harvested differential-algebraic prey-predator model, International Journal of Biomathematics, 2, 4, 463-482 (2009) · Zbl 1342.92185
[24] Liu, C.; Zhang, Q.; Zhang, X.; Duan, X., Dynamical behavior in a stage-structured differential-algebraic prey-predator model with discrete time delay and harvesting, Journal of Computational and Applied Mathematics, 231, 2, 612-625 (2009) · Zbl 1176.34101
[25] Liu, C.; Zhang, Q.; Zhang, Y., Bifurcation and control in a differential-algebraic harvested prey-predator model with stage structure for predator, International Journal of Bifurcation and Chaos, 18, 10, 3159-3168 (2008) · Zbl 1165.93329
[26] Clark, C. W., Mathematical Bioeconomics: The Optimal Management of Renewable Resources (1990), John Wiley and Sons: John Wiley and Sons New York · Zbl 0712.90018
[27] Venkatasubramanian, V.; Schattler, H.; Zaborszky, J., Local bifurcations and feasibility regions in differential-algebraic systems, IEEE Transactions on Automatic Control, 40, 12, 1992-2013 (1995) · Zbl 0843.34045
[28] Dai, L., Singular Control System (1989), Springer: Springer New York
[29] Gopalswamy, K., Stability and Oscillation in Delay Differential Equations of Population Dynamics (1992), Kluwer Academic Publisher: Kluwer Academic Publisher The Netherlands
[30] Kot, M., Element of Mathematical Biology (2001), Cambridge University Press: Cambridge University Press Cambridge
[31] Freedman, W.; Rao, V. S.H., The trade-off between mutual interference and time lags in predator-prey systems, Bulletin of Mathematical Biology, 45, 991-1004 (1983) · Zbl 0535.92024
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