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.


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
Full Text: DOI


[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)
[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)
[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 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 New York
[29] Gopalswamy, K., Stability and oscillation in delay differential equations of population dynamics, (1992), Kluwer Academic Publisher The Netherlands
[30] Kot, M., Element of mathematical biology, (2001), 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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