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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:
92D40Ecology
91B76Environmental economics (natural resource models, harvesting, pollution, etc.)
34K18Bifurcation theory of functional differential equations
34K60Qualitative investigation and simulation of models
65C20Models (numerical methods)
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)
[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, No. 2, 243-254 (2006) · Zbl 1105.92040 · doi:10.1142/S0218339006001775
[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 · doi:10.1016/j.nonrwa.2006.01.004
[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 · doi:http://www.aimsciences.org/journals/redirecting.jsp?paperID=2819
[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 · doi:10.1137/S0036139994275799
[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 · doi:10.1007/BF00173294
[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 · doi:10.1016/S0895-7177(03)90099-9
[12]Martin, A.; Ruan, S.: Predator-prey models with delay and prey harvesting, Journal of mathematical biology 43, 247-267 (2001) · Zbl 1008.34066 · doi:10.1007/s002850100095
[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, No. 2, 140-188 (2009) · Zbl 1172.34046 · doi:10.1051/mmnp/20094207
[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, No. 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, No. 3, 1309-1318 (2009) · Zbl 1197.37129 · doi:10.1016/j.chaos.2007.09.010
[17]Zhang, G.; Zhu, L.; Chen, B.: Hopf bifurcation and stability for a differential–algebraic biological economic system, Applied mathematics and computation 217, No. 1, 330-338 (2010) · Zbl 1197.92051 · doi:10.1016/j.amc.2010.05.065
[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, No. 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, No. 4, 759-777 (2009)
[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, No. 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, No. 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, No. 10, 1038-1059 (2009) · Zbl 1185.49043 · doi:10.1016/j.jfranklin.2009.06.004
[23]Liu, C.; Zhang, Q.; Zhang, X.; Duan, X.: Dynamical behavior in a harvested differential–algebraic prey–predator model, International journal of biomathematics 2, No. 4, 463-482 (2009)
[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, No. 2, 612-625 (2009) · Zbl 1176.34101 · doi:10.1016/j.cam.2009.04.011
[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, No. 10, 3159-3168 (2008) · Zbl 1165.93329 · doi:10.1142/S0218127408022329
[26]Clark, C. W.: Mathematical bioeconomics: the optimal management of renewable resources, (1990) · 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, No. 12, 1992-2013 (1995) · Zbl 0843.34045 · doi:10.1109/9.478226
[28]Dai, L.: Singular control system, (1989)
[29]Gopalswamy, K.: Stability and oscillation in delay differential equations of population dynamics, (1992) · Zbl 0752.34039
[30]Kot, M.: Element of mathematical biology, (2001)
[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