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Hopf bifurcation for a predator-prey biological economic system with Holling type II functional response. (English) Zbl 1225.34055
A biological economic system which considers a prey-predator system with Holling type II functional response and harvesting on the prey is proposed. The model is described by differential-algebraic equations. By using the theory of differential-algebraic systems and the theory of Hopf bifurcations, the Hopf bifurcation of the proposed system is investigated. The economic profit is chosen as a positive bifurcation parameter here. It is found that Hopf bifurcation occurs as the economic profit increases beyond a certain threshold. Numerical simulations are carried out to demonstrate the effectiveness of our results. Some interesting open problems are given.

MSC:
34C60Qualitative investigation and simulation of models (ODE)
34C23Bifurcation (ODE)
92D25Population dynamics (general)
34A09Implicit equations, differential-algebraic equations
92D40Ecology
Software:
Matlab
WorldCat.org
Full Text: DOI
References:
[1] Chen, B. S.; Liao, X. X.; Liu, Y. Q.: Normal forms and bifurcations for the differential-algebraic systems, Acta math. Appl. sinica 23, 429-433 (2000) · Zbl 0960.34004
[2] Gordon, H. S.: Economic theory of a common property resource: the fishery, J. polit. Econ. 62, No. 2, 124-142 (1954)
[3] Lucas, W. F.: Modules in applied mathematics: differential equation models, (1983)
[4] Kot, M.: Elements of mathematical biology, (2001)
[5] Huang, X. C.: Stability of a general predator--prey model, Journal of the franklin institute 327, 751-769 (1990) · Zbl 0719.34093 · doi:10.1016/0016-0032(90)90081-S
[6] Krajewski, W.; Viaro, U.: Locating the equilibrium points of a predator--prey model by means of affine state feedback, Journal of the franklin institute 345, 489-498 (2008) · Zbl 1167.49035 · doi:10.1016/j.jfranklin.2008.02.001
[7] Kumar, S.; Srivastava, S. K.; Chingakham, P.: Hopf bifurcation and stability analysis in a harvested one-predator--two-prey model, Appl. math. Comput. 129, No. 1, 107-118 (2002) · Zbl 1017.92041 · doi:10.1016/S0096-3003(01)00033-9
[8] Qu, Y.; Wei, J. J.: Bifurcation analysis in a predator--prey system with stage-structure and harvesting, Journal of the franklin institute 347, 1097-1113 (2010) · Zbl 1209.34101 · doi:10.1016/j.jfranklin.2010.03.017
[9] Zhang, X.; Zhang, Q. L.; Sreeram, V.: Bifurcation analysis and control of a discrete harvested prey--predator system with beddington--deangelis functional response, Journal of the franklin institute 347, 1076-1096 (2010) · Zbl 1210.92062 · doi:10.1016/j.jfranklin.2010.03.016
[10] Xiao, D. M.; Li, W. X.; Han, M. A.: Dynamics in ratio-dependent predator--prey model with predator harvesting, J. math. Anal. appl. 324, No. 1, 14-29 (2006) · Zbl 1122.34035 · doi:10.1016/j.jmaa.2005.11.048
[11] Kar, T. K.; Matsuda, H.: Global dynamics and controllability of a harvested prey--predator system with Holling type III functional response, Nonlinear anal. HS 1, No. 1, 59-67 (2007) · Zbl 1117.93311 · doi:10.1016/j.nahs.2006.03.002
[12] Liu, C.; Zhang, Q. L.; Duan, X. D.: Dynamical behavior in a harvested differential-algebraic prey--predator model with discrete time delay and stage structure, Journal of the franklin institute 346, 1038-1059 (2009) · Zbl 1185.49043 · doi:10.1016/j.jfranklin.2009.06.004
[13] Zhang, Y.; Zhang, Q. L.; Zhao, L. C.: Bifurcations and control in singular biological economical model with stage structure, Journal of systems engineering 22, No. 3, 232-238 (2007) · Zbl 1153.93398
[14] Zhang, Y.; Zhang, Q. L.: Chaotic control based on descriptor bioeconomic systems, Control and decision 22, No. 4, 445-452 (2007)
[15] Zhang, X.; Zhang, Q. L.; Zhang, Y.: Bifurcations of a class of singular biological economic models, Chaos, solitons and fractals 40, 1309-1318 (2009) · Zbl 1197.37129 · doi:10.1016/j.chaos.2007.09.010
[16] Guckenheimer, J.; Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1983) · Zbl 0515.34001