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Hopf bifurcation for a differential-algebraic biological economic system with time delay. (English) Zbl 1238.92058
Summary: We consider a differential-algebraic biological economic system with time delays and harvesting where the dynamics is logistic with the carrying capacity proportional to the prey population. By considering the time delay as bifurcation parameter, we analyze the stability and the Hopf bifurcations of the differential-algebraic biological economic system based on the normal form approach and center manifold theory. Finally numerical simulations are performed to illustrate the analytical results.
MSC:
92D40Ecology
12H05Differential algebra
34C23Bifurcation (ODE)
References:
[1]Sadhukhan, D.; Mondal, B.; Maiti, M.: Discrete age-structured population model with age dependent harvesting and its stability analysis, Appl. math. Comput. 201, 631-639 (2008) · Zbl 1143.92041 · doi:10.1016/j.amc.2007.12.063
[2]Berezansky, L.; Idels, L.: Periodic fox production harvesting models with delay, Appl. math. Comput. 195, 142-153 (2008) · Zbl 1128.92042 · doi:10.1016/j.amc.2007.04.084
[3]Yang, Yu: Hopf bifurcation in a two-competitor, one-prey system with time delay, Appl. math. Comput. 214, 228-235 (2009) · Zbl 1181.34090 · doi:10.1016/j.amc.2009.03.078
[4]Zhao, H.; Wang, L.; Ma, C.: Hopf bifurcation in a delayed Lotka – Volterra predator – prey system, Nonlinear anal. RWA 9, No. 1, 114-127 (2008) · Zbl 1149.34048 · doi:10.1016/j.nonrwa.2006.09.007
[5]Kumar, S.; Srivastava, S. K.; Chingakham, P.: Hopf bifurcation and stability analysis in a harvested one-predator-two-prey model, Appl. math. Comput. 129, 107-118 (2002) · Zbl 1017.92041 · doi:10.1016/S0096-3003(01)00033-9
[6]Fan, Y. H.; Li, W. T.: Permanence in delayed ratio-dependent predator – prey models with monotonic functional responses, Nonlinear anal. RWA 8, No. 2, 424-434 (2007) · Zbl 1152.34368 · doi:10.1016/j.nonrwa.2005.12.003
[7]&ccedil, C.; Elik: The stability and Hopf bifurcation for a predator – prey system with time delay, Chaos solitons fract. 37, 87-99 (2008)
[8]Mugisha, J. Y. T.; Ddumba, H.: The dynamics of a fisheries model with feeding patterns and harvesting: lates niloticus and oreochromis niloticus in lake Victoria, Appl. math. Comput 186, 142-158 (2007) · Zbl 1112.92066 · doi:10.1016/j.amc.2006.07.095
[9]Xiao, M.; Cao, J. D.: Hopf bifurcation and non-hyperbolic equilibrium in a ratio-dependent predator – prey model with linear harvesting rate: analysis and computation, Math. comput. Model 50, 360-379 (2009) · Zbl 1185.34047 · doi:10.1016/j.mcm.2009.04.018
[10]Huo, H. F.; Li, W. T.: Existence and global stability of periodic solutions of a discrete predator – prey system with delays, Appl. math. Comput 153, 337-351 (2004) · Zbl 1043.92038 · doi:10.1016/S0096-3003(03)00635-0
[11]Gordon, H. S.: Economic theory of a common property resource: the fishery, J. polit. Econ 62, No. 2, 124-142 (1954)
[12]Zhang, X.; Zhang, Q. L.: Bifurcation analysis and control of a class of hybrid biological economic models, Nonlinear anal.: hybrid syst. 3, 578-587 (2009) · Zbl 1194.93092 · doi:10.1016/j.nahs.2009.04.009
[13]Zhang, X.; Zhang, Q. L.; Zhang, Y.: Bifurcations of a class of singular biological economic models, Chaos solitons fract. 40, 1309-1318 (2009) · Zbl 1197.37129 · doi:10.1016/j.chaos.2007.09.010
[14]Zhang, G. D.; Zhu, L. L.; Chen, B. S.: Hopf bifurcation and stability for a differential – algebraic biological economic system, Appl. math. Comput 217, No. 1, 330-338 (2010) · Zbl 1197.92051 · doi:10.1016/j.amc.2010.05.065
[15]Chen, B. S.; Liao, X. X.; Liu, Y. Q.: Normal forms and bifurcations for the difference-algebraic systems, Acta math. Appl. sin. 23, 429-443 (2000) · Zbl 0960.34004
[16]Cooke, K. L.; Grossman, Z.: Discrete delay, distributed delay and stability switches, J. math. Anal. appl 86, No. 2, 592-627 (1982) · Zbl 0492.34064 · doi:10.1016/0022-247X(82)90243-8
[17]Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[18]Hassard, B.; Kazarinoff, D.; Wan, Y.: Theory and applications of Hopf bifurcation, (1981)