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Stability and Hopf bifurcation analysis in a prey-predator system with stage-structure for prey and time delay. (English) Zbl 1152.34370
Summary: A delay-differential modelling with stage-structure for prey is investigated. Its dynamics are studied in terms of local analysis and Hopf bifurcation theory, and its linear stability and Hopf bifurcation are demonstrated by studying the characteristic equation. The stability and direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold argument, and numerical simulations are given to illustrate the analytical result.

MSC:
34K20 Stability theory of functional-differential equations
37N25 Dynamical systems in biology
92D25 Population dynamics (general)
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[1] Hassard, B.; Kazarinoff, D.; Wan, Y.H., Theory and application of Hopf bifurcation, (1981), Cambrige University Press Cambrige · Zbl 0474.34002
[2] Rui, Xu; Chaplain, M.A.J.; Davidson, F.A., A lotka – volterra type food chain model with stage structure and time delays, Math anal appl, 315, 90-105, (2006) · Zbl 1096.34055
[3] Zhang, Zhengqiu; Zheng, Xianwu, A periodic stage-structure model, Appl math lett, 16, 1053-1061, (2003) · Zbl 1050.34061
[4] Wendi, Wang; Lansun, Chen, A predator – prey system with stage-structure for predator, Comput math appl, 33, 83-91, (1997)
[5] Chen, Fengde, Periodicity in a ratio-dependent predator – prey system with stage structure for predator, J appl math, 2, 153-169, (2005) · Zbl 1103.34060
[6] Rui, Xu; Chaplain, M.A.J.; Davidson, F.A., Global stability of a lotka – volterra type predator – prey model with stage structure and time delay, Appl math comput, 159, 863-880, (2004) · Zbl 1056.92063
[7] Rui, Xu; Chaplain, M.A.J.; Davidson, F.A., Periodic solutions of a predator – prey model with stage structure for predator, Appl math comput, 154, 847-870, (2004) · Zbl 1048.92035
[8] Zhou, L.; Tang, Y., Stability and Hopf bifurcation for a delay competition diffusion system, Chaos, solitons & fractals, 14, 1201-1225, (2002) · Zbl 1038.35147
[9] Krises; Choudhury, S.R., Bifurcations and chaos in a predator – prey model with delay and a laser-diode system with self-sustained pulsations, Chaos, solitons & fractals, 16, 59-77, (2003) · Zbl 1033.37048
[10] Yuan, S.; Han, M.; Ma, Z., Competition in the chemostat: convergence of a model with delayed response in growth, Chaos, solitons & fractals, 17, 659-667, (2003) · Zbl 1036.92037
[11] Saito, Y., The necessary and sufficient condition for global stability of a lotka – volterra cooperative or competition system with delays, J math anal appl, 268, 109-124, (2002) · Zbl 1012.34072
[12] Hale, J.; Lunel, S., Introduction to functional differential equations, (1993), Springer New York · Zbl 0787.34002
[13] Yuanyuan, Chen; Jiang, Yu; Chengjun, Sun, Stability and Hopf bifurcation analysis in a three-level food chain system with delay, Chaos, solitons & fractals, 31, 683-694, (2007) · Zbl 1146.34051
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