zbMATH — the first resource for mathematics

Stability and Hopf bifurcation in a ratio-dependent predator-prey system with stage structure. (English) Zbl 1146.34323
Summary: A ratio-dependent predator-prey model with stage structure for the predator and time delay due to the gestation of the predator is investigated. By analyzing the characteristic equations, the local stability of a positive equilibrium and a boundary equilibrium is discussed, respectively. Further, it is proved that the system undergoes a Hopf bifurcation at the positive equilibrium when \(\tau = \tau _{0}\). By using an iteration technique, sufficient conditions are derived for the global attractivity of the positive equilibrium. By comparison arguments, sufficient conditions are obtained for the global stability of the boundary equilibrium. Numerical simulations are carried out to illustrate the main results.

34D23 Global stability of solutions to ordinary differential equations
34K18 Bifurcation theory of functional-differential equations
37N25 Dynamical systems in biology
92D25 Population dynamics (general)
Full Text: DOI
[1] Abdallah, Sabah Hafez, Stability and persistence in plankton models with distributed delays, Chaos, solitons & fractals, 17, 879-884, (2003) · Zbl 1033.92032
[2] Arditi, R.; Ginzburg, L.R., Coupling in predator – prey dynamics: ratio-dependence, J theor biol, 139, 311-326, (1989)
[3] Arditi, R.; Perrin, N.; Saiah, H., Functional response and heterogeneities: an experiment test with cladocerans, Oikos, 60, 69-75, (1991)
[4] Arditi, R.; Saiah, H., Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73, 1544-1551, (1992)
[5] Beretta, E.; Kuang, Y., Global analyses in some delayed ratio-dependent predator – prey systems, Nonlinear anal, T.M.A., 32, 381-408, (1998) · Zbl 0946.34061
[6] Chen, L.; Song, X.; Lu, Z., Mathematical models and methods in ecology, (2003), Sichuan Science and Technology Press
[7] Chen, Y.; Yu, J.; Sun, C., Stability and Hopf bifurcation analysis in a three-level food chain system with delay, Chaos, solitons & fractals, 31, 683-694, (2007) · Zbl 1146.34051
[8] El-Sheikh, M.M.A.; Mahrouf, S.A.A., Stability and bifurcation of a simple food chain in a chemostat with removal rates, Chaos, solitons & fractals, 23, 1475-1489, (2005) · Zbl 1062.92068
[9] Gutierrez, A.P., The physiological basis of ratio-dependent predator – prey theory: a metabolic pool model of nicholson’s blowflies as an example, Ecology, 73, 1552-1563, (1992)
[10] Hale, J., Theory of functional differential equations, (1977), Springer-Verlag, Heidelberg
[11] Hanski, I., The functional response of predator: worries about scale, Tree, 6, 141-142, (1991)
[12] Jing, Z.; Yang, J., Bifurcation and chaos in discrete-time predator – prey system, Chaos, solitons & fractals, 27, 259-277, (2006) · Zbl 1085.92045
[13] Krise, S.; Roy Choudhury, S., 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
[14] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press New York · Zbl 0777.34002
[15] Kuang, Y.; Beretta, E., Global qualitative analyses of a ratio-dependent predator – prey system, J math biol, 36, 389-406, (1998) · Zbl 0895.92032
[16] Kuang, Y.; So, J.W.H., Analysis of a delayed two-stage population with space-limited recruitment, SIAM J appl math, 55, 1675-1695, (1995) · Zbl 0847.34076
[17] Lancaster, P.; Tismenetsky, M., The theory of matrices, (1985), Academic Press New York · Zbl 0516.15018
[18] Li, S.; Liao, X.; Li, C., Hopf bifurcation in a Volterra prey – predator model with strong kernel, Chaos, solitons & fractals, 22, 713-722, (2004) · Zbl 1073.34086
[19] Liu, Z.; Yuan, R., Stability and bifurcation in a delayed predator – prey system with beddingtoncdeangelis functional response, J math anal appl, 296, 521-537, (2004) · Zbl 1051.34060
[20] Liu, Z.; Yuan, R., Stability and bifurcation in a harvested one-predator-two-prey model with delays, Chaos, solitons & fractals, 27, 1395-1407, (2006) · Zbl 1097.34051
[21] Meng, X.; Han, M.; Song, Y., Stability and bifurcation in a non-Kolmogorov type prey – predator system with time delay, Math comput modell, 41, 1445-1455, (2005) · Zbl 1198.34157
[22] Smith, Hal, Monotone dynamical system: an introduction to the theory of competitive and cooperative system, (1995), American Mathematical Society Providence, RI · Zbl 0821.34003
[23] Song, Y.; Han, M.; Peng, Y., Stability and Hopf bifurcations in a competitive lotka – volterra system with two delays, Chaos, solitons & fractals, 22, 1139-1148, (2004) · Zbl 1067.34075
[24] Song, Y.; Wei, J., Local Hopf bifurcation and global periodic solutions in a delayed predator – prey system, J math anal appl, 301, 1-21, (2005) · Zbl 1067.34076
[25] Song, Y.; Yuan, S., Bifurcation analysis in a predator – prey system with time delay, Nonlinear anal RWA, 7, 265-284, (2006) · Zbl 1085.92052
[26] Sun, C.; Han, M.; Lin, Y., Analysis of stability and Hopf bifurcation for a delayed logistic equation, Chaos, solitons & fractals, 31, 672-682, (2007) · Zbl 1336.34101
[27] Sun, C.; Lin, Y.; Han, M., Stability and Hopf bifurcation for an epidemic disease model with delay, Chaos, solitons & fractals, 30, 204-216, (2007) · Zbl 1165.34048
[28] Wang, W.; Chen, L., A predator – prey system with stage structure for predator, Comput math appl, 33, 83-91, (1997)
[29] Wang, W.; Fergola, P.; Tenneriello, C., Global attractivity of periodic solutions of population models, J math anal appl, 211, 498-511, (1997) · Zbl 0879.92027
[30] Yuan, S.; Han, M., Bifurcation analysis of a chemostat model with two distributed delays, Chaos, solitons & fractals, 20, 995-1004, (2004) · Zbl 1059.34059
[31] Yuan, S.; Song, Y.; Han, M., Direction and stability of bifurcating periodic solutions of a chemostat model with two distributed delays, Chaos, solitons & fractals, 21, 1109-1123, (2004) · Zbl 1129.34328
[32] Zhou, L.; Tang, Y.; Hussein, S., Stability and Hopf bifurcation for a delay competition diffusion system, Chaos, solitons & fractals, 14, 1201-1225, (2002) · Zbl 1038.35147
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.