Stability and Hopf bifurcation in a predator-prey model with stage structure for the predator. (English) Zbl 1154.92327

Summary: A predator-prey system with stage structure for the predator and time delay due to the gestation of the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive equilibrium and two boundary equilibria of the system is discussed, respectively. Further, the existence of a Hopf bifurcation at the positive equilibrium is also studied. By using an iteration technique and comparison argument, respectively, sufficient conditions are derived for the global stability of the positive equilibrium and one of the boundary equilibria of the proposed system. As a result, the threshold is obtained for the permanence and extinction of the system. Numerical simulations are carried out to illustrate the main results.


92D40 Ecology
34K18 Bifurcation theory of functional-differential equations
34K20 Stability theory of functional-differential equations
37N25 Dynamical systems in biology
Full Text: DOI


[1] Chen, L.; Song, X.; Lu, Z., Mathematical Models and Methods in Ecology (2003), Sichuan Science and Technology Press
[2] Hale, J., Theory of Functional Differential Equations (1977), Springer: Springer Heidelberg
[3] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press New York · Zbl 0777.34002
[4] 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
[5] Lancaster, P.; Tismenetsky, M., The Theory of Matrices (1985), Academic Press: Academic Press New York · Zbl 0516.15018
[6] Smith, H., Monotone Dynamical System: An Introduction to the Theory of Competitive and Cooperative System (1995), American Mathematical Society: American Mathematical Society Providence, RI
[7] Song, X.; Chen, L., Optimal harvesting and stability for a two-species competitive system with stage structure, Math. Biosci., 170, 173-186 (2001) · Zbl 1028.34049
[8] Wang, W.; Chen, L., A predator-prey system with stage structure for predator, Comput. Math. Appl., 33, 83-91 (1997)
[9] 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
[10] Xiao, Y.; Chen, L., Effects of toxicants on a stage-structured population growth model, Appl. Math. Comput., 123, 63-73 (2001) · Zbl 1017.92044
[11] Xiao, Y.; Chen, L., Global stability of a predator-prey system with stage structure for the predator, ACTA Math. Sin. Engl. Ser., 20, 63-70 (2004) · Zbl 1062.34056
[12] Zhang, X.; Chen, L.; Neumann, A. U., The stage-structured predator-prey model and optimal havesting policy, Math. Biosci., 168, 201-210 (2000) · Zbl 0961.92037
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.