Stability and global Hopf bifurcation in a delayed predator-prey system. (English) Zbl 1183.37156

Nonlinear Anal., Real World Appl. 11, No. 2, 959-977 (2010); retraction note ibid. 47, 496 p. (2019).
Editorial remark: This article has been retracted at the request of the Editors-in-Chief due to its high similarity to [S. Yuan and Y. Song, IMA J. Appl. Math. 74, No. 4, 574–603 (2009; Zbl 1201.34132)]; see the retraction notice [S. Yuan and F. Zhang, Nonlinear Anal., Real World Appl. 47, 496 p. (2019; Zbl 1407.37124)].
Summary: We consider a delayed predator-prey system with same feedback delays of predator and prey species to their growth, respectively. Using the delay as a bifurcation parameter, we investigate the stability of the positive equilibrium and existence of Hopf bifurcation of the model. It is shown that Hopf bifurcations can occur as the delay crosses some critical values. Moreover, the model can exhibit an interesting property, that is, under certain conditions, the positive equilibrium may switch finite times from stability to instability to stability, and becomes unstable eventually. By deriving the equation describing the flow on the center manifold, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global Hopf bifurcation result of Wu [J. Wu, Trans. Amer. Math. Soc. 350, 4799–4838 (1998; Zbl 0905.34034)] for functional differential equations, we may show the global existence of periodic solutions. Computer simulations illustrate the results.


37N25 Dynamical systems in biology
34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
92D25 Population dynamics (general)
34K17 Transformation and reduction of functional-differential equations and systems, normal forms
34K19 Invariant manifolds of functional-differential equations
37G40 Dynamical aspects of symmetries, equivariant bifurcation theory
Full Text: DOI


[1] Hale, J.; Lunel, S. V.: Introduction to functional differential equations, (1993) · Zbl 0787.34002
[2] Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[3] Wu, J.: Theory and applications of partial functional differential equations, (1996) · Zbl 0870.35116
[4] Hutchinson, G. E.: Circular cause systems in ecology, Ann. New York acad. Sci. 50, 221-246 (1948)
[5] May, R. M.: Time delay versus stability in population models with two and three trophic levels, Ecology 4, 315-325 (1973)
[6] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H.: Theory and applications of Hopf bifurcation, (1981) · Zbl 0474.34002
[7] 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
[8] Wu, J.: Symmetric functional differential equations and neural networks with memory, Trans. amer. Math. soc. 350, 4799-4838 (1998) · Zbl 0905.34034
[9] Faria, T.; Magalháes, L. T.: Normal form for retarded functional differential equations and applications to bogdanov-Takens singularity, J. differential equations 122, 201-224 (1995) · Zbl 0836.34069
[10] Faria, T.; Magalháes, L. T.: Normal form for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. differential equations 122, 181-200 (1995) · Zbl 0836.34068
[11] Cooke, K.; Grossman, Z.: Discrete delay, distributed delay and stability switches, J. math. Anal. appl. 86, 592-627 (1982) · Zbl 0492.34064
[12] Chow, S. -N.; Hale, J. K.: Methods of bifurcation theory, (1982) · Zbl 0487.47039
[13] Cushing, J. M.: Periodic time-dependent predator–prey systems, SIAM J. Appl. math. 32, 82-95 (1977) · Zbl 0348.34031
[14] He, X.: Stability and delays in a predator–prey system, J. math. Anal. appl. 198, 355-370 (1996) · Zbl 0873.34062
[15] Ruan, S.: Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator–prey systems with discrete delays, Quart. appl. Math. 59, 159-173 (2001) · Zbl 1035.34084
[16] Ruan, S.; Wei, J.: Periodic solutions of planar systems with two delays, Proc. roy. Soc. Edinburgh sect. A 129, 1017-1032 (1999) · Zbl 0946.34062
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.