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**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Editorial remark: This article has been retracted at the request of the Editors-in-Chief due to its high similarity to [S. Yuan and Y. Song, IMA J. Appl.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.

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.

### MSC:

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 |

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\textit{S. Yuan} and \textit{F. Zhang}, Nonlinear Anal., Real World Appl. 11, No. 2, 959--977 (2010; Zbl 1183.37156)

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