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**Hopf bifurcations in a predator-prey system with multiple delays.**
*(English)*
Zbl 1198.34143

Summary: This paper is concerned with a two species Lotka-Volterra predator-prey system with three discrete delays. By regarding the gestation period of two species as the bifurcation parameter, the stability of positive equilibrium and Hopf bifurcations of nonconstant periodic solutions are investigated. Furthermore, the direction of Hopf bifurcations and the stability of bifurcated periodic solutions are determined by applying the normal form theory and the center manifold reduction for functional differential equations (FDEs). In addition, the global existence of bifurcated periodic solutions are also established by employing the topological global Hopf bifurcation theorem, which shows that the local Hopf bifurcations imply the global ones after the second critical value of parameter. Finally, to verify our theoretical predictions, some numerical simulations are also included.

Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

### MSC:

34K18 | Bifurcation theory of functional-differential equations |

37N25 | Dynamical systems in biology |

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\textit{G.-P. Hu} et al., Chaos Solitons Fractals 42, No. 2, 1273--1285 (2009; Zbl 1198.34143)

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### References:

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