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Stability and global Hopf bifurcation in a delayed food web consisting of a prey and two predators. (English) Zbl 1219.92065

Summary: This paper is concerned with a predator-prey system with Holling II functional response and hunting delay and gestation. By regarding the sum of delays as the bifurcation parameter, the local stability of the positive equilibrium and the existence of Hopf bifurcations are investigated. We obtained explicit formulas to determine the properties of Hopf bifurcations by using the normal form method and center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcations. Using a global Hopf bifurcation result of J. Wu [Symmetric functional differential equations and neural networks with memory. Trans. Am. Math. Soc. 350, No. 12, 4799–4838 (1998; Zbl 0905.34034)] for functional differential equations, we show the global existence of the periodic solutions. Finally, several numerical simulations illustrating the theoretical analysis are also given.

MSC:

92D40 Ecology
34K18 Bifurcation theory of functional-differential equations
34K20 Stability theory of functional-differential equations
65C20 Probabilistic models, generic numerical methods in probability and statistics

Citations:

Zbl 0905.34034
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References:

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