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Bifurcations for a predator-prey system with two delays. (English) Zbl 1132.34053

Authors’ abstract: In this paper, a predator-prey system with two delays is investigated. By choosing the sum \(\tau\) of two delays as a bifurcation parameter, we show that Hopf bifurcations can occur as \(\tau\) crosses some critical values. 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 bifurcation results of J. Wu [Trans. Am. Math. Soc. 350, 4799–4838 (1998; Zbl 0905.34034)], we may show the global existence of periodic solutions.

MSC:

34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
92D25 Population dynamics (general)

Citations:

Zbl 0905.34034
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References:

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