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Zbl 1123.34055
Yan, Xiang-Ping; Li, Wan-Tong
Bifurcation and global periodic solutions in a delayed facultative mutualism system. (English)
[J] Physica D 227, No. 1, 51-69 (2007). ISSN 0167-2789

Authors' summary: A facultative mutualism system with a discrete delay is considered. By analyzing the associated characteristic equation, its linear stability is investigated and Hopf bifurcations are demonstrated. Some explicit formulae are obtained by applying the normal form theory and center manifold reduction. Such formulae enable us to determine the stability and the direction of the bifurcating periodic solutions bifurcating from Hopf bifurcations. Furthermore, a global Hopf bifurcation result due to [{\it J. Wu}, Trans. Am. Math. Soc. 350, 4799--4838 (1998; Zbl 0905.34034)] is employed to study the global existence of periodic solutions. It is shown that the local Hopf bifurcation implies the global Hopf bifurcation after the third critical value $\tau_1^{(1)}$ of delay. Finally, numerical simulations supporting the theoretical analysis are given.
[Yuming Chen (Waterloo)]

MSC 2000:: *34K18 Bifurcation theory of functional differential equations
92D25 Population dynamics
34K19 Invariant manifolds
34K13 Periodic solutions of functional differential equations
34K20 Stability theory of functional-differential equations
34K60 Applications of functional-differential equations

Keywords: Facultative mutualism system; functional differential equation; stability; local Hopf bifurcation; global Hopf bifurcation; normal form theory; center manifold reduction

Citations: Zbl 0905.34034

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