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Bifurcation and global periodic solutions in a delayed facultative mutualism system. (English) Zbl 1123.34055
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 [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 ${\tau }_{1}^{\left(1\right)}$ of delay. Finally, numerical simulations supporting the theoretical analysis are given.
##### MSC:
 34K18 Bifurcation theory of functional differential equations 92D25 Population dynamics (general) 34K19 Invariant manifolds (functional-differential equations) 34K13 Periodic solutions of functional differential equations 34K20 Stability theory of functional-differential equations 34K60 Qualitative investigation and simulation of models