Bifurcation and global periodic solutions in a delayed facultative mutualism system

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 Wu [J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799–4838] 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 τ 1 ( 1 ) of delay. Finally, numerical simulations supporting the theoretical analysis are given.