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
of delay. Finally, numerical simulations supporting the theoretical analysis are given.