Authors’ abstract: In this paper, a predator-prey system with two delays is investigated. By choosing the sum
of two delays as a bifurcation parameter, we show that Hopf bifurcations can occur as
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.