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In this paper, a predator-prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we show that Hopf bifurcations can occur as τ 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 Hopf bifurcation result of [J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799-4838], we may show the global existence of periodic solutions.

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In this paper, we study the persistence, stability and Hopf bifurcation in a ratio-dependent predator–prey model with diffusion and delay. Sufficient conditions independent of diffusion and delay are obtained for the persistence of the system and global stability of the boundary equilibrium. The local stability of the positive constant equilibrium and delay-induced Hopf bifurcation are investigated by analyzing the corresponding characteristic equation. We show that delay can destabilize the positive equilibrium and induce spatially homogeneous and inhomogeneous periodic solutions. By calculating the normal form on the center manifold, the formulae determining the direction and the stability of Hopf bifurcations are explicitly derived. The numerical simulations are carried out to illustrate and extend our theoretical results.

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