Stability and bifurcation for a delayed predator-prey model and the effectof diffusion

We consider a predator-prey system with one or two delays and a unique posi tive equilibrium E-*. Its dynamics are studied in terms of the local stabil ity of E-* and of the description of the Hopf bifurcation that is proven to exist as one of the delays (taken as a parameter) crosses some critical va lues. We also consider a reaction-diffusion system with Neumann conditions, resulting from adding one spatial variable and diffusion terms in the prev ious model. The spectral and bifurcation analysis in the neighborhood of E- *, now as a stationary point of this latter system, is addressed and the re sults obtained for the case without diffusion are applied. (C) 2001 Academi c Press.