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

We consider a predator-prey system with one or two delays and a unique positive equilibrium E-*. Its dynamics are studied in terms of the local stability 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 values. We also consider a reaction-diffusion system with Neumann conditions, resulting from adding one spatial variable and diffusion terms in the previous model. The spectral and bifurcation analysis in the neighborhood of E-*, now as a stationary point of this latter system, is addressed and the results obtained for the case without diffusion are applied. (C) 2001 Academic Press.