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 delaysŽtaken 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.