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.