Abstract
In this paper, we consider a predator-prey model with Beddington-DeAngelis functional response with or without diffusion. For this system, we give a complete and rigorous analysis of the dynamics including the existence of a global positive solution, the stability/Turing instability and the Hopf bifurcation. In the meanwhile, we show, via numerical simulations, that there appears Hopf bifurcation, steady state solution and Turing-Hopf bifurcation with the changes of some parameters of the system.