Global stability and Hopf bifurcations of a Beddington–DeAngelis type predator–prey system with diffusion and delays

• Dynamics of a diffusive Beddington–DeAngelis system with two delays are studied.
• The direction and stability of Hopf bifurcating periodic orbits are determined.
• The global stability of the positive existence is investigated.
• Numerical simulations are illustrated for the theoretical results.

The dynamics of a diffusive predator–prey system with the Beddington–DeAngelis functional response and two delays are considered. By choosing the sum of two delays as a bifurcation parameter, the stability of the constant equilibria and the existence of Hopf bifurcations are investigated by analyzing the characteristic equations. And the formulas determining the direction and the stability of the bifurcating periodic solutions are derived by using the center manifold and the normal form theory of partial functional differential equations. Particularly, the sufficient conditions of the global stability of the positive equilibrium are given by the upper–lower solutions method.