Hopf bifurcation and Turing instability in spatial homogeneous and inhomogeneous predator-prey models

This paper is concerned with two-species spatial homogeneous and inhomogeneous predator-prey models with Beddington-DeAngelis functional response. For the spatial homogeneous model, the asymptotic behavior of the interior equilibrium and the existence of Hopf bifurcation of nonconstant periodic solutions surrounding the interior equilibrium are considered. Furthermore, the direction of Hopf bifurcation and the stability of bifurcated periodic solutions are investigated. For the model with no-flux boundary conditions, Turing instability of the interior equilibrium solution is studied. In particular, Turing instability region regarding the parameters is established. Finally, to verify our theoretical results, some numerical simulations are also included.