Bifurcation analysis of a predator-prey model with cross-diffusion and two delays

In this paper, a predator-prey model with cross-diffusion and two delays is investigated. First, the conditions for local stability and Turing instability of positive steady-state solution are studied separately when the system was without and with diffusion. Second, the existence and the stability of Hopf bifurcation were investigated by computing stability switching curves in the parameter plane with two delays. Moreover, explicit formulas for determining the stability and the direction of the bifurcation periodic solutions were derived using the normal form theory and the center manifold theorem. Finally, the theoretical results were verified by numerical simulations.