Bifurcation analysis of a diffusive predator-prey system with Crowley-Martin functional response and delay

In this paper, we investigate the dynamics of a diffusive predator-prey system with Crowley-Martin functional response and delay subject to Neumann boundary condition. More precisely, we study the stability and Turing instability of positive equilibrium for non-delay system, instability and Hopf bifurcation induced by time delay for delay system. In addition, by the theory of normal form and center manifold method, we derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution.