Abstract
In this paper, we study the effect of time delay on the dynamics of a diffusive predator–prey model with predator-taxis under Neumann boundary condition. The joint effect of predator-taxis and delay can lead to spatially nonhomogeneous periodic patterns via spatially nonhomogeneous Hopf bifurcations. It is also shown that there exist double Hopf bifurcations due to the interaction either between homogeneous and nonhomogeneous or between nonhomogeneous Hopf bifurcations with different modes, which cannot occur for the system with only either delay or predator-taxis diffusion.
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