Spatiotemporal patterns of a delayed diffusive prey-predator model with prey-taxis

This paper explored a delayed diffusive prey-predator model with prey-taxis involving the volume-filling mechanism subject to homogeneous Neumann boundary condition. To figure out the impact on the dynamic of the prey-predator model due to prey-taxis and time delay, we treated the prey-tactic coefficient $ \chi $ and time delay $ \tau $ as the bifurcating parameters and did stability and bifurcation analysis. It showed that the time delay will induce Hopf bifurcations in the absence of prey-taxis, and the bifurcation periodic solution at the first critical value of $ \tau $ was spatially homogeneous. Hopf bifurcations occurred in the model when the prey-taxis and time delay coexisted, and at the first critical value of $ \tau $, spatially homogeneous or nonhomogeneous periodic solutions might emerge. It was also discovered that the bifurcation curves will intersect, which implied that Hopf-Hopf bifurcations can occur. Finally, we did numerical simulations to validate our outcomes.