Spatiotemporal patterns of a delayed diffusive ff usive 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.