Spatiotemporal inhomogeneous pattern of a predator–prey model with delay and chemotaxis

In this paper, we investigate the spatiotemporal inhomogeneous pattern phenomenon of a predator–prey model with chemotaxis and time delay. The precise intervals of the Turing instability are yielded so that we can obtain sufficient conditions to ensure the existence of the Turing instability by adjusting the ranges of the control parameters of time delay and chemotaxis. Also, we perform the occurrence conditions of the Turing–Hopf bifurcation by employing the time delay control parameter and chemotaxis sensitivity coefficient as the bifurcation parameter. Theoretically, there are no Hopf bifurcation, Turing instability, and Turing–Hopf bifurcation as the chemotaxis effect and time delay are absent. Of course, complicated spatiotemporal inhomogeneous patterns are displayed with the help of numerical experiments. Particularly, we find that both the time delay control parameter and chemotaxis sensitivity coefficient can affect the formation of the spatiotemporal inhomogeneous patterns. Certainly, the method could be applied in other reaction–diffusion models with the chemotaxis and full/non-full time delay in reaction kinetics terms.