Hopf and Turing-Hopf bifurcation analysis of a delayed predator-prey model with schooling behavior

In the present research, we investigate the relevant dynamical mechanisms of a reflection-diffusion predator-prey model that involves time delay and schooling behavior. The existence and local stability of the positive equilibrium are discussed. In addition, the necessary and sufficient conditions regarding Turing instability and delay-induced Hopf bifurcation are obtained via investigating the relevant characteristic equation. Furthermore, by calculating and discussing the normal forms on the center manifold corresponding to the Turing-Hopf bifurcation, we discover the multitude of spatiotemporal dynamics close to the Turing-Hopf bifurcation point subject to appropriate conditions. The numerical simulations are carried out to confirm and expand our theoretical conclusions, which demonstrate that time delay reflects a crucial influence on the spatiotemporal dynamics of the system and will lead to spatially homogeneous and inhomogeneous periodic solutions.