Spatiotemporal dynamics induced by delay and diffusion in a predator-prey model with mutual interference among the predator

In this work, we study a reaction-diffusion predator-prey model with mutual interference among the predators while searching for food. We prove that the model exhibits bistability, which indicates that there are no patterns for our model. When time delay is incorporated into the model, multiple stability switches phenomenon of positive constant steady state emerged. By taking delay as a bifurcation parameter, the Hopf bifurcations at the positive constant steady state are proved to occur for a sequence of critical values of the delay. The algorithm for determining the direction and the stability of the bifurcating periodic solutions is also derived. The delay-diffusion driven Turing instability of the positive constant steady state is investigated. Our results show that delay and diffusion can create periodic oscillatory patterns of spatially homogeneous and inhomogeneous and Turing patterns.