Turing-Hopf bifurcation of a delayed predator-prey system with prey-taxis incorporating a Beddington-DeAngelis functional response

Abstract

In this paper, we investigate the spatiotemporal pattern dynamics in a delayed predator-prey model with prey-taxi. First, Turing bifurcation for the reaction-diffusion-prey-taxis model are analyzed and the conditions for it is obtained. By deriving an amplitude equation, we then classify the bifurcation as supercritical and subcritical at the bifurcation onset. Furthermore, using the prey-taxis sensitivity coefficient and time delay as bifurcation parameters, we determine the conditions for a Turing-Hopf bifurcation. Numerical simulations validate our theoretical analysis, revealing complex spatiotemporal patterns, including homogeneous and non-homogeneous steady patterns, non-homogeneous periodic patterns, and mixed wave patterns, resulting from the combined effect of prey-taxis and time delay. Interestingly, numerical results show that increased predator interference in Beddington-DeAngelis functional response leads to a more even spatial distribution. Certainly, the obtained results enhance the understanding of pattern dynamics in models with taxi effect and time delay.