Spatio-temporal patterns and global bifurcation of a nonlinear cross-diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses

The aim of this article is investigating the spatio-temporal patterns of a nonlinear cross-diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses. First, by utilizing user-friendly version of Crandall–Rabinowitz bifurcation theory as an analytical method, the spatio-temporal patterns of positive steady state are obtained. Then, by regarding the cross-diffusion coefficient $d_1$ as a bifurcation parameter, we will derive a bifurcation theorem for the corresponding nonlinear cross-diffusion model. Moreover, by applying spectrum theory, perturbation of simple eigenvalues and linearized stability, it is discovered that the bifurcation solutions possess local stability near the bifurcating point in proper conditions. These theoretical results mean that the cross-diffusion mechanism can create a coexistence environment for the preys and predator under some circumstances. Finally, a numerical example is proposed to verify our results.