High codimension bifurcations and complex pattern selection in a diffusive Leslie–Gower predator–prey model with Bazykin functional response

Abstract

Most existing studies on diffusive predator–prey systems mainly focus on codimension-1 Turing and Hopf bifurcations, while the role of higher codimension bifurcations in complex pattern selection and spatiotemporal dynamics remains far from fully understood. Therefore, we investigate a generally diffusive Leslie–Gower predator–prey model with Bazykin functional response and focus on the formation mechanism of complex spatiotemporal patterns induced by diffusion and nonlinear interactions. We establish the existence conditions of codimension-2 Turing–Hopf and Turing-Turing bifurcations, as well as codimension-3 Turing-Turing–Hopf bifurcation, which significantly enriches the bifurcation structure of the system. By combining linear stability analysis, center manifold reduction and normal form theory, we derive the amplitude equations near the Turing–Hopf bifurcation point and obtain the corresponding two-parameter bifurcation diagrams and dispersion relations in the δ − d plane. Numerical simulations further demonstrate abundant dynamical behaviors, including spatially homogeneous periodic solutions, stationary spatial patterns, transient spatiotemporal periodic solutions, the multiple spatiotemporal steady solutions induced by different initial conditions, and superposed patterns generated by different Turing modes. In particular, the coexistence of multiple spatiotemporal stable states and transient pattern transitions highlights the important role of high codimension bifurcations in pattern selection and ecological self-organization. These results provide new theoretical insights into the nonlinear mechanism of spatiotemporal dynamics in predator–prey systems.