Emergence of spatiotemporal patterns beyond Turing-Hopf bifurcation in semi-arid vegetation systems

Previous studies have mainly focused on spatiotemporal dynamics near the Turing-Hopf bifurcation point, typically employing normal form theory to partition parameter space and identify classic solutions. However, both Hopf and Turing bifurcations can be classified into supercritical and subcritical types, and the combined effects of their criticality on vegetation dynamics remain unclear. To address this issue, we construct a diffusive vegetation-water model under semi-arid grazing conditions. Our analysis shows that the system admits a unique vegetation equilibrium once the grazing intensity reaches a threshold, and we derive the conditions for the occurrence of both Turing and Hopf bifurcations. By taking the water reduction rate and the water diffusion rate as control parameters, we apply normal form theory to determine the Hopf bifurcation type and employ weakly nonlinear analysis to derive the amplitude equation at the critical wavenumber, thereby identifying the Turing bifurcation type. Numerical simulations further reveal that the Turing and Hopf bifurcations partition the parameter plane into four regions: the vegetation equilibrium stability, pure Turing instability, Turing and Hopf instabilities, and pure Hopf instability. In the supercritical-supercritical case, the system sequentially exhibits a homogeneous steady state, spatial periodic patterns and spatiotemporal periodic solutions. In the subcritical-subcritical case, a bistable region arises where vegetation homogeneous state and spatially periodic state coexist, accompanied by a snaking bifurcation structure that induces localized patterns. Along this snaking bifurcation, a Hopf bifurcation emerges, generating more complex oscillatory localized states. Moreover, under the subcritical-subcritical case, the final state is highly sensitive to initial conditions. Overall, this study not only uncovers rich dynamical behaviors arising from combination of different type of bifurcations, but also provides new theoretical insights into the spatiotemporal evolution and ecological stability of vegetation systems.