Novel pattern dynamics in a vegetation-water reaction–diffusion model

This paper investigates the pattern dynamics of a four-variable vegetation-water reaction–diffusion model. By incorporating inhibitory factors and promoting factors, the model provides a more comprehensive framework for describing the interaction mechanisms between vegetation growth and environmental factors. Through linear stability analysis and Turing bifurcation theory, the amplitude equation near the Turing bifurcation point is derived. Furthermore, multi-scale analysis and weakly nonlinear analysis are employed to elucidate the pattern selection mechanism, revealing that diverse vegetation patterns emerge under different parameter conditions. For numerical simulations, a new high-precision Fourier spectral method is constructed to simulate the model with varying parameter conditions. The results validate the theoretical analysis and demonstrate the emergence of multiple novel pattern morphologies. Additionally, the study extends the classical model by introducing a fractional-order Laplacian operator, constructing a spatiotemporal fractional-order diffusion model to explore the effects of sub-diffusion and super-diffusion on vegetation pattern formation.