Analyzing diffusive vegetation-sand model: Instability, bifurcation, and pattern formation

In this study, we explored a diffusive vegetation-sand model with Neumann boundary conditions, investigating the role of Turing instability in vegetation pattern formation. A priori estimates for steady-state solutions were established using the maximum principle and Poincaré inequality. Bifurcation analysis was performed for simple and double eigenvalue cases. By employing bifurcation theory, a local bifurcation was extended globally, and the direction of bifurcation was characterized. Double eigenvalue cases were analyzed through spatial decomposition and the implicit function theorem. Finally, numerical simulations validated and complemented the theoretical results.