Stability and Turing-like instability analysis of a general reaction-diffusion-advection system: theory and application to sand-vegetation model

In this paper, we investigate the influence of transport terms on the stability of a general reaction-diffusion-advection system. The interaction between diffusion and advection within a system can lead to the emergence of complex patterns. In some cases, diffusion and advection can work together synergistically to enhance pattern formation, while in other cases, they may counteract each other, leading to complex dynamics and behavior. Thus, transport terms exhibit a dual effect by inducing instability and restoring stability. By systematically analyzing the linear stability of constant steady solutions with transport terms, we can identify critical thresholds and parameter regimes where instabilities emerge. This approach offers valuable insights into the underlying mechanisms driving pattern formation, transitions between different states, and the overall behavior of dynamical systems influenced by advection and other transport processes. Finally, we apply our theoretical findings to a sand-vegetation system with cross-diffusion and cross-advection terms and validate them through numerical simulations.