Multiple diffusion scales and diffusion-driven instability: Emergence of near- and far-from-equilibrium patterns

Abstract

This paper investigates pattern formation in reaction--diffusion systems with both diffusive and nondiffusive components, providing necessary and sufficient conditions for diffusion-driven instability (DDI) and establishing the existence of far-from-equilibrium patterns. While previous work has linked DDI to instability in the purely nondiffusive subsystem -- thereby destabilizing all regular Turing patterns -- we show that DDI can also arise from subsystems involving nondiffusive and slow-diffusive components using three different spatial scales. This leads to simple sufficient conditions for DDI in systems with arbitrary numbers of components. Moreover, we fully classify all possible sources of DDI in the case of two diffusive and one nondiffusive component. Further, we prove the existence of far-from-equilibrium patterns exhibiting branch-switching and discontinuities in the nondiffusive components, which cannot occur in classical reaction--diffusion equations. We illustrate our results with a receptor-based model supported by numerical bifurcation analysis and simulations. These findings extend the theoretical foundations of pattern formation, demonstrating how coupling between diffusive and nondiffusive dynamics can generate patterns beyond the reach of the classical reaction--diffusion framework.