Stability analysis and simulations of coupled bulk-surface reaction-diffusion systems

TL;DR

This paper develops models for coupled bulk-surface reaction-diffusion systems, analyzes their stability, and demonstrates how bulk patterns can induce surface patterns, with numerical simulations supporting the theoretical analysis.

Contribution

It introduces new coupled models with Robin boundary conditions and provides a decoupled stability analysis, revealing how bulk patterns influence surface patterning.

Findings

Bulk reaction-diffusion can induce surface patterning independently.

Surface patterns cannot form without bulk patterning in the absence of bulk patterns.

Robin boundary conditions create boundary layer effects coupling bulk and surface dynamics.

Abstract

In this article we formulate new models for coupled systems of bulk-surface reaction-diffusion equations on stationary volumes. The bulk reaction-diffusion equations are coupled to the surface reaction-diffusion equations through linear Robin-type boundary conditions. We then state and prove the necessary conditions for diffusion-driven instability for the coupled system. Due to the nature of the coupling between bulk and surface dynamics, we are able to decouple the stability analysis of the bulk and surface dynamics. Under a suitable choice of model parameter values, the bulk reaction-diffusion system can induce patterning on the surface independent of whether the surface reaction-diffusion system produces or not, patterning. On the other hand, the surface reaction-diffusion system can not generate patterns everywhere in the bulk in the absence of patterning from the bulk reaction-diffusion system. For this case, patterns can only be induced in regions close to the surface membrane. Various numerical experiments are presented to support our theoretical findings. Our most revealing numerical result is that, Robin-type boundary conditions seem to introduce a boundary layer coupling the bulk and surface dynamics.