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

Anotida Madzvamuse*, Andy H. W. Chung, Chandrasekhar Venkataraman

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

35 Citations (Scopus)


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. Owing 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 cannot generate patterns everywhere in the bulk in the absence of patterning from the bulk reaction-diffusion system. For this case, patterns can be induced only 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.

Original languageEnglish
Article number20140546
Number of pages18
JournalProceedings of the Royal Society A - Mathematical, Physical & Engineering Sciences
Issue number2175
Publication statusPublished - 8 Mar 2015


  • Bulk-surface reaction-diffusion equations
  • Bulk-surface finite-elements
  • Turing diffusively driven instability
  • Linear stability
  • Pattern formation
  • Robin-type boundary conditions


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