Analysis and Simulations of Coupled Bulk-surface Reaction-Diffusion Systems on Exponentially Evolving Volumes

¹ University of Sussex, School of Mathematical and Physical Sciences, Department of Mathematics, Pevensey III, 5C15, Brighton, BN1 9QH, UK
² 34-36 St. Georges Road, Brighton, BN2 1ED, UK

^* Corresponding author. E-mail: a.madzvamuse@sussex.ac.uk

Abstract

In this article we present a system of coupled bulk-surface reaction-diffusion equations on exponentially evolving volumes. Detailed linear stability analysis of the homogeneous steady state is carried out. It turns out that due to the nature of the coupling (linear Robin-type boundary conditions) the characterisation of the dispersion relation in the absence and presence of spatial variation (i.e. diffusion), can be decomposed as a product of the dispersion relation of the bulk and surface models thereby allowing detailed analytical tractability. As a result we state and prove the conditions for diffusion-driven instability for systems of coupled bulk-surface reaction-diffusion equations. Furthermore, we plot explicit evolving parameter spaces for the case of an exponential growth. By selecting parameter values from the parameter spaces, we exhibit pattern formation in the bulk and on the surface in complete agreement with theoretical predictions.