Pattern formation of bulk-surface reaction-diffusion systems in a ball

TL;DR

This paper derives amplitude equations for pattern formation in multi-component bulk-surface reaction-diffusion systems within a ball, analyzing stability and bifurcations using weakly nonlinear theory and numerical simulations.

Contribution

It provides a general framework for analyzing pattern bifurcations in bulk-surface systems with Robin boundary conditions, extending prior normal form results.

Findings

Conditions for pattern mode instability depend on wavenumber parity.

Derived amplitude equations match known normal forms with O(3) symmetry.

Numerical simulations confirm theoretical predictions in two example models.

Abstract

Weakly nonlinear amplitude equations are derived for the onset of spatially extended patterns on a general class of n-component bulk-surface reaction-diffusion systems in a ball, under the assumption of linear kinetics in the bulk and coupling Robin-type boundary conditions. Linear analysis shows conditions under which various pattern modes can become unstable to either generalised pitchfork or transcritical bifurcations depending on the parity of the spatial wavenumber. Weakly nonlinear analysis is used to derive general expressions for the multi-component amplitude equations of different patterned states. These reduced-order systems are found to agree with prior normal forms for pattern formation bifurcations with O(3) symmetry and they provide information on the stability of bifurcating patterns of different symmetry types. The analysis is complemented with numerical results using a dedicated bulk-surface finite-element method. The theory is illustrated in two examples: a bulk-surface version of the Brusselator and a four-component bulk-surface cell-polarity model.