Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces

TL;DR

This paper introduces computational methods for analyzing nonlinear reaction-diffusion systems on arbitrary surfaces, enabling the study of pattern formation on complex geometries using spectral techniques, surface finite element methods, and numerical continuation.

Contribution

It extends existing reaction-diffusion analysis techniques to arbitrary surfaces with large-scale meshes, incorporating spectral analysis, surface finite element methods, and multiresolution continuation.

Findings

Demonstrated framework on biological pattern models

Enabled analysis of pattern formation on complex geometries

Provided tools for stability and bifurcation analysis

Abstract

In this paper we present computational techniques to investigate the solutions of two-component, nonlinear reaction-diffusion (RD) systems on arbitrary surfaces. We build on standard techniques for linear and nonlinear analysis of RD systems, and extend them to operate on large-scale meshes for arbitrary surfaces. In particular, we use spectral techniques for a linear stability analysis to characterize and directly compose patterns emerging from homogeneities. We develop an implementation using surface finite element methods and a numerical eigenanalysis of the Laplace-Beltrami operator on surface meshes. In addition, we describe a technique to explore solutions of the nonlinear RD equations using numerical continuation. Here, we present a multiresolution approach that allows us to trace solution branches of the nonlinear equations efficiently even for large-scale meshes. Finally, we demonstrate the working of our framework for two RD systems with applications in biological pattern formation: a Brusselator model that has been used to model pattern development on growing plant tips, and a chemotactic model for the formation of skin pigmentation patterns. While these models have been used previously on simple geometries, our framework allows us to study the impact of arbitrary geometries on emerging patterns.