Bulletin of Mathematical Biology

, Volume 79, Issue 4, pp 788–827 | Cite as

Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces

  • Daljit Singh J. Dhillon
  • Michel C. Milinkovitch
  • Matthias Zwicker
Original Article

Abstract

In this paper, we present computational techniques to investigate the effect of surface geometry on biological pattern formation. In particular, we study 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 characterise 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.

Keywords

Reaction diffusion Pattern formation Bifurcation analysis Linear stability analysis Marginal stability analysis Branch tracing Nonlinear PDEs Surface FEMs Large-scale systems Multigrid approach Cross-diffusion 

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Copyright information

© Society for Mathematical Biology 2017

Authors and Affiliations

  1. 1.Institute of Computer ScienceUniversity of BernBernSwitzerland
  2. 2.Department of ComputingImperial College LondonLondonUK
  3. 3.Laboratory of Artificial & Natural Evolution (LANE), Department of Genetics and EvolutionUniversity of GenevaGenevaSwitzerland
  4. 4.SIB Swiss Institute of BioinformaticsGenevaSwitzerland

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