Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces
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.
KeywordsReaction diffusion Pattern formation Bifurcation analysis Linear stability analysis Marginal stability analysis Branch tracing Nonlinear PDEs Surface FEMs Large-scale systems Multigrid approach Cross-diffusion
MCM was supported by grants from the Swiss National Science Foundation (FNSNF, grants 31003A_140785 and SINERGIA CRSII3_132430), and the SystemsX.ch initiative (project EpiPhysX) for this work. We thank Liana Manukyan from LANE, University of Geneva for 3D scanning a gecko surface and providing us with the point-cloud data. We used Meshlab software for surface reconstruction with this point-cloud. DSD was also supported by the FNSNF grant SINERGIA CRSII3_132430 for this work. He is thankful to his colleague Shihao Wu at CGG, Univ. of Bern for uniform re-sampling of the gecko surface mesh, for level \(L_2\) in Fig. 17. DSD thanks Prof. Dr. T. Wihler at Mathematical Institute, Univ. of Bern for suggesting Deal.II library.
- Bonito A, Pauletti S, Bangerth W (2013) The step-38 tutorial program, Reference documentation for Deal.II version 8.4.0. https://dealii.org/8.4.0/doxygen/deal.II/step_38.html, Accessed 11 May 2016
- Cignoni P, Callieri M, Corsini M, Dellepiane M, Ganovelli F, Ranzuglia G (2008) MeshLab: an open-source mesh processing tool. In: Scarano V, Chiara RD, Erra U (eds) Sixth Eurographics Italian Chapter Conference, The Eurographics Association, pp 129–136Google Scholar
- Garland M, Heckbert PS (1997) Surface simplification using quadric error metrics. Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH ’97). ACM Press, New York, pp 209–216Google Scholar
- Gary, Julia (2013) [photograph] A Gecko lizard with small spots. Retrieved from http://www.westcoastleopardgecko.com/images/00451.JPG, Accessed 07 July 2015
- Lévy B, Zhang HR (2010) Spectral mesh processing. In: ACM SIGGRAPH 2010 Courses, ACM Press, New York, pp 8:1–8:312. doi: 10.1145/1837101.1837109
- Locke T (2007) [photograph] A Gecko lizard with an aligned spot pattern. Retrieved from http://www.geckotime.com/breeding-leopard-geckos-on-a-small-scale/, Accessed 07 July 2015
- Psyon (2009) [photograph] A Gecko lizard with bands on its back. Retrieved from https://commons.wikimedia.org/wiki/File:Juvenile-leopard-gecko-2.jpg, Accessed 07 July 2015
- Raven (2009) [photograph] An Eublepharis macularius Gecko lizard with irregular spots. Retrieved from https://plus.google.com/communities/107069611951806704428, Accessed 07 July 2015
- Rooster (2009) [photograph] A juvenile Gecko lizard with a stripe pattern. Retrieved from http://www.captivebredreptileforums.co.uk/members/rooster/albums/rooster-s-geckos/4223-jungle-leopard-gecko-baby-pic-hes-bigger-now/, Accessed 07 July 2015
- Salinger AG, Bou-Rabee NM, Pawlowski RP, Wilkes ED, Burroughs EA, Lehoucq RB, Romero LA (2002) LOCA 1.0 Library of continuation algorithms: theory and implementation manual. Sandia National Laboratories, Albuquerque, NM, Technical Report No SAND2002-0396Google Scholar