Dhillon, Daljit Singh Joginder Singh; Zwicker, Matthias (2016). Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces arXiv
1605.01583v1.pdf - Submitted Version
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In this article we present a computational framework for isolating spatial patterns arising in the steady states of reaction-diffusion systems. Such systems have been used to model many different phenomena in areas such as developmental and cancer biology, cell motility and material science. Often one is interested in identifying parameters which will lead to a particular pattern. To attempt to answer this, we compute eigenpairs of the Laplacian on a variety of domains and use linear stability analysis to determine parameter values for the system that will lead to spatially inhomogeneous steady states whose patterns correspond to particular eigenfunctions. This method has previously been used on domains and surfaces where the eigenvalues and eigenfunctions are found analytically in closed form. Our contribution to this methodology is that we numerically compute eigenpairs on arbitrary domains and surfaces. Here we present various examples and demonstrate that mode isolation is straightforward especially for low eigenvalues. Additionally we see that if two or more eigenvalues are in a permissible range then the inhomogeneous steady state can be a linear combination of the respective eigenfunctions. Finally we show an example which suggests that pattern formation is robust on similar surfaces in cases that the surface either has or does not have a boundary.
|Item Type:||Report (Report)|
|Division/Institute:||08 Faculty of Science > Institute of Computer Science (INF) > Computer Graphics Group (CGG)
08 Faculty of Science > Institute of Computer Science (INF)
|UniBE Contributor:||Dhillon, Daljit Singh Joginder Singh and Zwicker, Matthias|
|Subjects:||000 Computer science, knowledge & systems
500 Science > 510 Mathematics
500 Science > 570 Life sciences; biology
|Date Deposited:||20 Jul 2016 09:53|
|Last Modified:||19 Aug 2016 07:37|