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A three level finite element approximation of a pattern formation model in developmental biology

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Abstract

This paper concerns a second-order, three level piecewise linear finite element scheme 2-SBDF (Ruuth, in J Math Biol 34:148–176, 1995) for approximating the stationary (Turing) patterns of a well-known experimental substrate-inhibition reaction-diffusion (‘Thomas’) system (Thomas, in Analysis and control of immobilized enzyme systems, pp 115–150, 1975). A numerical analysis of the semi-discrete in time approximations leads to semi-discrete a priori bounds and an optimal error estimate. The analysis highlights the technical challenges in undertaking the numerical analysis of multi-level (\({\ge } 3\)) schemes. We illustrate the effectiveness of the numerical method by repeating an important classical experiment in mathematical biology, namely, to approximate the Turing patterns of the Thomas system over a schematic mammal skin domain with fixed geometry at various scales. We also make some comments on the correct procedure for simulating Turing patterns in general reaction-diffusion systems.

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  2. The physics review paper [20] has been cited 4068 times (ISI Web of Knowledge).

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Acknowledgments

We thank James Blowey (University of Durham, UK) for some helpful comments during the preparation of this manuscript and for the comments of the anonymous reviewers.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Guelph, Guelph, ON, N1G 2W1, Canada

    Marcus R. Garvie

  2. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Catalin Trenchea

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Correspondence to Marcus R. Garvie.

Appendix: Pseudo random number generation

Appendix: Pseudo random number generation

In the interests of repeatability, we give details of the pseudo random number generator D_UNIFORM_01 [37] used to perturb the coefficients \(z_{rs}\) of the double Fourier series (4.4). It is not the most efficient random number generator, but it is simple enough to be easily implemented in different languages.

We take \(z_{rs}\) equal to the \(n\)th random number \(r_n\) drawn from D_UNIFORM_01, where \(n = r + 20(s - 1)\). The random numbers are calculated recursively via

$$\begin{aligned} r_n&= s_n / (2^{31}-1),\\ s_n&= 16807 *s_{n-1}*\mod (2^{31} - 1), \end{aligned}$$

for \(n=1,2,\dots \) ‘seeded’ with \(s_0\). In all our simulations we used \(s_0=4\).

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Garvie, M.R., Trenchea, C. A three level finite element approximation of a pattern formation model in developmental biology. Numer. Math. 127, 397–422 (2014). https://doi.org/10.1007/s00211-013-0591-z

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