DYNAMICS ANALYSIS OF THREE-SPECIES REACTION-DIFFUSION SYSTEM VIA THE MULTIPLE SCALE PERTURBATION METHOD

The amplitude equation describes a reduced form of a reaction–diffusion system, yet still retains its essential dynamical features. By approximating the analytic solution, the amplitude equation allows the examination of mode instability when the system is near a bifurcation point. Multiple-scale expansion (MSE) offers a straightforward way to systematically derive the amplitude equations. The method expresses the single independent variable as an asymptotic power series consisting of newly introduced independent variables with differing time and space scales. The amplitude equations are then formulated under the solvability conditions which remove secular terms. To our knowledge, there is little information in the research literature that explains how the exhaustive workflow of MSE is applied to a reaction–diffusion system. In this paper, detailed mathematical operations underpinning the MSE are elucidated, and the practical ways of encoding these operations using MAPLE are discussed. A semi-automated MSE computer algorithm Amp_solving is presented for deriving the amplitude equations in this research. Amp_solving has been applied to the classical Brusselator model for the derivation of amplitude equations when the system is in the vicinity of a Turing codimension-1 and a Turing–Hopf codimension-2 bifurcation points. Full open-source Amp_solving codes for the derivation are comprehensively demonstrated and available to the public domain.

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Stabilization of Dominant Structures in an Ionic Reaction-Diffusion System

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19980761 ◽

1998 ◽

Vol 63 (6) ◽

pp. 761-769 ◽

Cited By ~ 1

Author(s):

Roland Krämer ◽

Arno F. Münster

Keyword(s):

Reaction Diffusion ◽

Dominant Mode ◽

Reaction Diffusion System ◽

Diffusion System ◽

Local Perturbation ◽

Dimensional System ◽

One Dimensional ◽

Brusselator Model ◽

The One ◽

Dominant Structure

We describe a method of stabilizing the dominant structure in a chaotic reaction-diffusion system, where the underlying nonlinear dynamics needs not to be known. The dominant mode is identified by the Karhunen-Loeve decomposition, also known as orthogonal decomposition. Using a ionic version of the Brusselator model in a spatially one-dimensional system, our control strategy is based on perturbations derived from the amplitude function of the dominant spatial mode. The perturbation is used in two different ways: A global perturbation is realized by forcing an electric current through the one-dimensional system, whereas the local perturbation is performed by modulating concentrations of the autocatalyst at the boundaries. Only the global method enhances the contribution of the dominant mode to the total fluctuation energy. On the other hand, the local method leads to simple bulk oscillation of the entire system.

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The human EDAR 370V/A polymorphism affects tooth root morphology potentially through the modification of a reaction–diffusion system

Scientific Reports ◽

10.1038/s41598-021-84653-4 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Keiichi Kataoka ◽

Hironori Fujita ◽

Mutsumi Isa ◽

Shimpei Gotoh ◽

Akira Arasaki ◽

...

Keyword(s):

Molecular Mechanisms ◽

Computational Study ◽

Reaction Diffusion ◽

Human Migration ◽

Reaction Diffusion System ◽

Diffusion System ◽

Tooth Root ◽

Computed Tomography Images ◽

Diffusion Dynamics ◽

Root Morphogenesis

AbstractMorphological variations in human teeth have long been recognized and, in particular, the spatial and temporal distribution of two patterns of dental features in Asia, i.e., Sinodonty and Sundadonty, have contributed to our understanding of the human migration history. However, the molecular mechanisms underlying such dental variations have not yet been completely elucidated. Recent studies have clarified that a nonsynonymous variant in the ectodysplasin A receptor gene (EDAR370V/A; rs3827760) contributes to crown traits related to Sinodonty. In this study, we examined the association between theEDARpolymorphism and tooth root traits by using computed tomography images and identified that the effects of theEDARvariant on the number and shape of roots differed depending on the tooth type. In addition, to better understand tooth root morphogenesis, a computational analysis for patterns of tooth roots was performed, assuming a reaction–diffusion system. The computational study suggested that the complicated effects of theEDARpolymorphism could be explained when it is considered that EDAR modifies the syntheses of multiple related molecules working in the reaction–diffusion dynamics. In this study, we shed light on the molecular mechanisms of tooth root morphogenesis, which are less understood in comparison to those of tooth crown morphogenesis.

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