Modeling of Turing Structures in the Chlorite—Iodide—Malonic Acid—Starch Reaction System

Lengyel, Istivan; Epstein, Irving R.

doi:10.1126/science.251.4994.650

Cited by 481 publications

(304 citation statements)

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“…For the CIMA reaction experimentally found pattern formation [2], [6], [51], [52] could successfully be explained by reaction-diffusion modelling [30].…”

Section: Previous Results On Peaked Solutionsmentioning

confidence: 86%

Stationary multiple spots for reaction–diffusion systems

Wei

Winter

2007

J. Math. Biol.

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Abstract. In this paper, we review analytical methods for a rigorous study of the existence and stability of stationary, multiple spots for reaction-diffusion systems. We will consider two classes of reactiondiffusion systems: activator-inhibitor systems (such as the Gierer-Meinhardt system) and activatorsubstrate systems (such as the Gray-Scott system or the Schnakenberg model).The main ideas are presented in the context of the Schnakenberg model, and these results are new to the literature.We will consider the systems in a two-dimensional, bounded and smooth domain for small diffusion constant of the activator.Existence of multi-spots is proved using tools from nonlinear functional analysis such as LiapunovSchmidt reduction and fixed-point theorems. The amplitudes and positions of spots follow from this analysis.Stability is shown in two parts, for eigenvalues of order one and eigenvalues converging to zero, respectively. Eigenvalues of order one are studied by deriving their leading-order asymptotic behavior and reducing the eigenvalue problem to a nonlocal eigenvalue problem (NLEP). A study of the NLEP reveals a condition for the maximal number of stable spots.Eigenvalues converging to zero are investigated using a projection similar to Liapunov-Schmidt reduction and conditions on the positions for stable spots are derived. The Green's function of the Laplacian plays a central role in the analysis.The results are interpreted in the biological, chemical and ecological contexts. They are confirmed by numerical simulations.

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“…For the CIMA reaction experimentally found pattern formation [2], [6], [51], [52] could successfully be explained by reaction-diffusion modelling [30].…”

Section: Previous Results On Peaked Solutionsmentioning

confidence: 86%

Stationary multiple spots for reaction–diffusion systems

Wei

Winter

2007

J. Math. Biol.

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“…27 We further use the LE model to study numerically the interactions between pairs of Ising and Bloch fronts and relate the results of this study to the existence of rectangular and oblique patterns.…”

Section: Introductionmentioning

confidence: 99%

Fronts and patterns in a spatially forced CDIMA reaction

Haim

Hagberg

Nagao

et al. 2014

Phys. Chem. Chem. Phys.

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We use the CDIMA chemical reaction and the Lengyel-Epstein model of this reaction to study resonant responses of a pattern-forming system to time-independent spatial periodic forcing. We focus on the 2 : 1 resonance, where the wavenumber of a one-dimensional periodic forcing is about twice the wavenumber of the natural stripe pattern that the unforced system tends to form. Within this resonance, we study transverse fronts that shift the phase of resonant stripe patterns by p. We identify phase fronts that shift the phase discontinuously, and pairs of phase fronts that shift the phase continuously, clockwise and anti-clockwise. We further identify a front bifurcation that destabilizes the discontinuous front and leads to a pair of continuous fronts. This bifurcation is the spatial counterpart of the nonequilibrium Ising-Bloch (NIB) bifurcation in temporally forced oscillatory systems. The spatial NIB bifurcation that we find occurs as the forcing strength is increased, unlike earlier studies of the NIB bifurcation. Furthermore, the bifurcation is subcritical, implying a range of forcing strength where both discontinuous Ising fronts and continuous Bloch fronts are stable. Finally, we find that both Ising fronts and Bloch fronts can form discrete families of bound pairs, and we relate arrays of these front pairs to extended rectangular and oblique patterns.

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“…This perspective not only highlights a major source of complexity and hierarchy currently missing from synthetic strategies, but also suggests we might be able to actively shape selfassembling structures by manipulating the environment as they grow. This possibility is especially compelling for systems that develop through reaction-diffusion processes, where dynamic feedback between the reaction front and the solution is the central mechanism of pattern evolution (34)(35)(36)(37)(38)(39)(40). Such feedback mechanisms are not only proposed to produce a variety of biological morphologies, but can also generate patterned precipitation in synthetic systems (41).…”

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confidence: 99%