Simple chemical reaction systems with limit cycle behaviour

Schnakenberg, J.

doi:10.1016/0022-5193(79)90042-0

Cited by 587 publications

(416 citation statements)

References 15 publications

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“…We used several initial distributions with different random deviations and confirmed the robustness of the result. We set the reaction terms as: Schnackenberg, 1979), based on an activator-depleted substrate model that was originally proposed by Gierer and Meinhardt (1972). In all simulations, except those in Figure 4i The method used to identify the direction of the stripes is shown in the accompanying study (Shoji et al, 2002).…”

Section: Discussionmentioning

confidence: 99%

Origin of directionality in the fish stripe pattern

Shoji

Mochizuki

Iwasa

et al. 2003

Developmental Dynamics

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The formation of stripe patterns in animal skin has been explained by the reaction-diffusion (RD) system, a hypothetical chemical reaction proposed by A. Turing. Although animal stripes usually have directionality, the RD model alone cannot explain how the direction is specified. To investigate the mechanism regulating the direction of stripes, we studied stripe pattern formation in two species of Genicanthus during sexual conversion. These species share almost identical morphologic properties, except for their stripe direction. In both species, spots transiently arise at random positions and then combine and rearrange to form directional stripes. Computational analysis has shown that diffusion anisotropy is very effective at specifying the direction of stripes formed by the RD system. Model simulations reproduce the transient dynamics of directional pattern formation observed in fish as well as the resulting stripes. In cases where the magnitude and direction of diffusion anisotropy of the substances are identical, the resulting stripes are not directional. However, if they differ, stripes become directional. As only a small difference in anisotropy is required for this effect, any kind of structure with directional conformation might cause a marked change in stripe direction. Scales are the most likely candidate structure for generating anisotropic interactions in skin. Developmental Dynamics 226:627-633, 2003.

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Section: Discussionmentioning

confidence: 99%

Origin of directionality in the fish stripe pattern

Shoji

Mochizuki

Iwasa

et al. 2003

Developmental Dynamics

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“…Then two examples are considered in detail. The first is an activator-depleted model known to satisfy the cross/pure kinetics conditions (Gierer and Meinhardt 1972;Prigogine and Lefever 1968;Schnakenberg 1979). The second example consists of reaction kinetics which do not satisfy any of the cross/pure kinetics conditions.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…For illustrative purposes let us consider the activator-depleted substrate model (Gierer and Meinhardt 1972;Prigogine and Lefever 1968;Schnakenberg 1979) also known as the Brusselator model given by…”

Section: Activator-depleted Model: An Illustrative Examplementioning

confidence: 99%

Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains

2009

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By using asymptotic theory, we generalise the Turing diffusively-driven instability conditions for reaction-diffusion systems with slow, isotropic domain growth. There are two fundamental biological differences between the Turing conditions on fixed and growing domains, namely: (i) we need not enforce cross nor pure kinetic conditions and (ii) the restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Our theoretical findings are confirmed and reinforced by numerical simulations for the special cases of isotropic linear, exponential and logistic growth profiles. In particular we illustrate an example of a reaction-diffusion system which cannot exhibit a diffusively-driven instability on a fixed domain but is unstable in the presence of slow growth.

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“…We illustrate this with a concrete example for c = (c 1 , c 2 ), using the two-component Schnakenberg kinetic scheme (Schnakenberg, 1979) (where c 2 is the activator):…”

Section: Reaction and Diffusion On Growing Domainsmentioning

confidence: 99%

Reaction and Diffusion on Growing Domains: Scenarios for Robust Pattern Formation

Crampin¹,

Gaffney²,

Maini³

1999

Bulletin of Mathematical Biology

333

388

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We investigate the sequence of patterns generated by a reaction-diffusion system on a growing domain. We derive a general evolution equation to incorporate domain growth in reaction-diffusion models and consider the case of slow and isotropic domain growth in one spatial dimension. We use a self-similarity argument to predict a frequency-doubling sequence of patterns for exponential domain growth and we find numerically that frequency-doubling is realized for a finite range of exponential growth rate. We consider pattern formation under different forms for the growth and show that in one dimension domain growth may be a mechanism for increased robustness of pattern formation.

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Simple chemical reaction systems with limit cycle behaviour

Cited by 587 publications

References 15 publications

Origin of directionality in the fish stripe pattern

Origin of directionality in the fish stripe pattern

Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains

Reaction and Diffusion on Growing Domains: Scenarios for Robust Pattern Formation

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