Turing model for the patterns of lady beetles

Liaw, Sy-Sang; Yang, C. C.; Liu, R. T.; Hong, Jing-Yuan

doi:10.1103/physreve.64.041909

Cited by 43 publications

(31 citation statements)

References 10 publications

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“…Varea et al (1999) have considered pattern formation on spherical surfaces while Chaplain et al (2001) investigate patterns on the radially growing sphere. Liaw et al (2001) show that the patterns exhibited by a Turing reaction-diffusion system on a fixed portion of a sphere resemble closely those observed on lady beetles (ladybirds). Moreover, their preliminary study suggests that curvature does not alter significantly the results for the case of planar domains.…”

Section: Discussionsupporting

confidence: 52%

Pattern Formation in Reaction–Diffusion Models with Nonuniform Domain Growth

Crampin

Hackborn²,

Maini

2002

Bulletin of Mathematical Biology

165

192

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Recent examples of biological pattern formation where a pattern changes qualitatively as the underlying domain grows have given rise to renewed interest in the reaction-diffusion (Turing) model for pattern formation. Several authors have now reported studies showing that with the addition of domain growth the Turing model can generate sequences of patterns consistent with experimental observations. These studies demonstrate the tendency for the symmetrical splitting or insertion of concentration peaks in response to domain growth. This process has also been suggested as a mechanism for reliable pattern selection. However, thus far authors have only considered the restricted case where growth is uniform throughout the domain. In this paper we generalize our recent results for reaction-diffusion pattern formation on growing domains to consider the effects of spatially nonuniform growth. The purpose is twofold: firstly to demonstrate that the addition of weak spatial heterogeneity does not significantly alter pattern selection from the uniform case, but secondly that sufficiently strong nonuniformity, for example where only a restricted part of the domain is growing, can give rise to sequences of patterns not seen for the uniform case, giving a further mechanism for controlling pattern selection. A framework for modelling is presented in which domain expansion and boundary * Author to whom correspondence should be addressed. E-mail: crampin@maths.ox.ac. (apical) growth are unified in a consistent manner. The results have implications for all reaction-diffusion type models subject to underlying domain growth.

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Section: Discussionsupporting

confidence: 52%

Pattern Formation in Reaction–Diffusion Models with Nonuniform Domain Growth

Crampin

Hackborn²,

Maini

2002

Bulletin of Mathematical Biology

165

192

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“…In contrast to the case of a fixed domain, numerous different stripe and spot patterns occur depending on perturbations of the initial conditions for the model. Further examples of studies of RDEs illustrating the role of domain growth can be found in papers by Varea et al (1999), Chaplain et al (2001), Liaw et al (2001), Painter et al (1999), Crampin et al ( , 1999, Oster and Bressloff (2006), Madzvamuse et al (2003Madzvamuse et al ( , 2005, Madzvamuse (2005) and for a review see Plaza et al (2004).…”

Section: Introductionmentioning

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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“…Turing patterns on a growing sphere were recently investigated numerically by Chaplain et al [8] in an application to metastasis in a growing tumor. Liaw et al [17] simulated a Turing model on part of a spherical surface and showed that they could reproduce various patterns that are exhibited on the hard wings of lady beetles.…”

Section: Introductionmentioning

confidence: 99%

The Effect of Growth and Curvature on Pattern Formation

et al. 2004

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Based on first principles, we derive a general model to describe the spatio-temporal dynamics of two morphogens. The diffusive part of the model incorporates the dynamics, growth and curvature of one-and two-dimensional domains embedded in R 3 . Our generalized diffusion process includes spatio-temporal varying diffusion coefficients, advection, and dilution terms. We present specific examples by analyzing a third order activator-inhibitor mechanism for the kinetic part. We carry out illustrative numerical simulations on two-dimensional growing domains having different geometries. Comparisons with former results on fixed domains show the crucial role of growth and curvature of pattern selection. Evidence is given that both effects might be biologically relevant in explaining the selection of some observed patterns and in changing or enhancing their stability.

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Turing model for the patterns of lady beetles

Cited by 43 publications

References 10 publications

Pattern Formation in Reaction–Diffusion Models with Nonuniform Domain Growth

Pattern Formation in Reaction–Diffusion Models with Nonuniform Domain Growth

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

The Effect of Growth and Curvature on Pattern Formation

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