Mode-doubling and tripling in reaction-diffusion patterns on growing domains: A piecewise linear model

Crampin, EJ; Gaffney, Eamonn A.; Maini, P. K.

doi:10.1007/s002850100112

Cited by 89 publications

(101 citation statements)

References 28 publications

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“…For reaction-diffusion systems on growing domains, which is a good model for the growth of organisms, we mention [3], [4], [34], [35]. Recently in [61] hair follicle arrangements in mice have been modelled by a reaction-diffusion system, where the WNT and DKK proteins serve as an activator and inhibitor, respectively, and experiments are combined with numerical computations.…”

Section: Previous Results On Peaked Solutionsmentioning

confidence: 99%

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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Section: Previous Results On Peaked Solutionsmentioning

confidence: 99%

Stationary multiple spots for reaction–diffusion systems

Wei

Winter

2007

J. Math. Biol.

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“…It is worth remarking here that all previous works except [17] and [20] have exclusively considered dilatation dynamics. Even in the different field of reaction-diffusion dynamics in which the dilution term was derived, the focus was on the limit in which it was irrelevant [18]. This work is devoted to exploring further the consequences of dilution, dilatation, and decorrelation and their effects on scaling of radial interfaces.…”

Section: Introductionmentioning

confidence: 99%

Statistics of interfacial fluctuations of radially growing clusters

Escudero

2011

Phys. Rev. E

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The dynamics of fluctuating radially growing interfaces is approached using the formalism of stochastic growth equations on growing domains. This framework reveals a number of dynamic features arising during surface growth. For fast growth, dilution, which spatially reorders the incoming matter, is responsible for the transmission of correlations. Its effects include the erasing of memory with respect to the initial condition, a partial attenuation of geometrically originated instabilities, and the restoration of universality in some special cases in which the critical exponents depend on the parameters of the equation of motion. In this sense, dilution rends the dynamics more similar to the usual one of planar systems. This fast growth regime is also characterized by the spatial decorrelation of the interface, which, in the case of radially growing interfaces, naturally originates rapid roughening and scale-dependent fractality, and suggests the advent of a self-similar fractal dimension. The center-of-mass fluctuations of growing clusters are also studied, and our analysis suggests the possible nonapplicability of usual scalings to the long-range surface fluctuations of the radial Eden model. In fact, our study points to the fact that this model belongs to a dilution-free universality class.

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“…These mechanisms for pattern transitions have been investigated in a class of reactiondiffusion systems by Crampin et al (2002), wherein a third possibility is also described in which splitting and insertion occur simultaneously. Schnakenberg (1979) proposed a hypothetical series of trimolecular autocatalytic reactions, for which the nondimensionalized concentrations of self-activating, v, and self-inhibiting, u, chemicals give the kinetic scheme…”

Section: Reaction Schemesmentioning

confidence: 99%

“…Transitions between patterns arise through one of two mechanisms: insertion (as for the fish) or splitting of activator peaks (Meinhardt, 1982). Under certain circumstances these mechanisms can be made to operate simultaneously [mode-tripling, see Crampin et al (2002)]. Patterns evolving on the growing domain follow a sequence which depends on the first pattern to form but is otherwise insensitive to initial conditions, and hence it has been suggested that this is a mechanism for reliable pattern formation (Crampin et al, 1999).…”

Section: Introductionmentioning

confidence: 99%

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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Mode-doubling and tripling in reaction-diffusion patterns on growing domains: A piecewise linear model

Cited by 89 publications

References 28 publications

Stationary multiple spots for reaction–diffusion systems

Stationary multiple spots for reaction–diffusion systems

Statistics of interfacial fluctuations of radially growing clusters

Pattern Formation in Reaction–Diffusion Models with Nonuniform Domain Growth

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