Pattern formation driven by cross-diffusion in a 2D domain

Gambino, G.; Lombardo, Maria Carmela; Sammartino, M.

doi:10.1016/j.nonrwa.2012.11.009

Cited by 122 publications

(75 citation statements)

References 48 publications

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“…In order to accurately predict the spatial features of the expected Turing patterns, non-linear bifurcation analysis and the amplitude equations formalism must be used, see i.e. [19,22,29,30]. Nevertheless, this kind of analysis is beyond the scope of the present paper, for this reason we resorted to numerical investigation of pattern selection issues.…”

Section: Numerical Investigationsmentioning

confidence: 99%

Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay

2014

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In this paper, we investigate from a theoretical point of view the 2D reaction-diffusion system for electrodeposition coupling morphology and surface chemistry, presented and experimentally validated in Bozzini et al. (2013 J. Solid State Electr. 17, 467-479). We analyse the mechanisms responsible for spatio-temporal organization. As a first step, spatially uniform dynamics is discussed and the occurrence of a supercritical Hopf bifurcation for the local kinetics is proved. In the spatial case, initiation of morphological patterns induced by diffusion is shown to occur in a suitable region of the parameter space. The intriguing interplay between Hopf and Turing instability is also considered, by investigating the spatio-temporal behaviour of the system in the neighbourhood of the codimensiontwo Turing-Hopf bifurcation point. An ADI (Alternating Direction Implicit) scheme based on high-order finite differences in space is applied to obtain numerical approximations of Turing patterns at the steady state and for the simulation of the oscillating Turing-Hopf dynamics.

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Section: Numerical Investigationsmentioning

confidence: 99%

Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay

2014

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“…In developmental biology, recent experimental findings demonstrate that cross-diffusion can be quite significant in generating spatial structure [10]. The effects of cross-diffusion on models for pattern formation (i.e., reaction-diffusion type) have been studied in many theoretical papers [13][14][15][16][17][18][19][20][21][22][23][24]. Recently, in [25] we showed that introducing cross-diffusion to a system of reaction-diffusion equations results in further relaxation of the conditions necessary for the emergency of patterns.…”

Section: Introductionmentioning

confidence: 99%

Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces

Madzvamuse

Barreira

2014

Phys. Rev. E

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Article (Published Version) http://sro.sussex.ac.uk Madzvamuse, A and Barreira, R (2014) Exhibiting cross-diffusion-induced patterns for reactiondiffusion systems on evolving domains and surfaces. Physical Review E, 90. 043307-1-043307-14. ISSN 1539-3755 This version is available from Sussex Research Online: http://sro.sussex.ac.uk/51334/ This document is made available in accordance with publisher policies and may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the URL above for details on accessing the published version. Copyright and reuse:Sussex Research Online is a digital repository of the research output of the University.Copyright and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable, the material made available in SRO has been checked for eligibility before being made available.Copies of full text items generally can be reproduced, displayed or performed and given to third parties in any format or medium for personal research or study, educational, or not-for-profit purposes without prior permission or charge, provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.PHYSICAL REVIEW E 90, 043307 (2014) Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces The aim of this manuscript is to present for the first time the application of the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces. Furthermore we present pattern formation generated by the reaction-diffusion system with cross-diffusion on evolving domains and surfaces. A two-component reaction-diffusion system with linear cross-diffusion in both u and v is presented. The finite element method is based on the approximation of the domain or surface by a triangulated domain or surface consisting of a union of triangles. For surfaces, the vertices of the triangulation lie on the continuous surface. A finite element space of functions is then defined by taking the continuous functions which are linear affine on each simplex of the triangulated domain or surface. To demonstrate the role of cross-diffusion to the theory of pattern formation, we compute patterns with model kinetic parameter values that belong only to the cross-diffusion parameter space; these do not belong to the standard parameter space for classical reaction-diffusion systems. Numerical results exhibited show the robustness, flexibility, versatility, and generality of our methodology; the methodology can deal with complicated evolution laws of the domain and surface, and these include uniform isotropic and anisotropic growth profiles as well as those profiles driven by chemical concentrations residing in...

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“…An important number of contributions have been proposed to treat systems like (1.1) from a numerical perspective, either considering or not the cross-diffusion effect (see [2,5,6,8,14,18,21,39] for finite differences, finite volumes, spectral and finite element methods for the spatial discretization). Here, and following [11,27,37], we propose a new finite volume element (FVE) method for the numerical approximation of the underlying reaction-cross-diffusion system.…”

Section: Introductionmentioning

confidence: 99%

Finite volume element approximation of an inhomogeneous Brusselator model with cross-diffusion

Lin

Ruiz–Baier

Tian

2014

Journal of Computational Physics

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This paper is concerned with the study of pattern formation for an inhomogeneous Brusselator model with cross-diffusion, modeling an autocatalytic chemical reaction taking place in a three-dimensional domain. For the spatial discretization of the problem we develop a novel finite volume element (FVE) method associated to a piecewise linear finite element approximation of the cross-diffusion system. We study the main properties of the unique equilibrium of the related dynamical system. A rigorous linear stability analysis around the spatially homogeneous steady state is provided and we address in detail the formation of Turing patterns driven by the cross-diffusion effect. In addition we focus on the spatial accuracy of the FVE method, and a series of numerical simulations confirm the expected behavior of the solutions. In particular we show that, depending on the spatial dimension, the magnitude of the cross-diffusion influences the selection of spatial patterns.

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Pattern formation driven by cross-diffusion in a 2D domain

Cited by 122 publications

References 48 publications

Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay

Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay

Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces

Finite volume element approximation of an inhomogeneous Brusselator model with cross-diffusion

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