Mathematical analysis and numerical simulation of pattern formation under cross-diffusion

Ruiz–Baier, Ricardo; Tian, Canrong

doi:10.1016/j.nonrwa.2012.07.020

Cited by 63 publications

(31 citation statements)

References 46 publications

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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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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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“…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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“…That is, we require that the positive equilibrium point u * is linearly stable in the absence of the cross-diffusion but unstable in the presence of cross-diffusion. For this, we first introduce the following notations [Tian et al, 2011;Ruiz-Baier & Tian, 2013]. Let 0 = µ 1 < µ 2 < · · · → ∞ be the eigenvalues of −∇ 2 on Ω under no-flux boundary conditions and E(µ i ) be the space of eigenfunctions corresponding to eigenvalue µ i .…”

Section: Theoretical Analysis Of Dynamical Behaviorsmentioning

confidence: 99%

“…A two-species model with cross-diffusion in the context of plasma physics and predator-prey dynamics was also proposed and studied [del-Castillo-Negrete et al, 2002;del-Castillo-Negrete & Carreras, 2002]. It was shown that in the predator-prey system, mutual diffusion can induce Turing instability to produce spatial patterns even though a spatial homogeneous equilibrium state in the absence of one cross-diffusion is stable [Tian et al, 2011;Ruiz-Baier & Tian, 2013;Xue, 2012;Wang et al, 2010]. In addition, in chemical reaction systems, cross-diffusion can arise from interactions between the species and occurs in, e.g.…”

Section: Introductionmentioning

confidence: 99%

Turing Pattern Formation in a Two-Species Negative Feedback System With Cross-Diffusion

Yang

Yuan

2013

Int. J. Bifurcation Chaos

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Pattern formation is a ubiquitous phenomenon in the natural world. Previous studies showed that for an activator-inhibitor system without cross-diffusion, spatial patterns can be formed only when the diffusion of the inhibitor is significantly faster than that of the activator. However, cross-diffusion exists extensively in real systems, especially in biological systems. Here, we study a prototypic two-species negative feedback system with cross-diffusion. By performing stability analysis of equilibrium state, we find sufficient conditions for Turing instability. Both analytical and numerical results demonstrate that mutual diffusions of the two species can lead to the Turing pattern formation regardless of differences in self-diffusion coefficients. However, in the absence of the mutual diffusion or even if there is the cross-diffusion of only one species, the system cannot exhibit Turing patterns. Our results reveal the mechanism of Turing pattern formation in a class of reaction-diffusion systems, where mutual diffusion between species plays a key role.

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Mathematical analysis and numerical simulation of pattern formation under cross-diffusion

Cited by 63 publications

References 46 publications

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

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

Turing Pattern Formation in a Two-Species Negative Feedback System With Cross-Diffusion

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