Amplitude equations for reaction-diffusion systems with cross diffusion

Zemskov, E. P.; Vanag, Vladimir K.; Epstein, Irving R.

doi:10.1103/physreve.84.036216

Cited by 43 publications

(25 citation statements)

References 34 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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“…The effects of cross-diffusion on models for pattern formation (i.e. reaction-diffusion type) have been studied in many theoretical papers [5,6,8,9,12,14,33,34,36,41,42,43]. Recently, in Madzvamuse [26] we showed that introducing cross-diffusion to a system of reactiondiffusion equations results in further relaxation of the conditions necessary for the emergency of patterns.…”

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confidence: 99%

Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion

Madzvamuse¹,

Ndakwo²,

Barreira³

2015

DCDS-A

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This article presents stability analytical results of a two component reaction-diffusion system with linear cross-diffusion posed on continuously evolving domains. First the model system is mapped from a continuously evolving domain to a reference stationary frame resulting in a system of partial differential equations with time-dependent coefficients. Second, by employing appropriately asymptotic theory, we derive and prove cross-diffusion-driven instability conditions for the model system for the case of slow, isotropic domain growth. Our analytical results reveal that unlike the restrictive diffusion-driven instability conditions on stationary domains, in the presence of cross-diffusion coupled with domain evolution, it is no longer necessary to enforce cross nor pure kinetic conditions. The restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Reaction-cross-diffusion models with equal diffusion coefficients in the principal components as well as those of the short-range inhibition, long-range activation and activator-activator form can generate patterns only in the presence of cross-diffusion coupled with domain evolution. To confirm our theoretical findings, detailed parameter spaces are exhibited for the special cases of isotropic exponential, linear and logistic growth profiles. In support of our theoretical predictions, we present evolving or moving finite element solutions exhibiting patterns generated by a short-range inhibition, long-range activation reactiondiffusion model with linear cross-diffusion in the presence of domain evolution.

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“…This provides us to accurately distinguish the fundamental characteristics of the intrinsic dynamics and the modeling approaches enriche the spiking network level activities such as synchronization and its measure with synchronization factor [8,25,44]. For Hopf and Turing instabilities, responsible for close to the homogeneous fluctuations and the emergence of stationary and spatial patterns respectively, the corresponding amplitude equations are referred to as Turing amplitude equation (TAE) [43,47] and complex Ginzburg-Landau equation (CGLE, it is responsible for the onset of Hopf instability in the occurrence of homogeneous fluctuations) [2,22,37].…”

Section: Introductionmentioning

confidence: 99%

Analysis of spatially extended excitable Izhikevich neuron model near instability

Mondal

Sharma

et al. 2021

Nonlinear Dyn

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The article focuses on the issue of a spatiotemporal excitable biophysical model that describes the propagation of electrical potential called spikes to model the diffusion induced dynamics based on an analytical development of amplitude equations. Considering the Izhikevich neuron model consisting of coupled systems of ODEs, we demonstrate various results of spatiotemporal architecture (PDEs) using a suitable parameter regime. We analytically perform the saddle node bifurcation and Hopf bifurcation analysis with bifurcating periodic solutions that show the transition phases in the system dynamics. We study different types of firing patterns both analytically and numerically by the formation of Riccati differential equation. To examine the characteristics of diffusive instabilities, we use Turing amplitude equations by multiscaling method and then expansion in powers of a small control parameter. The instabilities and Turing bifurcation are established using the theoretical analysis and numerical simulations. The spatial system has potential effects on the deterministic system as a result of the

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Amplitude equations for reaction-diffusion systems with cross diffusion

Cited by 43 publications

References 34 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

Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion

Analysis of spatially extended excitable Izhikevich neuron model near instability

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