Turing pattern amplitude equation for a model glycolytic reaction-diffusion system

Dutt, Arun K.

doi:10.1007/s10910-010-9699-x

Cited by 25 publications

(17 citation statements)

References 26 publications

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“…First, we prove Eqs. (17) and (18) that lead to the definition of the evolving domains of the HSM. Secondly, we compute the equations for the densities n 1 , n 2 and n in the limit → 0, m → +∞, α → 0.…”

Section: Formal Proof Of Theoremmentioning

confidence: 99%

Incompressible limit of a continuum model of tissue growth with segregation for two cell populations

Chertock

Degond

Hecht

et al. 2019

Mathematical Biosciences and Engineering

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This paper proposes a model for the growth two interacting populations of cells that do not mix. The dynamics is driven by pressure and cohesion forces on the one hand and proliferation on the other hand. Following earlier works on the single population case, we show that the model approximates a free boundary Hele Shaw type model that we characterise using both analytical and numerical arguments.

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Section: Formal Proof Of Theoremmentioning

confidence: 99%

Incompressible limit of a continuum model of tissue growth with segregation for two cell populations

Chertock

Degond

Hecht

et al. 2019

Mathematical Biosciences and Engineering

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“…From the characteristic equation (3.2) for nonzero k mode, the condition for Hopf-wave bifurcation is given by k 2 Hopf ≤ a 11 + a 22 …”

Section: Linear Stability Analysismentioning

confidence: 99%

“…For such a D-R model, 17,18 we have attempted to derive the AE, which interprets the stability of various forms of Turing patterns as well as the structural transitions between them. In a recent paper, 22 we have made a preliminary report of this derivation, but could not present in detail, particularly the nonlinear analysis and the amplitude equation derivation steps after incorporating the two solvability conditions -this has been accomplished in this manuscript. Also, a linear stability analysis of the amplitude equation together with some numerical results on Hopf and Turing bifurcations has been reported in detail, which we could not present before.…”

Section: Introductionmentioning

confidence: 99%

“…Also, a linear stability analysis of the amplitude equation together with some numerical results on Hopf and Turing bifurcations has been reported in detail, which we could not present before. 22 The other related sections have been included in this manuscript in condensed form. Section II has discussed the reversible Sel'kov model in brief.…”

Section: Introductionmentioning

confidence: 99%

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Amplitude equation for a diffusion-reaction system: The reversible Sel'kov model

Dutt

2012

AIP Advances

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For a model glycolytic diffusion-reaction system, an amplitude equation has been derived in the framework of a weakly nonlinear theory. The linear stability analysis of this amplitude equation interprets the structural transitions and stability of various forms of Turing structures. This amplitude equation also conforms to the expectation that time-invariant amplitudes in Turing structures are independent of complexing reaction with the activator species, whereas complexing reaction strongly influences Hopf-wave bifurcation.

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“…Method of continuation is a powerful method to study the PTW solutions of a R-D system [13]. In this paper, we consider an activator-inhibitor Brusselator model which represents an autocatalytic oscillating chemical reaction and the pattern formations for this model are discussed in [14,15,16,17]. In paper [18], author theoretically proved the existence of the Hopf bifurcation of the Brusselator model which reveals the existence of PTW solutions of the model.…”

Section: Introductionmentioning

confidence: 99%

Periodic Pattern Formation Analysis Numerically in a Chemical Reaction-Diffusion System

Nazimuddin¹,

Ali²

2019

IJMSC

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In this paper, we analyze the pattern formation in a chemical reaction-diffusion Brusselator model. Twocomponent Brusselator model in two spatial dimensions is studied numerically through direct partial differential equation simulation and we find a periodic pattern. In order to understand the periodic pattern, it is important to investigate our model in one-dimensional space. However, direct partial differential equation simulation in one dimension of the model is performed and we get periodic traveling wave solutions of the model. Then, the local dynamics of the model is investigated to show the existence of the limit cycle solutions. After that, we establish the existence of periodic traveling wave solutions of the model through the continuation method and finally, we get a good consistency among the results.

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Turing pattern amplitude equation for a model glycolytic reaction-diffusion system

Cited by 25 publications

References 26 publications

Incompressible limit of a continuum model of tissue growth with segregation for two cell populations

Incompressible limit of a continuum model of tissue growth with segregation for two cell populations

Amplitude equation for a diffusion-reaction system: The reversible Sel'kov model

Periodic Pattern Formation Analysis Numerically in a Chemical Reaction-Diffusion System

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