Steady state bifurcations for a glycolysis model in biochemical reaction

Wei, Meihua; Wu, Jianhua; Guo, Ge

doi:10.1016/j.nonrwa.2014.08.003

Cited by 23 publications

(6 citation statements)

References 22 publications

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“…In view of spatial diffusion, there are some existence and stability results of constant steady states (see [3,21]). Under the fixed Dirichlet boundary condition, the authors [36] discuss the stability of constant steady state solution and the existence of non-constant steady state solutions not only from a simple eigenvalue, but more difficultly from a double one, which are confirmed by numerical simulations. Based on the following condition (C), no Hopf bifurcation can occur for the model (1) by taking the bifurcation parameter d 1 .…”

mentioning

confidence: 78%

Stability and bifurcation with singularity for a glycolysis model under no-flux boundary condition

Wei¹,

Li²,

Xi³

2019

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper, a glycolysis model subject to no-flux boundary condition is considered. First, by discussing the corresponding characteristic equation, the stability of constant steady state solution is discussed, and the Turing's instability is shown. Next, based on Lyapunov-Schmidt reduction method and singularity theory, the multiple stationary bifurcations with singularity are analyzed. In particular, under no-flux boundary condition we show the existence of nonconstant steady state solution bifurcating from a double zero eigenvalue, which is always excluded in most existing works. Also, the stability, bifurcation direction and multiplicity of the bifurcation steady state solutions are investigated by the singularity theory. Finally, the theoretical results are confirmed by numerical simulations. It is also shown that there is no Hopf bifurcation on basis of the condition (C).

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mentioning

confidence: 78%

Stability and bifurcation with singularity for a glycolysis model under no-flux boundary condition

Wei¹,

Li²,

Xi³

2019

Discrete &Amp; Continuous Dynamical Systems - B

Self Cite

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“…The second example comes from a simplified model of glycolysis (check [34] for a more in-depth analysis) (18) x…”

Section: 3mentioning

confidence: 99%

Hilbert's 16th problem. II. Pfaffian equations and variational methods

Pedregal¹

2019

Preprint

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Starting from a Pfaffian equation in dimension N and focusing on compact solutions for it, we place in perspective the variational method used in [29] to solve Hilbert's 16th problem. In addition to exploring how this viewpoint can help in detecting and finding approximations for limit cycles of planar systems, we recall some of the initial important facts of the full program developed in [29] to motivate that same proposal could eventually be used in other situations. In particular, we make some initial interesting calculations in dimension N = 3 that lead to some similar initial conclusions as with the case N = 2. Contents 1. Introduction 2. The case N = 2 2.1. Quadratic functional for periodic paths 2.2. Optimality 2.3. The descent procedure 2.4. Some numerical tests 3. Hilbert's 16th problem 4. The case N = 3 4.1. The optimality system 4.2. The Pfaffian functional

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“…Lately, some spatial patterns arising from two interacting Turing modes with different wavelengths, have been theoretically investigated, see [7][8][9][10][11][12][13][14]. Yang and Song [7] investigated conditions of the occurrence of spatial resonance bifurcation and corresponding spatial patterns for a Gierer-Meinhardt system, utilizing linear stability analysis, center manifold theory and normal form method.…”

Section: Introductionmentioning

confidence: 99%

On Turing-Turing bifurcation of partial functional differential equations and its induced superposition patterns

Cao¹,

Jiang²

2022

Preprint

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When two Turing modes interact, i.e., Turing-Turing bifurcation occurs, superposition patterns revealing complex dynamical phenomena appear. In this paper, superposition patterns resulting from Turing-Turing bifurcation are investigated in theory. Firstly, the third-order normal form locally topologically equivalent to original partial functional differential equations (PFDEs) is derived. When selecting 1D domain and Neumann boundary conditions, three normal forms describing different spatial patterns are deduced from original third-order normal form. Also, formulas for computing coefficients of these normal forms are given, which are expressed in explicit form of original system parameters. With the aid of three normal forms, spatial patterns of a diffusive predator-prey system with Crowley-Martin functional response near Turing-Turing singularity are investigated. For one set of parameters, diffusive system supports the coexistence of four stable steady states with different single characteristic wavelengths, which demonstrates our previous conjecture. For another set of parameters, superposition patterns, tri-stable patterns that a pair of stable superposition steady states coexists with the stable coexistence equilibrium or another stable steady state, as well as quad-stable patterns that a pair of stable superposition steady states and another pair of stable steady states coexist, arise. Finally, numerical simulations are shown to support theory analysis.

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Steady state bifurcations for a glycolysis model in biochemical reaction

Cited by 23 publications

References 22 publications

Stability and bifurcation with singularity for a glycolysis model under no-flux boundary condition

Stability and bifurcation with singularity for a glycolysis model under no-flux boundary condition

Hilbert's 16th problem. II. Pfaffian equations and variational methods

On Turing-Turing bifurcation of partial functional differential equations and its induced superposition patterns

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