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

Meihua Wei; Yanling Li; Xi Wei

doi:10.3934/dcdsb.2019129

Article Contents

2019, Volume 24, Issue 9: 5203-5224. Doi: 10.3934/dcdsb.2019129

This issue Previous Article Stability of synchronization under stochastic perturbations in leaky integrate and fire neural networks of finite size Next Article

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

1.
School of Mathematics and Statistics, Yulin University, Yulin 719000, China
2.
College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China

^* Corresponding author: Meihua Wei
^* Corresponding author: Meihua Wei

Received: August 2017

Revised: January 2019

Early access: July 2019

Published: July 2019

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Abstract

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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Mathematics Subject Classification: Primary: 35B32, 35B35; Secondary: 37G05, 37G40.

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Figure 1. Local bifurcation of $ g(s,\lambda) = 0 $ at $ (s,\lambda) = (0,0) $. Here, (a) $ g(s,\lambda) = s^3+\lambda s $; (b) $ g(s,\lambda) = -s^3+\lambda s $

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Figure 2. The zero of $ C = 0 $

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Figure 3. The graph of (2) with $ k = 0.1, \delta = 3.0 $ and $ l = 6.0 $. Here, (a) $ d_2 = 0.105 $; (b) $ d_2 = \frac{\sqrt{(\lambda_3+\lambda_4)^2+4g_1\lambda_3\lambda_4} -(\lambda_3+\lambda_4)}{2\lambda_3\lambda_4} = 0.1200 $

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Figure 4. Numerical simulations of the steady state solution characterized by $ \phi_{4} $ for system (1) with $ k = 0.1, \delta = 3.0, l = 6.0, d_2 = 0.105 $ and $ d_1 = 5.8560 $

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Figure 5. Concentration profiles for $ u $ and $ v $ of (1) for $ k = 0.1, \delta = 3.0, l = 6.0, d_2 = 0.105 $ and $ d_1 = 5.8560 $

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Figure 6. Numerical simulations of the steady state solution involved two models $ \phi_{3} $ and $ \phi_{4} $ for system (1) with $ k = 0.1, \delta = 3.0, l = 6.0, d_2 = 0.1200 $ and $ d_1 = 7.0093 $

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Figure 7. Concentration profiles for $ u $ and $ v $ of (1) for $ k = 0.1, \delta = 3.0, l = 6.0, d_2 = 0.1200 $ and $ d_1 = 7.0093 $

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