Turing Instability and Turing–Hopf Bifurcation in Diffusive Schnakenberg Systems with Gene Expression Time Delay

The reaction time delay in the transcription process depends on the concentration of the protein because the transportation of mRNA from the nucleus to the cytoplasm becomes saturated. Thus the gene regulatory network is a state-dependent delayed model. This study aims to provide some mathematical explanations for the dynamics of the system, such as the linear stability and periodic oscillation, using mathematical techniques, such as formal linearization, linear stability analysis, the method of multiple scale (MMS), and the normal form. First, Hopf bifurcation of the state-dependent delayed gene regulatory networks model in the gene expression is analyzed by the method of multiple scales (MMS). Mechanism of periodic oscillations is obtained by Hopf bifurcation. The findings show that when degradation effects of the mRNA and protein are very strong, the oscillatory gene expression disappears. Then, a more realistic version of the aforementioned model with both constant and state-dependent time delays is established due to the existence of the constant time delay in the protein degradation process. Its nonresonant double Hopf bifurcation is found and analyzed using MMS. Interesting complex dynamic phenomena, such as periodic, quasi-periodic, and global period-[Formula: see text] solutions, are also discovered. These observations indicate that both state-dependent delay and constant delay could induce richer dynamics of the system, and the modified model may potentially describe the real dynamical mechanism (both the transcription process and the degradation process) more accurately in the gene expression. The findings may provide important guidance or hints to understand the real dynamic mechanism of the gene expression process.

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Hopf Bifurcation and Delay-Induced Turing Instability in a Diffusive lac Operon Model

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416501674 ◽

2016 ◽

Vol 26 (10) ◽

pp. 1650167 ◽

Cited By ~ 5

Author(s):

Xin Cao ◽

Yongli Song ◽

Tonghua Zhang

Keyword(s):

Hopf Bifurcation ◽

Time Delay ◽

Spatial Patterns ◽

Turing Instability ◽

Delayed Feedback ◽

Spatiotemporal Dynamics ◽

Positive Equilibrium ◽

Lac Operon ◽

Diffusion Effect ◽

And Diffusion

In this paper, we investigate the dynamics of a [Formula: see text] operon model with delayed feedback and diffusion effect. If the system is without delay or the delay is small, the positive equilibrium is stable so that there are no spatial patterns formed; while the time delay is large enough the equilibrium becomes unstable so that rich spatiotemporal dynamics may occur. We have found that time delay can not only incur temporal oscillations but also induce imbalance in space. With different initial values, the system may have different spatial patterns, for instance, spirals with one head, four heads, nine heads, and even microspirals.

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Stability analysis for Schnakenberg reaction-diffusion model with gene expression time delay

Chaos Solitons & Fractals ◽

10.1016/j.chaos.2021.111730 ◽

2022 ◽

Vol 155 ◽

pp. 111730

Author(s):

H.Y. Alfifi

Keyword(s):

Gene Expression ◽

Stability Analysis ◽

Time Delay ◽

Diffusion Model ◽

Reaction Diffusion ◽

Reaction Diffusion Model ◽

Expression Time

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Turing–Hopf bifurcation and spatiotemporal patterns in a Gierer–Meinhardt system with gene expression delay

Nonlinear Analysis Modelling and Control ◽

10.15388/namc.2021.26.23054 ◽

2021 ◽

Vol 26 (3) ◽

pp. 461-481

Author(s):

Shuangrui Zhao ◽

Hongbin Wang ◽

Weihua Jiang

Keyword(s):

Gene Expression ◽

Hopf Bifurcation ◽

Pattern Formation ◽

Time Delays ◽

Spatiotemporal Patterns ◽

Third Order ◽

Spatially Inhomogeneous ◽

Math Biol ◽

Formation Systems ◽

Expression Time

In this paper, we consider the dynamics of delayed Gierer–Meinhardt system, which is used as a classic example to explain the mechanism of pattern formation. The conditions for the occurrence of Turing, Hopf and Turing–Hopf bifurcation are established by analyzing the characteristic equation. For Turing–Hopf bifurcation, we derive the truncated third-order normal form based on the work of Jiang et al. [11], which is topologically equivalent to the original equation, and theoretically reveal system exhibits abundant spatial, temporal and spatiotemporal patterns, such as semistable spatially inhomogeneous periodic solutions, as well as tristable patterns of a pair of spatially inhomogeneous steady states and a spatially homogeneous periodic solution coexisting. Especially, we theoretically explain the phenomenon that time delay inhibits the formation of heterogeneous steady patterns, found by S. Lee, E. Gaffney and N. Monk [The influence of gene expression time delays on Gierer–Meinhardt pattern formation systems, Bull. Math. Biol., 72(8):2139–2160, 2010.]

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Dynamical analysis of a giving up smoking model with time delay

Advances in Difference Equations ◽

10.1186/s13662-019-2450-4 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 4

Author(s):

Zizhen Zhang ◽

Ruibin Wei ◽

Wanjun Xia

Keyword(s):

Hopf Bifurcation ◽

Time Delay ◽

Local Stability ◽

Center Manifold ◽

Dynamical Behavior ◽

Sufficient Conditions ◽

Dynamical Analysis ◽

Normal Form Theory ◽

Center Manifold Theorem ◽

Form Theory

AbstractIn this paper, we are concerned with a delayed smoking model in which the population is divided into five classes. Sufficient conditions guaranteeing the local stability and existence of Hopf bifurcation for the model are established by taking the time delay as a bifurcation parameter and employing the Routh–Hurwitz criteria. Furthermore, direction and stability of the Hopf bifurcation are investigated by applying the center manifold theorem and normal form theory. Finally, computer simulations are implemented to support the analytic results and to analyze the effects of some parameters on the dynamical behavior of the model.

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