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

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.]

OSCILLATORY DYNAMICS INDUCED BY MULTI-DELAYS IN GENE EXPRESSION

2011 ◽  
Vol 14 (03) ◽  
pp. 451-469 ◽  
Author(s):  
QI WANG ◽  
BO LIU ◽  
SHIWEI YAN
Keyword(s):  
Gene Expression ◽  
Mathematical Method ◽  
Hopf Bifurcation ◽  
Positive Feedback ◽  
Time Delays ◽  
Chaotic Dynamics ◽  
Biochemical Reaction ◽  
Reaction Systems ◽  
Oscillatory Dynamics ◽  
Model Equations

We discuss the existence of Hopf bifurcation, the dynamical stability breaking of the molecular concentrations of two generic biochemical reaction systems and their sensitivity to the changes of the parameters incorporated in the model equations. It is found that the oscillatory dynamics can be expected for the systems due to the inclusion of time delays. The chaotic dynamics and the periodic windows in the chaotic domains can exist in the case of the system with two time delays. The proposed mathematical method may have the significance in the problems where the negative and/or positive feedback dynamics, as well as the time delays have the characteristic physical and biological background.

Hopf Bifurcation Analysis of a Gene Regulatory Network Mediated by Small Noncoding RNA with Time Delays and Diffusion

2017 ◽  
Vol 27 (13) ◽  
pp. 1750194 ◽  
Author(s):  
Chengxian Li ◽  
Haihong Liu ◽  
Tonghua Zhang ◽  
Fang Yan
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Time Delays ◽  
Noncoding Rna ◽  
Small Noncoding Rna ◽  
Spatially Inhomogeneous ◽  
Gene Regulatory ◽  
Spatially Homogeneous ◽  
The Stability ◽  
And Diffusion

In this paper, a gene regulatory network mediated by small noncoding RNA involving two time delays and diffusion under the Neumann boundary conditions is studied. Choosing the sum of delays as the bifurcation parameter, the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated by analyzing the corresponding characteristic equation. It is shown that the sum of delays can induce Hopf bifurcation and the diffusion incorporated into the system can effect the amplitude of periodic solutions. Furthermore, the spatially homogeneous periodic solution always exists and the spatially inhomogeneous periodic solution will arise when the diffusion coefficients of protein and mRNA are suitably small. Particularly, the small RNA diffusion coefficient is more robust and its effect on model is much less than protein and mRNA. Finally, the explicit formulae for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by employing the normal form theory and center manifold theorem for partial functional differential equations. Finally, numerical simulations are carried out to illustrate our theoretical analysis.

scholarly journals Stability and Hopf Bifurcation Analysis of a Gene Expression Model with Diffusion and Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Yahong Peng ◽  
Tonghua Zhang
Keyword(s):  
Gene Expression ◽  
Hopf Bifurcation ◽  
Local Stability ◽  
Time Delays ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Expression Model ◽  
The Stability ◽  
And Diffusion ◽  
Gene Expression Model

We consider a model for gene expression with one or two time delays and diffusion. The local stability and delay-induced Hopf bifurcation are investigated. We also derive the formulas determining the direction and the stability of Hopf bifurcations by calculating the normal form on the center manifold.

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