scholarly journals Diffusion‐driven codimension‐2 Turing–Hopf bifurcation in the general Brusselator model

2021 ◽  
Author(s):  
Lei Kong ◽  
Changrong Zhu
Keyword(s):  
Hopf Bifurcation ◽  
Brusselator Model

Oscillatory Dynamics of P53 Network With Time Delays

2018 ◽  
Vol 21 (6) ◽  
pp. 411-419 ◽  
Author(s):  
Conghua Wang ◽  
Fang Yan ◽  
Yuan Zhang ◽  
Haihong Liu ◽  
Linghai Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Cell Fate ◽  
Time Delays ◽  
Sustained Oscillation ◽  
Multiple Time ◽  
Cell Fate Decisions ◽  
Oscillatory Dynamics ◽  
Main Components ◽  
Parameter Values ◽  
P53 Network

Aims and Objective: A large number of experimental evidences report that the oscillatory dynamics of p53 would regulate the cell fate decisions. Moreover, multiple time delays are ubiquitous in gene expression which have been demonstrated to lead to important consequences on dynamics of genetic networks. Although delay-driven sustained oscillation in p53-based networks is commonplace, the precise roles of such delays during the processes are not completely known. Method: Herein, an integrated model with five basic components and two time delays for the network is developed. Using such time delays as the bifurcation parameter, the existence of Hopf bifurcation is given by analyzing the relevant characteristic equations. Moreover, the effects of such time delays are studied and the expression levels of the main components of the system are compared when taking different parameters and time delays. Result and Conclusion: The above theoretical results indicated that the transcriptional and translational delays can induce oscillation by undergoing a super-critical Hopf bifurcation. More interestingly, the length of these delays can control the amplitude and period of the oscillation. Furthermore, a certain range of model parameter values is essential for oscillation. Finally, we illustrated the main results in detail through numerical simulations.

Hopf bifurcation for a discontinuous HTLV-1 model

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6247-6267 ◽  
Author(s):  
Elham Shamsara ◽  
Zahra Afsharnezhad ◽  
Reihaneh Mostolizadeh
Keyword(s):  
Hopf Bifurcation ◽  
Sliding Mode ◽  
Early Stage ◽  
Time Lag ◽  
Discontinuous System ◽  
Steady State Condition ◽  
Ctl Response ◽  
Optimal Drug ◽  
Long Time ◽  
Immunosuppressive Diseases

Developing accurate mathematical models for host immune response in immunosuppressive diseases such as HIV and HTLV-1 are essential to achieve an optimal drug therapy regime. Since for HTLV-1 specific CTL response typically occurs after a time lag, we consider a discontinuous response function to better describe this lagged response during the early stage of the infectious, thus the system of HTLV-1 model will be a discontinuous system. For analyzing the dynamic of the system we use Filippov theory and find conditions in which the Filippov system undergoes a Hopf bifurcation. The Hopf bifurcation help us to find stable and unstable periodic oscillations and can be used to predict whether the CTL response can return to a steady state condition. Also, Hopf bifurcation in sliding mode is investigated. In this case the solutions will remain in the hyper-surface of discontinuity and as a consequence the disease cannot progress, at least for a long time. Finally we use numerical simulations to demonstrate the results by example.

Stabilization of Dominant Structures in an Ionic Reaction-Diffusion System

1998 ◽  
Vol 63 (6) ◽  
pp. 761-769 ◽  
Author(s):  
Roland Krämer ◽  
Arno F. Münster
Keyword(s):  
Reaction Diffusion ◽  
Dominant Mode ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Local Perturbation ◽  
Dimensional System ◽  
One Dimensional ◽  
Brusselator Model ◽  
The One ◽  
Dominant Structure

We describe a method of stabilizing the dominant structure in a chaotic reaction-diffusion system, where the underlying nonlinear dynamics needs not to be known. The dominant mode is identified by the Karhunen-Loeve decomposition, also known as orthogonal decomposition. Using a ionic version of the Brusselator model in a spatially one-dimensional system, our control strategy is based on perturbations derived from the amplitude function of the dominant spatial mode. The perturbation is used in two different ways: A global perturbation is realized by forcing an electric current through the one-dimensional system, whereas the local perturbation is performed by modulating concentrations of the autocatalyst at the boundaries. Only the global method enhances the contribution of the dominant mode to the total fluctuation energy. On the other hand, the local method leads to simple bulk oscillation of the entire system.

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