scholarly journals Bifurcations and Dynamics of the Rb-E2F Pathway Involving miR449

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-20
Author(s):  
Lingling Li ◽  
Jianwei Shen
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Theory ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Delay Model ◽  
Dynamical Behaviors ◽  
Theoretical Results

We focused on the gene regulative network involving Rb-E2F pathway and microRNAs (miR449) and studied the influence of time delay on the dynamical behaviors of Rb-E2F pathway by using Hopf bifurcation theory. It is shown that under certain assumptions the steady state of the delay model is asymptotically stable for all delay values; there is a critical value under another set of conditions; the steady state is stable when the time delay is less than the critical value, while the steady state is changed to be unstable when the time delay is greater than the critical value. Thus, Hopf bifurcation appears at the steady state when the delay passes through the critical value. Numerical simulations were presented to illustrate the theoretical results.

scholarly journals Stability and Hopf Bifurcation in a Computer Virus Model with Multistate Antivirus

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Tao Dong ◽  
Xiaofeng Liao ◽  
Huaqing Li
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Computer Virus ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Dynamical Behaviors ◽  
Form Theory ◽  
Virus Model ◽  
Theoretical Results

By considering that people may immunize their computers with countermeasures in susceptible state, exposed state and using anti-virus software may take a period of time, a computer virus model with time delay based on an SEIR model is proposed. We regard time delay as bifurcating parameter to study the dynamical behaviors which include local asymptotical stability and local Hopf bifurcation. By analyzing the associated characteristic equation, Hopf bifurcation occurs when time delay passes through a sequence of critical value. The linerized model and stability of the bifurcating periodic solutions are also derived by applying the normal form theory and the center manifold theorem. Finally, an illustrative example is also given to support the theoretical results.

scholarly journals Chaos and Hopf Bifurcation Analysis of the Delayed Local Lengyel-Epstein System

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qingsong Liu ◽  
Yiping Lin ◽  
Jingnan Cao ◽  
Jinde Cao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Local Reaction ◽  
Reaction Diffusion ◽  
Critical Value ◽  
Hopf Bifurcations ◽  
The Stability

The local reaction-diffusion Lengyel-Epstein system with delay is investigated. By choosingτas bifurcating parameter, we show that Hopf bifurcations occur when time delay crosses a critical value. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, numerical simulations are performed to support the analytical results and the chaotic behaviors are observed.

GLOBAL DYNAMICS OF A CHOLERA MODEL WITH TIME DELAY

2013 ◽  
Vol 06 (01) ◽  
pp. 1250070 ◽  
Author(s):  
YUJIE WANG ◽  
JUNJIE WEI
Keyword(s):  
Time Delay ◽  
Numerical Simulations ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Threshold Value ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Ode Model ◽  
Periodic Oscillations ◽  
Theoretical Results

The global dynamics of a cholera model with delay is considered. We determine a basic reproduction number R0 which is chosen based on the relative ODE model, and establish that the global dynamics are determined by the threshold value R0. If R0 < 1, then the infection-free equilibrium is global asymptotically stable, that is, the cholera dies out; If R0 > 1, then the unique endemic equilibrium is global asymptotically stable, which means that the infection persists. The results obtained show that the delay does not lead to periodic oscillations. Finally, some numerical simulations support our theoretical results.

scholarly journals Bifurcation Analysis of Three-Strategy Imitative Dynamics with Mutations

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Wenjun Hu ◽  
Haiyan Tian ◽  
Gang Zhang
Keyword(s):  
Social Networks ◽  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Evolutionary Game ◽  
Critical Value ◽  
Important Research ◽  
Subcritical Hopf Bifurcation ◽  
Cooperative Behaviors ◽  
Theoretical Results

Evolutionary game dynamics is an important research, which is widely used in many fields such as social networks, biological systems, and cooperative behaviors. This paper focuses on the Hopf bifurcation in imitative dynamics of three strategies (Rock-Paper-Scissors) with mutations. First, we verify that there is a Hopf bifurcation in the imitative dynamics with no mutation. Then, we find that there is a critical value of mutation such that the system tends to an unstable limit cycle created in a subcritical Hopf bifurcation. Moreover, the Hopf bifurcation exists for other kinds of the considered mutation patterns. Finally, the theoretical results are verified by numerical simulations through Rock-Paper-Scissors game.

scholarly journals Dynamics of Fractional-Order Epidemic Models with General Nonlinear Incidence Rate and Time-Delay

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1829
Author(s):  
Ardak Kashkynbayev ◽  
Fathalla A. Rihan
Keyword(s):  
Steady State ◽  
Time Delay ◽  
Incidence Rate ◽  
Fractional Order ◽  
Asymptotically Stable ◽  
Nonlinear Incidence Rate ◽  
Globally Asymptotically Stable ◽  
Nonlinear Incidence ◽  
Theoretical Results ◽  
Disease Free

In this paper, we study the dynamics of a fractional-order epidemic model with general nonlinear incidence rate functionals and time-delay. We investigate the local and global stability of the steady-states. We deduce the basic reproductive threshold parameter, so that if R0<1, the disease-free steady-state is locally and globally asymptotically stable. However, for R0>1, there exists a positive (endemic) steady-state which is locally and globally asymptotically stable. A Holling type III response function is considered in the numerical simulations to illustrate the effectiveness of the theoretical results.

Bifurcation Analysis in a Lotka-Volterra Model with Delay

2012 ◽  
Vol 594-597 ◽  
pp. 2693-2696
Author(s):  
Chang Jin Xu
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Volterra Model ◽  
The Stability ◽  
Theoretical Results

In this paper, a Lotka-Volterra model with time delay is considered. The stability of the equilibrium of the model is investigated and the existence of Hopf bifurcation is proved. Numerical simulations are performed to justify the theoretical results. Finally, main conclusions are included.

scholarly journals Hopf Bifurcation of an Improved SLBS Model under the Influence of Latent Period

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Chunming Zhang ◽  
Wanping Liu ◽  
Jing Xiao ◽  
Yun Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Dynamical Behaviors ◽  
Latent Time ◽  
Normal Form Method ◽  
The Stability ◽  
Theoretical Results

A model applicable to describe the propagation of computer virus is developed and studied, along with the latent time incorporated. We regard time delay as a bifurcating parameter to study the dynamical behaviors including local asymptotical stability and local Hopf bifurcation. By analyzing the associated characteristic equation, Hopf bifurcation occurs when the time delay passes through a sequence of critical values. A formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions is given by using the normal form method and center manifold theorem. Finally, illustrative examples are given to support the theoretical results.

scholarly journals Analysis of an HIV - HCV simultaneous infection model with time delay

2021 ◽  
pp. 1-11
Author(s):  
Adamu Shitu Hassan ◽  
Nafiu Hussaini
Keyword(s):  
Hepatitis C Virus ◽  
Hepatitis C ◽  
Time Delay ◽  
Numerical Simulations ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Delay Model ◽  
Globally Asymptotically Stable ◽  
Simultaneous Infection ◽  
Disease Free

A novel mathematical delay model for simultaneous infection of HIV and hepatitis C virus is formulated and dynamically analyzed. Basic properties of the model are established and proved. Basic reproductive threshold is systematically calculated as the maximum of three subthreshold parameters. A disease free equilibrium is determined to be globally asymptotically stable for all values of the delay when the threshold is less than unity. However, when the threshold is greater than one, endemic equilibrium emerged which is shown to be locally asymptotically stable for any length of delay. Although the delay has no effect on stabilities of equilibria points, however, it is found to reduce the infectivity of the viruses as the length of the delay is increased. Epidemiological interpretations of the results and numerical simulations illustrating them are given.

scholarly journals Stability and Hopf Bifurcation Analysis of a Fractional-Order Epidemic Model with Time Delay

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Zhen Wang ◽  
Xinhe Wang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Equilibrium Point ◽  
Fractional Order ◽  
Bifurcation Analysis ◽  
Epidemic Model ◽  
Bifurcation Parameter ◽  
Theoretical Results ◽  
Disease Free

A fractional-order epidemic model with time delay is considered. Firstly, stability of the disease-free equilibrium point and endemic equilibrium point is studied. Then, by choosing the time delay as a bifurcation parameter, the existence of Hopf bifurcation is studied. Finally, numerical simulations are given to illustrate the effectiveness and feasibility of theoretical results.

Bifurcation Analysis of a Network of Three Neurons with Time Delays

2011 ◽  
Vol 105-107 ◽  
pp. 147-150
Author(s):  
Xiao Chen Mao
Keyword(s):  
Hopf Bifurcation ◽  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Center Manifold Theorem ◽  
Dynamical Behaviors ◽  
Trivial Equilibrium ◽  
Nice Agreement ◽  
Theoretical Results

The paper studies the dynamical behaviors of a recurrent neural network model consisting of three neurons with time delays through theoretical analysis and numerical simulations. The local stability of the trivial equilibrium of the network is analyzed and the sufficient conditions of the existence of Hopf bifurcation are given by discussing the associated characteristic equation. The direction and stability of the bifurcated periodic oscillations arising from Hopf bifurcation, which depend on the nonlinear terms of the network, are determined by means of the normal form and the center manifold theorem. Afterwards, numerical examples are performed to validate the theoretical analysis. The case studies of numerical simulations reach nice agreement with the theoretical results.

Sign in / Sign up

Export Citation Format

Share Document