scholarly journals Hopf bifurcation of an HIV-1 virus model with two delays and logistic growth

2020 ◽  
Vol 15 ◽  
pp. 16
Author(s):  
Caixia Sun ◽  
Lele Li ◽  
Jianwen Jia
Keyword(s):  
Hopf Bifurcation ◽  
Logistic Growth ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Humoral Immune ◽  
Virus Model ◽  
Pass Through ◽  
The Stability ◽  
Hiv 1

The paper establish and investigate an HIV-1 virus model with logistic growth, which also has intracellular delay and humoral immunity delay. The local stability of feasible equilibria are established by analyzing the characteristic equations. The globally stability of infection-free equilibrium and immunity-inactivated equilibrium are studied using the Lyapunov functional and LaSalles invariance principle. Besides, we prove that Hopf bifurcation will occur when the humoral immune delay pass through the critical value. And the stability of the positive equilibrium and Hopf bifurcations are investigated by using the normal form theory and the center manifold theorem. Finally, we confirm the theoretical results by numerical simulations.

scholarly journals Hopf Bifurcation Analysis of a Two-Delay HIV-1 Virus Model with Delay-Dependent Parameters

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Yu Xiao ◽  
Yunxian Dai ◽  
Jinde Cao
Keyword(s):  
Hopf Bifurcation ◽  
Positive Equilibrium ◽  
Reproduction Rate ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Virus Model ◽  
Delay Dependent ◽  
Theoretical Results ◽  
Hiv 1

In this paper, a two-delay HIV-1 virus model with delay-dependent parameters is considered. The model includes both virus-to-cell and cell-to-cell transmissions. Firstly, immune-inactivated reproduction rate R 0 and immune-activated reproduction rate R 1 are deduced. When R 1 > 1 , the system has the unique positive equilibrium E ∗ . The local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the characteristic equation at the positive equilibrium with the time delay as the bifurcation parameter and four different cases. Besides, we obtain the direction and stability of the Hopf bifurcation by using the center manifold theorem and the normal form theory. Finally, the theoretical results are validated by numerical simulation.

scholarly journals Stability and Hopf Bifurcation in a Delayed SEIRS Worm Model in Computer Network

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Computer Network ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Form Theory ◽  
Worm Model ◽  
The Stability

A delayed SEIRS epidemic model with vertical transmission in computer network is considered. Sufficient conditions for local stability of the positive equilibrium and existence of local Hopf bifurcation are obtained by analyzing distribution of the roots of the associated characteristic equation. Furthermore, the direction of the local Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. Finally, a numerical example is presented to verify the theoretical analysis.

Stability and Hopf Bifurcation in a Delayed SIS Epidemic Model with Double Epidemic Hypothesis

2018 ◽  
Vol 19 (6) ◽  
pp. 561-571
Author(s):  
Jiangang Zhang ◽  
Yandong Chu ◽  
Wenju Du ◽  
Yingxiang Chang ◽  
Xinlei An
Keyword(s):  
Hopf Bifurcation ◽  
Epidemic Model ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Characteristic Roots ◽  
Sis Epidemic Model ◽  
Form Theory ◽  
The Stability ◽  
Sis Epidemic

AbstractThe stability and Hopf bifurcation of a delayed SIS epidemic model with double epidemic hypothesis are investigated in this paper. We first study the stability of the unique positive equilibrium of the model in four cases, and we obtain the stability conditions through analyzing the distribution of characteristic roots of the corresponding linearized system. Moreover, we choosing the delay as bifurcation parameter and the existence of Hopf bifurcation is investigated in detail. We can derive explicit formulas for determining the direction of the Hopf bifurcation and the stability of bifurcation periodic solution by center manifold theorem and normal form theory. Finally, we perform the numerical simulations for justifying the theoretical results.

scholarly journals The Hopf Bifurcation for a Predator-Prey System with -Logistic Growth and Prey Refuge

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Shaoli Wang ◽  
Zhihao Ge
Keyword(s):  
Hopf Bifurcation ◽  
Logistic Growth ◽  
Positive Equilibrium ◽  
Prey Refuge ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Prey System ◽  
Form Theory ◽  
The Stability

The Hopf bifurcation for a predator-prey system with -logistic growth and prey refuge is studied. It is shown that the ODEs undergo a Hopf bifurcation at the positive equilibrium when the prey refuge rate or the index- passed through some critical values. Time delay could be considered as a bifurcation parameter for DDEs, and using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results.

Periodic Oscillations in the Quorum-Sensing System with Time Delay

2020 ◽  
Vol 30 (09) ◽  
pp. 2050127
Author(s):  
Menghan Chen ◽  
Jinchen Ji ◽  
Haihong Liu ◽  
Fang Yan
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Quorum Sensing ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Sensing System ◽  
Quorum Sensing System ◽  
The Stability ◽  
System With Time Delay

The main aim of this paper is to study the oscillatory behaviors of gene expression networks in quorum-sensing system with time delay. The stability of the unique positive equilibrium and the existence of Hopf bifurcation are investigated by choosing the time delay as the bifurcation parameter and by applying the bifurcation theory. The explicit criteria determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are developed based on the normal form theory and the center manifold theorem. Numerical simulations demonstrate good agreements with the theoretical results. Results of this paper indicate that the time delay plays a crucial role in the regulation of the dynamic behaviors of quorum-sensing system.

Oscillatory behavior of p53-Mdm2 system driven by transcriptional and translational time delays

2020 ◽  
Vol 13 (05) ◽  
pp. 2050034
Author(s):  
Chunyan Gao ◽  
Haihong Liu ◽  
Zengrong Liu ◽  
Yuan Zhang ◽  
Fang Yan
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Cell Fate ◽  
Time Delays ◽  
Oscillatory Behavior ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Positive Equilibrium Point ◽  
The Stability

Biological experiments clarify that p53-Mdm2 module is the core of tumor network and p53 oscillation plays an important role in determining the tumor cell fate. In this paper, we investigate the effect of time delay on the oscillatory behavior induced by Hopf bifurcation in p53-Mdm2 system. First, the stability of the unique positive equilibrium point and the existence of Hopf bifurcation are investigated by using the time delay as the bifurcation parameter and by applying the bifurcation theory. Second, the explicit criteria determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are developed based on the normal form theory and the center manifold theorem. In addition, the combination of numerical simulation results and theoretical calculation results indicates that time delays in p53-Mdm2 system are critical for p53 oscillations. The results may help us to better understand the biological functions of p53 pathway and provide clues for treatment of cancer.

scholarly journals Dynamic Analysis of a Delayed Reaction-Diffusion Predator-Prey System with Modified Holling-Tanner Functional Response

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Xinhong Pan ◽  
Min Zhao ◽  
Chuanjun Dai ◽  
Yapei Wang
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Reaction Diffusion ◽  
Positive Equilibrium ◽  
Delayed Reaction ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
The Stability

A predator-prey model with modified Holling-Tanner functional response and time delays is considered. By regarding the delays as bifurcation parameters, the local and global asymptotic stability of the positive equilibrium are investigated. The system has been found to undergo a Hopf bifurcation at the positive equilibrium when the delays cross through a sequence of critical values. In addition, the direction of the Hopf bifurcation and the stability of bifurcated periodic solutions are also studied, and an explicit algorithm is obtained by applying normal form theory and the center manifold theorem. The main results are illustrated by numerical simulations.

scholarly journals Dynamical Analysis of a Viral Infection Model with Delays in Computer Networks

2015 ◽  
Vol 2015 ◽  
pp. 1-15 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Propagation Model ◽  
Dynamical Analysis ◽  
Infection Model ◽  
Virus Propagation ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Two Delays ◽  
The Stability

This paper is devoted to the study of an SIRS computer virus propagation model with two delays and multistate antivirus measures. We demonstrate that the system loses its stability and a Hopf bifurcation occurs when the delay passes through the corresponding critical value by choosing the possible combination of the two delays as the bifurcation parameter. Moreover, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by means of the center manifold theorem and the normal form theory. Finally, some numerical simulations are performed to illustrate the obtained results.

scholarly journals Bifurcation Analysis in a Delayed Diffusive Leslie-Gower Model

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Shuling Yan ◽  
Xinze Lian ◽  
Weiming Wang ◽  
Youbin Wang
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Neumann Boundary ◽  
Positive Equilibrium ◽  
Center Manifold Theorem ◽  
Normal Form Theorem ◽  
The Stability

We investigate a modified delayed Leslie-Gower model under homogeneous Neumann boundary conditions. We give the stability analysis of the equilibria of the model and show the existence of Hopf bifurcation at the positive equilibrium under some conditions. Furthermore, we investigate the stability and direction of bifurcating periodic orbits by using normal form theorem and the center manifold theorem.

STABILITY AND BIFURCATION ANALYSIS ON A DELAYED NEURAL NETWORK MODEL

2005 ◽  
Vol 15 (09) ◽  
pp. 2883-2893 ◽  
Author(s):  
XIULING LI ◽  
JUNJIE WEI
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Network Model ◽  
Neural Network Model ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Delayed Neural Network ◽  
The Stability

A simple delayed neural network model with four neurons is considered. Linear stability of the model is investigated by analyzing the associated characteristic equation. It is found that Hopf bifurcation occurs when the sum of four delays varies and passes a sequence of critical values. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. An example is given and numerical simulations are performed to illustrate the obtained results. Meanwhile, the bifurcation set is provided in the appropriate parameter plane.

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