Hopf bifurcation analysis in a model of oscillatory gene expression with delay

2009 ◽  
Vol 139 (4) ◽  
pp. 879-895 ◽  
Author(s):  
Junjie Wei ◽  
Chunbo Yu
Keyword(s):  
Gene Expression ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Normal Forms ◽  
Higher Dimensional ◽  
Expression Model ◽  
Oscillatory Gene Expression ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Gene Expression Model

The dynamics of a gene expression model with time delay are investigated. The investigation confirms that a Hopf bifurcation occurs due to the existence of stability switches when the delay varies. An explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions has been derived by using the theory of the centre manifold and the normal forms method. The global existence of periodic solutions has been established using a global Hopf bifurcation result by Wu and a Bendixson criterion for higher-dimensional ordinary differential equations due to Li and Muldowney.

GLOBAL EXISTENCE OF PERIODIC SOLUTIONS IN THE LINEARLY COUPLED MACKEY–GLASS SYSTEM

2011 ◽  
Vol 21 (03) ◽  
pp. 711-724 ◽  
Author(s):  
YANQIU LI ◽  
WEIHUA JIANG
Keyword(s):  
Hopf Bifurcation ◽  
Global Existence ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Glass System ◽  
Positive Equilibrium ◽  
Higher Dimensional ◽  
Distribution Of Eigenvalues ◽  
Existence Of Periodic Solutions ◽  
The Stability

The dynamics of a linearly coupled Mackey–Glass system with delay are investigated. Based on the distribution of eigenvalues, we prove that a sequence of Hopf bifurcation occurs at the positive equilibrium as the delay increases and obtain the bifurcation set in the parameter plane. The explicit algorithm for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived, using the theories of normal form and center manifold. The global existence of periodic solutions is established using a global Hopf bifurcation result due to Wu [1998] and a Bendixson's criterion for higher dimensional ordinary differential equations due to [Li & Muldowney, 1993].

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.

HOPF BIFURCATION ANALYSIS IN A MACKEY–GLASS SYSTEM

2007 ◽  
Vol 17 (06) ◽  
pp. 2149-2157 ◽  
Author(s):  
JUNJIE WEI ◽  
DEJUN FAN
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Higher Dimensional ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Equation With Delay ◽  
Bendixson Criterion

The dynamics of a Mackey–Glass equation with delay are investigated. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold. Global existence of periodic solutions are established using a global Hopf bifurcation result due to Wu [1998] and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney [1994].

A NONLINEAR DELAY MODEL DESCRIBING THE GROWTH OF TUMOR CELLS UNDER IMMUNE SURVEILLANCE AGAINST CANCER AND ITS STABILITY ANALYSIS

2012 ◽  
Vol 05 (03) ◽  
pp. 1260017 ◽  
Author(s):  
LING CHEN ◽  
WANBIAO MA
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Tumor Cells ◽  
Local Stability ◽  
Immune Surveillance ◽  
Delay Model ◽  
Existence Of Periodic Solutions ◽  
Growth Of Tumor ◽  
Nonlinear Delay

In this paper, based on some biological meanings and a model which was proposed by Lefever and Garay (1978), a nonlinear delay model describing the growth of tumor cells under immune surveillance against cancer is given. Then, boundedness of the solutions, local stability of the equilibria and Hopf bifurcation of the model are discussed in details. The existence of periodic solutions explains the restrictive interactions between immune surveillance and the growth of the tumor cells.

On the Stability Properties of a Periodically Forced, Time-Varying Mass System

2009 ◽  
Author(s):  
W. T. van Horssen ◽  
O. V. Pischanskyy ◽  
J. L. A. Dubbeldam
Keyword(s):  
Periodic Solutions ◽  
Forced Vibrations ◽  
Time Varying ◽  
Mass System ◽  
Single Degree Of Freedom ◽  
Stability Properties ◽  
Single Degree ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Varying Mass

In this paper the forced vibrations of a linear, single degree of freedom oscillator (sdofo) with a time-varying mass will be studied. The forced vibrations are due to small masses which are periodically hitting and leaving the oscillator with different velocities. Since these small masses stay for some time on the oscillator surface the effective mass of the oscillator will periodically vary in time. Not only solutions of the oscillator equation will be constructed, but also the stability properties, and the existence of periodic solutions will be discussed.

Periodic Solutions for a Class of Functional Differential Equations with State-Dependent Delay Close to Zero

2003 ◽  
Vol 13 (06) ◽  
pp. 807-841 ◽  
Author(s):  
R. Ouifki ◽  
M. L. Hbid
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Delay Equation ◽  
Constant Delay ◽  
State Dependent ◽  
State Dependent Delay ◽  
Existence Of Periodic Solutions ◽  
Functional Differential

The purpose of the paper is to prove the existence of periodic solutions for a functional differential equation with state-dependent delay, of the type [Formula: see text] Transforming this equation into a perturbed constant delay equation and using the Hopf bifurcation result and the Poincaré procedure for this last equation, we prove the existence of a branch of periodic solutions for the state-dependent delay equation, bifurcating from r ≡ 0.

Hopf Bifurcation Induced by Delay Effect in a Diffusive Tumor-Immune System

2018 ◽  
Vol 28 (11) ◽  
pp. 1850136 ◽  
Author(s):  
Ben Niu ◽  
Yuxiao Guo ◽  
Yanfei Du
Keyword(s):  
Immune System ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Immune Cell ◽  
Tumor Treatment ◽  
Delay Effect ◽  
Stable Periodic ◽  
The Stability ◽  
Bifurcation And Stability

Tumor-immune interaction plays an important role in the tumor treatment. We analyze the stability of steady states in a diffusive tumor-immune model with response and proliferation delay [Formula: see text] of immune system where the immune cell has a probability [Formula: see text] in killing tumor cells. We find increasing time delay [Formula: see text] destabilizes the positive steady state and induces Hopf bifurcations. The criticality of Hopf bifurcation is investigated by deriving normal forms on the center manifold, then the direction of bifurcation and stability of bifurcating periodic solutions are determined. Using a group of parameters to simulate the system, stable periodic solutions are found near the Hopf bifurcation. The effect of killing probability [Formula: see text] on Hopf bifurcation values is also discussed.

TWO DIFFERENT APPROACHES FOR GENE EXPRESSION MODEL CONTROL

BIOMAT 2011 ◽  
2012 ◽  
pp. 153-177
Author(s):  
N. A. BARBOSA ◽  
H DÍAZ ◽  
A. RAMIREZ
Keyword(s):  
Gene Expression ◽  
Model Control ◽  
Expression Model ◽  
Gene Expression Model
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