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.

scholarly journals Stability and Hopf Bifurcation Analysis on a Bazykin Model with Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Jianming Zhang ◽  
Lijun Zhang ◽  
Chaudry Masood Khalique
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Finite Delay ◽  
The Stability ◽  
Predator System

The dynamics of a prey-predator system with a finite delay is investigated. We show that a sequence of Hopf bifurcations occurs at the positive equilibrium as the delay increases. By using the theory of normal form and center manifold, explicit expressions for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.

scholarly journals Stability and Hopf Bifurcation Analysis of a Nutrient-Phytoplankton Model with Delay Effect

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Xinhong Pan ◽  
Min Zhao ◽  
Chuanjun Dai ◽  
Yapei Wang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Center Manifold Theory ◽  
Delay Effect ◽  
Delay Differential System ◽  
The Stability ◽  
Delay Differential ◽  
Manifold Theory ◽  
Two Equilibria

A delay differential system is investigated based on a previously proposed nutrient-phytoplankton model. The time delay is regarded as a bifurcation parameter. Our aim is to determine how the time delay affects the system. First, we study the existence and local stability of two equilibria using the characteristic equation and identify the condition where a Hopf bifurcation can occur. Second, the formulae that determine the direction of the Hopf bifurcation and the stability of periodic solutions are obtained using the normal form and the center manifold theory. Furthermore, our main results are illustrated using numerical simulations.

scholarly journals Hopf Bifurcation Control in a Delayed Predator-Prey System with Prey Infection and Modified Leslie-Gower Scheme

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
State Feedback ◽  
Center Manifold ◽  
Controlled System ◽  
Predator Prey ◽  
Hybrid Controller ◽  
Prey System ◽  
The Stability ◽  
Manifold Theory ◽  
Bifurcation And Stability

Hopf bifurcation of a delayed predator-prey system with prey infection and the modified Leslie-Gower scheme is investigated. The conditions for the stability and existence of Hopf bifurcation of the system are obtained. The state feedback and parameter perturbation are used for controlling Hopf bifurcation in the system. In addition, direction of Hopf bifurcation and stability of the bifurcated periodic solutions of the controlled system are obtained by using normal form and center manifold theory. Finally, numerical simulation results are presented to show that the hybrid controller is efficient in controlling Hopf bifurcation.

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 of a Computer Virus Model with Infection Delay and Recovery Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Haitao Song ◽  
Qiaochu Wang ◽  
Weihua Jiang
Keyword(s):  
Hopf Bifurcation ◽  
Virus Infection ◽  
Periodic Orbits ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Computer Virus ◽  
Stable Periodic ◽  
Virus Model ◽  
Manifold Theory ◽  
Bifurcation And Stability

A computer virus model with infection delay and recovery delay is considered. The sufficient conditions for the global stability of the virus infection equilibrium are established. We show that the time delay can destabilize the virus infection equilibrium and give rise to Hopf bifurcations and stable periodic orbits. By the normal form and center manifold theory, the direction of the Hopf bifurcation and stability of the bifurcating periodic orbits are determined. Numerical simulations are provided to support our theoretical conclusions.

Numerical Approximation of Hopf Bifurcation for Tumor-Immune System Competition Model with Two Delays

2013 ◽  
Vol 5 (2) ◽  
pp. 146-162
Author(s):  
Jing-Jun Zhao ◽  
Jing-Yu Xiao ◽  
Yang Xu
Keyword(s):  
Immune System ◽  
Hopf Bifurcation ◽  
Numerical Approximation ◽  
Center Manifold ◽  
Competition Model ◽  
Center Manifold Theory ◽  
Two Delays ◽  
Normal Form Method ◽  
The Stability ◽  
Manifold Theory

AbstractThis paper is concerned with the Hopf bifurcation analysis of tumor-immune system competition model with two delays. First, we discuss the stability of state points with different kinds of delays. Then, a sufficient condition to the existence of the Hopf bifurcation is derived with parameters at different points. Furthermore, under this condition, the stability and direction of bifurcation are determined by applying the normal form method and the center manifold theory. Finally, a kind of Runge-Kutta methods is given out to simulate the periodic solutions numerically. At last, some numerical experiments are given to match well with the main conclusion of this paper.

scholarly journals Bifurcation of a Microelectromechanical Nonlinear Coupling System with Delay Feedback

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Yanqiu Li ◽  
Wei Duan ◽  
Shujian Ma ◽  
Pengfei Li
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Electromechanical Coupling ◽  
Critical Values ◽  
Nonlinear Coupling ◽  
Coupling System ◽  
Distribution Of Eigenvalues ◽  
The Stability

The dynamics of a kind of electromechanical coupling deformable micromirror device torsion micromirror with delay are investigated. Based on the distribution of eigenvalues, we prove that a sequence of Hopf bifurcation occurs at the equilibrium as the delay increases and obtain the critical values of Hopf bifurcation. Explicit algorithms for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived, using the theories of normal form and 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].

scholarly journals Mathematical analysis of a cancer model with time-delay in tumor-immune interaction and stimulation processes

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kaushik Dehingia ◽  
Hemanta Kumar Sarmah ◽  
Yamen Alharbi ◽  
Kamyar Hosseini
Keyword(s):  
Time Delay ◽  
Immune Responses ◽  
Periodic Solutions ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Cancer Model ◽  
Delay Effect ◽  
Discrete Time Delay ◽  
The Stability ◽  
Vital Parameters

AbstractIn this study, we discuss a cancer model considering discrete time-delay in tumor-immune interaction and stimulation processes. This study aims to analyze and observe the dynamics of the model along with variation of vital parameters and the delay effect on anti-tumor immune responses. We obtain sufficient conditions for the existence of equilibrium points and their stability. Existence of Hopf bifurcation at co-axial equilibrium is investigated. The stability of bifurcating periodic solutions is discussed, and the time length for which the solutions preserve the stability is estimated. Furthermore, we have derived the conditions for the direction of bifurcating periodic solutions. Theoretically, it was observed that the system undergoes different states if we vary the system’s parameters. Some numerical simulations are presented to verify the obtained mathematical results.

scholarly journals Stability and Hopf Bifurcation Analysis of a Vector-Borne Disease with Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yuanyuan Chen ◽  
Ya-Qing Bi
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Local Analysis ◽  
Normal Form Theory ◽  
Form Theory ◽  
Vector Borne ◽  
The Stability ◽  
Delay Differential

A delay-differential modelling of vector-borne is investigated. Its dynamics are studied in terms of local analysis and Hopf bifurcation theory, and its linear stability and Hopf bifurcation are demonstrated by studying the characteristic equation. The stability and direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold argument.

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