scholarly journals Hopf bifurcation in a delayed model for tumor-immune system competition with negative immune response

2006 ◽  
Vol 2006 ◽  
pp. 1-9 ◽  
Author(s):  
Radouane Yafia
Keyword(s):  
Immune Response ◽  
Asymptotic Behavior ◽  
Steady State ◽  
Immune System ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Limit Cycle ◽  
Numerical Study ◽  
Bifurcation Problem ◽  
Delayed Model

The dynamics of the model for tumor-immune system competition with negative immune response and with one delay investigated. We show that the asymptotic behavior depends crucially on the time delay parameter. We are particularly interested in the study of the Hopf bifurcation problem to predict the occurrence of a limit cycle bifurcating from the nontrivial steady state, by using the delay as a parameter of bifurcation. The obtained results provide the oscillations given by the numerical study in M. Gałach (2003), which are observed in reality by Kirschner and Panetta (1998).

scholarly journals Stability of limit cycle in a delayed model for tumor immune system competition with negative immune response

2006 ◽  
Vol 2006 ◽  
pp. 1-13 ◽  
Author(s):  
Radouane Yafia
Keyword(s):  
Immune Response ◽  
Immune System ◽  
Hopf Bifurcation ◽  
Discrete Delay ◽  
Local Asymptotic Stability ◽  
Nonlinear Delay Differential Equations ◽  
The Stability ◽  
Delay Differential ◽  
Delayed Model ◽  
Nonlinear Delay

This paper is devoted to the study of the stability of limit cycles of a system of nonlinear delay differential equations with a discrete delay. The system arises from a model of population dynamics describing the competition between tumor and immune system with negative immune response. We study the local asymptotic stability of the unique nontrivial equilibrium of the delay equation and we show that its stability can be lost through a Hopf bifurcation. We establish an explicit algorithm for determining the direction of the Hopf bifurcation and the stability or instability of the bifurcating branch of periodic solutions, using the methods presented by Diekmann et al.

Dynamics in a Van Der Pol Model with Delay

2012 ◽  
Vol 204-208 ◽  
pp. 4586-4589
Author(s):  
Chang Jin Xu ◽  
Pei Luan Li ◽  
Ling Yun Yao
Keyword(s):  
Asymptotic Behavior ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Parameter ◽  
Bifurcation Problem ◽  
Van Der Pol ◽  
Van Der Pol Model ◽  
The Stability

In this paper, the dynamics of a van der pol model with delay are considered. It is shown that the asymptotic behavior depends crucially on the time delay parameter. By regarded the delay as a bifurcation parameter, we are particularly interested in the study of the Hopf bifurcation problem. The length of delay which preserves the stability of the equilibrium is calculated. Some numerical simulations for justifying the analytical findings are included.

Analytical Study of the Existence of a Hopf Bifurcation in the Tumor Cell Growth Model with Time Delay

2021 ◽  
Vol 3 (1) ◽  
pp. 20-28
Author(s):  
A. Yusnaeni ◽  
Kasbawati Kasbawati ◽  
Toaha Syamsuddin
Keyword(s):  
Differential Equation ◽  
Steady State ◽  
Immune System ◽  
Hopf Bifurcation ◽  
Delay Differential Equation ◽  
Immune Cells ◽  
Steady State Solution ◽  
State Solution ◽  
Activation Process ◽  
Delay Differential

AbstractIn this paper, we study a mathematical model of an immune response system consisting of a number of immune cells that work together to protect the human body from invading tumor cells. The delay differential equation is used to model the immune system caused by a natural delay in the activation process of immune cells. Analytical studies are focused on finding conditions in which the system undergoes changes in stability near a tumor-free steady-state solution. We found that the existence of a tumor-free steady-state solution was warranted when the number of activated effector cells was sufficiently high. By considering the lag of stimulation of helper cell production as the bifurcation parameter, a critical lag is obtained that determines the threshold of the stability change of the tumor-free steady state. It is also leading the system undergoes a Hopf bifurcation to periodic solutions at the tumor-free steady-state solution.Keywords: tumor–immune system; delay differential equation; transcendental function; Hopf bifurcation. AbstrakDalam makalah ini, dikaji model matematika dari sistem respon imun yang terdiri dari sejumlah sel imun yang bekerja sama untuk melindungi tubuh manusia dari invasi sel tumor. Persamaan diferensial tunda digunakan untuk memodelkan sistem kekebalan yang disebabkan oleh keterlambatan alami dalam proses aktivasi sel-sel imun. Studi analitik difokuskan untuk menemukan kondisi di mana sistem mengalami perubahan stabilitas di sekitar solusi kesetimbangan bebas tumor. Diperoleh bahwa solusi kesetimbangan bebas tumor dijamin ada ketika jumlah sel efektor yang diaktifkan cukup tinggi. Dengan mempertimbangkan tundaan stimulasi produksi sel helper sebagai parameter bifurkasi, didapatkan lag kritis yang menentukan ambang batas perubahan stabilitas dari solusi kesetimbangan bebas tumor. Parameter tersebut juga mengakibatkan sistem mengalami percabangan Hopf untuk solusi periodik pada solusi kesetimbangan bebas tumor.Kata kunci: sistem tumor–imun; persamaan differensial tundaan; fungsi transedental; bifurkasi Hopf.

Limit Cycles in a Class of Quartic Kolmogorov Model with Three Positive Equilibrium Points

2015 ◽  
Vol 25 (06) ◽  
pp. 1550080 ◽  
Author(s):  
Chaoxiong Du ◽  
Yirong Liu ◽  
Qi Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Equilibrium Points ◽  
Positive Equilibrium ◽  
Bifurcation Problem ◽  
Kolmogorov Model ◽  
Limit Cycle Bifurcation ◽  
Perturbed Model

Limit cycle bifurcation problem of Kolmogorov model is interesting and significant both in theory and applications. In this paper, we will focus on investigating limit cycles for a class of quartic Kolmogorov model with three positive equilibrium points. Perturbed model can bifurcate three small limit cycles near (1, 2) or (2, 1) under a certain condition and can bifurcate one limit cycle near (1, 1). In addition, we have given some examples of simultaneous Hopf bifurcation and the structure of limit cycles bifurcated from three positive equilibrium points. The limit cycle bifurcation problem for Kolmogorov model with several positive equilibrium points are less seen in published references. Our result is good and interesting.

Hopf Bifurcation in PD Controlled Pendulum or Manipulator

2002 ◽  
Vol 124 (2) ◽  
pp. 327-332 ◽  
Author(s):  
Tom Bucklaew ◽  
Ching-Shi Liu
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Dynamic Behavior ◽  
Limit Cycle ◽  
Initial Conditions ◽  
Pendulum System ◽  
Periodic Attractors ◽  
Control Failure ◽  
Damped Systems ◽  
Strongly Damped

In this brief the dynamic behavior of a parametrically forced manipulator, or pendulum, system with PD control is examined. For an excitation of sufficient amplitude or frequency a Hopf bifurcation to a steady-state limit cycle is shown to result, appearing as a precursor to instability. The parameter space is mapped in order to illustrate regions where control failure will likely occur, even in the strongly damped case. For weakly damped systems, the Hopf bifurcation can additionally exhibit a dependence on initial conditions. The resulting case of competing point and periodic attractors is discussed.

scholarly journals EFFECT OF TIME DELAY ON SPATIAL PATTERNS IN A AIRAL INFECTION MODEL WITH DIFFUSION

2016 ◽  
Vol 21 (2) ◽  
pp. 143-158
Author(s):  
Jia Liu ◽  
Qunying Zhang ◽  
Canrong Tian
Keyword(s):  
Immune Response ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Viral Infection ◽  
Spatial Patterns ◽  
Infection Model ◽  
The Stability ◽  
Manifold Theory ◽  
Viral Infection Model ◽  
Theoretical Results

This paper is concerned with the dynamics of a viral infection model with diffusion under the assumption that the immune response is retarded. A time delay is incorporated into the model described the delayed immune response after viral infection. Based upon a stability analysis, we demonstrate that the appearance, or the absence, of spatial patterns is determined by the delay under some conditions. Moreover, the spatial patterns occurs as a consequence of Hopf bifurcation. By applying the normal form and the center manifold theory, the direction as well as the stability of the Hopf bifurcation is explored. In addition, a series of numerical simulations are performed to illustrate our theoretical results.

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 Hopf Bifurcation and Stability of Periodic Solutions for Delay Differential Model of HIV Infection of CD4+T-cells

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
P. Balasubramaniam ◽  
M. Prakash ◽  
Fathalla A. Rihan ◽  
S. Lakshmanan
Keyword(s):  
T Cells ◽  
Steady State ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Hiv Infection ◽  
Logistic Growth ◽  
Differential Model ◽  
Discrete Time Delay ◽  
The Stability ◽  
Delay Differential

This paper deals with stability and Hopf bifurcation analyses of a mathematical model of HIV infection ofCD4+T-cells. The model is based on a system of delay differential equations with logistic growth term and antiretroviral treatment with a discrete time delay, which plays a main role in changing the stability of each steady state. By fixing the time delay as a bifurcation parameter, we get a limit cycle bifurcation about the infected steady state. We study the effect of the time delay on the stability of the endemically infected equilibrium. We derive explicit formulae to determine the stability and direction of the limit cycles by using center manifold theory and normal form method. Numerical simulations are presented to illustrate the results.

scholarly journals Anticontrol of Hopf Bifurcation and Control of Chaos for a Finance System through Washout Filters with Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Huitao Zhao ◽  
Mengxia Lu ◽  
Junmei Zuo
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Feedback Control ◽  
Price Index ◽  
Financial System ◽  
Control Of Chaos ◽  
Control Laws ◽  
Washout Filter ◽  
And Control

A controlled model for a financial system through washout-filter-aided dynamical feedback control laws is developed, the problem of anticontrol of Hopf bifurcation from the steady state is studied, and the existence, stability, and direction of bifurcated periodic solutions are discussed in detail. The obtained results show that the delay on price index has great influences on the financial system, which can be applied to suppress or avoid the chaos phenomenon appearing in the financial system.

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