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.

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).

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.

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