Bifurcations in delay differential equations and applications to tumor and immune system interaction models

Ping Bi, Shigui Ruan

Research output: Contribution to journalArticle

24 Citations (Scopus)

Abstract

In this paper, we consider a two-dimensional delay differential system with two delays. By analyzing the distribution of eigenvalues, linear stability of the equilibria and existence of Hopf, Bautin, and Hopf-Hopf bifurcations are obtained in which the time delays are used as the bifurcation parameter. General formula for the direction, period, and stability of the bifurcated periodic solutions are given for codimension one and codimension two bifurcations, including Hopf bifurcation, Bautin bifurcation, and Hopf-Hopf bifurcation. As an application, we study the dynamical behaviors of a model describing the interaction between tumor cells and effector cells of the immune system. Numerical examples and simulations are presented to illustrate the obtained results.

Original languageEnglish (US)
Pages (from-to)1847-1888
Number of pages42
JournalSIAM Journal on Applied Dynamical Systems
Volume12
Issue number4
DOIs
StatePublished - 2013

Fingerprint

Hopf bifurcation
Bifurcation (mathematics)
Immune system
Immune System
Delay Differential Equations
Hopf Bifurcation
Tumors
Tumor
Differential equations
Bifurcation
Codimension
Interaction
Delay-differential Systems
Cell
Two-dimensional Systems
Linear Stability
Dynamical Behavior
Time Delay
Time delay
Periodic Solution

Keywords

  • Bautin bifurcation
  • Delay differential equations
  • Hopf bifurcation
  • Hopf-Hopf bifurcation
  • Tumor-immune system interaction

ASJC Scopus subject areas

  • Analysis
  • Modeling and Simulation

Cite this

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abstract = "In this paper, we consider a two-dimensional delay differential system with two delays. By analyzing the distribution of eigenvalues, linear stability of the equilibria and existence of Hopf, Bautin, and Hopf-Hopf bifurcations are obtained in which the time delays are used as the bifurcation parameter. General formula for the direction, period, and stability of the bifurcated periodic solutions are given for codimension one and codimension two bifurcations, including Hopf bifurcation, Bautin bifurcation, and Hopf-Hopf bifurcation. As an application, we study the dynamical behaviors of a model describing the interaction between tumor cells and effector cells of the immune system. Numerical examples and simulations are presented to illustrate the obtained results.",
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T1 - Bifurcations in delay differential equations and applications to tumor and immune system interaction models

AU - Bi, Ping

AU - Ruan, Shigui

PY - 2013

Y1 - 2013

N2 - In this paper, we consider a two-dimensional delay differential system with two delays. By analyzing the distribution of eigenvalues, linear stability of the equilibria and existence of Hopf, Bautin, and Hopf-Hopf bifurcations are obtained in which the time delays are used as the bifurcation parameter. General formula for the direction, period, and stability of the bifurcated periodic solutions are given for codimension one and codimension two bifurcations, including Hopf bifurcation, Bautin bifurcation, and Hopf-Hopf bifurcation. As an application, we study the dynamical behaviors of a model describing the interaction between tumor cells and effector cells of the immune system. Numerical examples and simulations are presented to illustrate the obtained results.

AB - In this paper, we consider a two-dimensional delay differential system with two delays. By analyzing the distribution of eigenvalues, linear stability of the equilibria and existence of Hopf, Bautin, and Hopf-Hopf bifurcations are obtained in which the time delays are used as the bifurcation parameter. General formula for the direction, period, and stability of the bifurcated periodic solutions are given for codimension one and codimension two bifurcations, including Hopf bifurcation, Bautin bifurcation, and Hopf-Hopf bifurcation. As an application, we study the dynamical behaviors of a model describing the interaction between tumor cells and effector cells of the immune system. Numerical examples and simulations are presented to illustrate the obtained results.

KW - Bautin bifurcation

KW - Delay differential equations

KW - Hopf bifurcation

KW - Hopf-Hopf bifurcation

KW - Tumor-immune system interaction

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