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 language | English (US) |
|---|---|
| Pages (from-to) | 1847-1888 |
| Number of pages | 42 |
| Journal | SIAM Journal on Applied Dynamical Systems |
| Volume | 12 |
| Issue number | 4 |
| DOIs | |
| State | Published - 2013 |
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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
Bifurcations in delay differential equations and applications to tumor and immune system interaction models. / Bi, Ping; Ruan, Shigui.
In: SIAM Journal on Applied Dynamical Systems, Vol. 12, No. 4, 2013, p. 1847-1888.Research output: Contribution to journal › Article
}
TY - JOUR
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
UR - http://www.scopus.com/inward/record.url?scp=84891422410&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84891422410&partnerID=8YFLogxK
U2 - 10.1137/120887898
DO - 10.1137/120887898
M3 - Article
AN - SCOPUS:84891422410
VL - 12
SP - 1847
EP - 1888
JO - SIAM Journal on Applied Dynamical Systems
JF - SIAM Journal on Applied Dynamical Systems
SN - 1536-0040
IS - 4
ER -