TY - JOUR
T1 - Nonlinear dynamics in tumor-immune system interaction models with delays
AU - Ruan, Shigui
N1 - Funding Information:
2020 Mathematics Subject Classification. Primary: 34K18, 92C37; Secondary: 37N25. Key words and phrases. Delay differential equations, tumor-immune system interaction, Hopf bifurcation, Bautin bifurcation, Fold-Hopf (zero-Hopf) bifurcation, Hopf-Hopf bifurcation, Chaos. Research was partially supported by National Science Foundation grant (DMS-1853622).
Publisher Copyright:
© 2021 American Institute of Mathematical Sciences. All rights reserved.
PY - 2021/1
Y1 - 2021/1
N2 - In this paper, we review some recent results on the nonlinear dynamics of delayed differential equation models describing the interaction between tumor cells and effector cells of the immune system, in which the delays represent times necessary for molecule production, proliferation, differentiation of cells, transport, etc. First we consider a tumor-immune system interaction model with a single delay and present results on the existence and local stability of equilibria as well as the existence of Hopf bifurcation in the model when the delay varies. Second we investigate a tumor-immune system interaction model with two delays and show that the model undergoes various possible bifurcations including Hopf, Bautin, Fold-Hopf (zero-Hopf), and Hopf-Hopf bifurcations. Finally we discuss a tumor-immune system interaction model with three delays and demonstrate that the model exhibits more complex behaviors including chaos. Numerical simulations are provided to illustrate the nonlinear dynamics of the delayed tumor-immune system interaction models. More interesting issues and questions on modeling and analyzing tumor-immune dynamics are given in the discussion section.
AB - In this paper, we review some recent results on the nonlinear dynamics of delayed differential equation models describing the interaction between tumor cells and effector cells of the immune system, in which the delays represent times necessary for molecule production, proliferation, differentiation of cells, transport, etc. First we consider a tumor-immune system interaction model with a single delay and present results on the existence and local stability of equilibria as well as the existence of Hopf bifurcation in the model when the delay varies. Second we investigate a tumor-immune system interaction model with two delays and show that the model undergoes various possible bifurcations including Hopf, Bautin, Fold-Hopf (zero-Hopf), and Hopf-Hopf bifurcations. Finally we discuss a tumor-immune system interaction model with three delays and demonstrate that the model exhibits more complex behaviors including chaos. Numerical simulations are provided to illustrate the nonlinear dynamics of the delayed tumor-immune system interaction models. More interesting issues and questions on modeling and analyzing tumor-immune dynamics are given in the discussion section.
KW - Bautin bifurcation
KW - Chaos
KW - Delay differential equations
KW - Fold-Hopf (zero-Hopf) bifurcation
KW - Hopf bifurcation
KW - Hopf-Hopf bifurcation
KW - Tumor-immune system interaction
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U2 - 10.3934/dcdsb.2020282
DO - 10.3934/dcdsb.2020282
M3 - Review article
AN - SCOPUS:85101383769
VL - 26
SP - 541
EP - 602
JO - Discrete and Continuous Dynamical Systems - Series B
JF - Discrete and Continuous Dynamical Systems - Series B
SN - 1531-3492
IS - 1
ER -