| Citation: |
Jingnan Wang, Lu Zang, Li Xu. HOPF BIFURCATION AND CHAOS OF COMBINATIONAL IMMUNE ANTI-TUMOR MODEL WITH DOUBLE DELAYS[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2682-2702. doi: 10.11948/20220534
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HOPF BIFURCATION AND CHAOS OF COMBINATIONAL IMMUNE ANTI-TUMOR MODEL WITH DOUBLE DELAYS
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Abstract
In order to investigate the relations between tumor species growth and T cell activation assisted by dendritic cells, we establish a combinational immune anti-tumor model with double delays. Taking the activation rate of T cell and two time delays of tumor species growth and dendritic cell activation as parameters, we investigate the dynamical properties of the double delayed model, including the stability switches and the Hopf bifurcations of tumor-escape equilibrium and tumor-present equilibrium. With Hopf bifurcation curves, the center manifold theory and the normal form method, we find bi-stability states, the coexistence of two periodic solutions with different stabilities, two double Hopf bifurcation points, and use numerical simulations to show rich dynamic behaviors around the double Hopf bifurcation points, including the phase portraits and the corresponding Poincaré maps of chaotic attractors, as well as the progress transmission of unstable-oscillation-stable-oscillation. The theoretical and numerical results reveal the new methods of controlling tumor cells.
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Keywords:
- Tumor-immune model /
- double time delays /
- chaotic attractor /
- stability switch /
- Hopf bifurcation
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References
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Jingnan Wang, Lu Zang, Li Xu. HOPF BIFURCATION AND CHAOS OF COMBINATIONAL IMMUNE ANTI-TUMOR MODEL WITH DOUBLE DELAYS[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2682-2702. doi: 10.11948/20220534
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Figure 1.
The phase plane and solution curves of model (1.5) with
$ \tau_1=0 $ -
Figure 2.
The phase plane and solution curves of model (1.5) with
$ \tau_1=0.2, \tau_2=0 $ -
Figure 3.
An unstable periodic oscillation and a stable periodic oscillation of model (1.5).
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Figure 4.
The phase plane and solution curves of model (1.5) with
$ \tau_1=0.2 $ -
Figure 5.
The stability switch curves on
$ \tau_{1} $ $ \tau_{2} $ -
Figure 6.
The phase portraits of chaotic attractors of model (1.5), and the corresponding Poincar
$ \acute{e} $ $ \acute{e} $ $ z(t)=z* $ -
Figure 7.
The phase portraits of chaotic attractors of model (1.5), and the corresponding Poincar
$ \acute{e} $ $ \acute{e} $ $ x(t)=x* $ -
Figure 8.
Chaotic attractors of model (1.5) with coefficient condition (3.1)
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Figure 9.
The stability switch curves on
$ \tau_{1} $ $ \tau_{2} $ $ b=0.25 $ -
Figure 10.
The phase plane and solution curves of model (1.5) with
$ b=0.25 $ -
Figure 11.
The phase diagrams of the chaos in model (1.5) with
$ K=260, \; a=0.02, \; b=0.25, \; c=1.2, \; r=6.2, \; \beta=1, \; \delta=0.9 $ -
Figure 12.
The phase diagrams of model (1.5) with
$ K=250, \; a=0.2, \; b=0.6, \; c=10, \; r=10, \; \beta=5, \; \delta=1 $