Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions
This paper presents qualitative and bifurcation analysis near the degenerate equilibrium in a two-stage cancer model of interactions betweenlymphocyte cells and solid tumor and contributes to a better understanding of the dynamics of tumor and immune system interactions.We first establish the existe...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2012-02-01
|
| Series: | Mathematical Biosciences and Engineering |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2012.9.347 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832590188696567808 |
|---|---|
| author | Dan Liu Shigui Ruan Deming Zhu |
| author_facet | Dan Liu Shigui Ruan Deming Zhu |
| author_sort | Dan Liu |
| collection | DOAJ |
| description | This paper presents qualitative and bifurcation analysis near the degenerate equilibrium in a two-stage cancer model of interactions betweenlymphocyte cells and solid tumor and contributes to a better understanding of the dynamics of tumor and immune system interactions.We first establish the existence of Hopf bifurcation in the 3-dimensional cancer model and rule out the occurrence of the degenerateHopf bifurcation. Then a general Hopf bifurcation formula is applied to determine the stability of the limit cycle bifurcated fromthe interior equilibrium. Sufficient conditions on the existence of stable periodic oscillations of tumor levels are obtained forthe two-stage cancer model. Numerical simulations are presented to illustrate the existence of stable periodic oscillations with reasonable parametersand demonstrate the phenomenon of long-term tumor relapse in the model. |
| format | Article |
| id | doaj-art-68b1ab3decc24eb4910a9dd49c2d856e |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2012-02-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-68b1ab3decc24eb4910a9dd49c2d856e2025-01-24T02:05:29ZengAIMS PressMathematical Biosciences and Engineering1551-00182012-02-019234736810.3934/mbe.2012.9.347Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactionsDan Liu0Shigui Ruan1Deming Zhu2Department of Mathematics, Xidian University, Xi’an, Shaanxi 710071Department of Mathematics, Xidian University, Xi’an, Shaanxi 710071Department of Mathematics, Xidian University, Xi’an, Shaanxi 710071This paper presents qualitative and bifurcation analysis near the degenerate equilibrium in a two-stage cancer model of interactions betweenlymphocyte cells and solid tumor and contributes to a better understanding of the dynamics of tumor and immune system interactions.We first establish the existence of Hopf bifurcation in the 3-dimensional cancer model and rule out the occurrence of the degenerateHopf bifurcation. Then a general Hopf bifurcation formula is applied to determine the stability of the limit cycle bifurcated fromthe interior equilibrium. Sufficient conditions on the existence of stable periodic oscillations of tumor levels are obtained forthe two-stage cancer model. Numerical simulations are presented to illustrate the existence of stable periodic oscillations with reasonable parametersand demonstrate the phenomenon of long-term tumor relapse in the model.https://www.aimspress.com/article/doi/10.3934/mbe.2012.9.347hopf bifurcationtumorlymphocyte.oscillationequilibrium |
| spellingShingle | Dan Liu Shigui Ruan Deming Zhu Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions Mathematical Biosciences and Engineering hopf bifurcation tumor lymphocyte. oscillation equilibrium |
| title | Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions |
| title_full | Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions |
| title_fullStr | Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions |
| title_full_unstemmed | Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions |
| title_short | Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions |
| title_sort | stable periodic oscillations in a two stage cancer model of tumor and immune system interactions |
| topic | hopf bifurcation tumor lymphocyte. oscillation equilibrium |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2012.9.347 |
| work_keys_str_mv | AT danliu stableperiodicoscillationsinatwostagecancermodeloftumorandimmunesysteminteractions AT shiguiruan stableperiodicoscillationsinatwostagecancermodeloftumorandimmunesysteminteractions AT demingzhu stableperiodicoscillationsinatwostagecancermodeloftumorandimmunesysteminteractions |