Mathematical modeling and Hopf bifurcation analysis of tumor macrophage interaction with polarization delay
Macrophages have both anti-tumor and pro-tumor effects. Time delay is commonly observed in real systems, yet its impact on tumor-macrophage dynamics remains unclear. This paper develops a new tumor-macrophage model with time delay. The model describes the interactions between tumor cells (T), the cl...
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| Format: | Article |
| Language: | English |
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Taylor & Francis Group
2025-12-01
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| Series: | Journal of Biological Dynamics |
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| Online Access: | https://www.tandfonline.com/doi/10.1080/17513758.2025.2508240 |
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| _version_ | 1850141314617704448 |
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| author | Jianping Li Nan Liu Danni Wang Hongli Yang |
| author_facet | Jianping Li Nan Liu Danni Wang Hongli Yang |
| author_sort | Jianping Li |
| collection | DOAJ |
| description | Macrophages have both anti-tumor and pro-tumor effects. Time delay is commonly observed in real systems, yet its impact on tumor-macrophage dynamics remains unclear. This paper develops a new tumor-macrophage model with time delay. The model describes the interactions between tumor cells (T), the classically activated macrophages (M1), the alternatively activated macrophages (M2), and the inactive macrophages (M0). The system's solution is computed, and equilibrium stability is analyzed. The existence of Hopf bifurcation is subsequently established. There exists bifurcating periodic solutions near the internal equilibrium, showing tumor cells and macrophages can coexist in the long term, as well as the potential for tumor relapse. Furthermore, the normal form and center manifold theorem are utilized to study the nature of Hopf bifurcation. Sensitivity analysis highlights the effect of parameters on tumor population dynamics. Numerical simulations validate the theory, elaborating the model can serve as a useful tool for tumor system analysis. |
| format | Article |
| id | doaj-art-e36660c6691346049344ee5a531beb87 |
| institution | OA Journals |
| issn | 1751-3758 1751-3766 |
| language | English |
| publishDate | 2025-12-01 |
| publisher | Taylor & Francis Group |
| record_format | Article |
| series | Journal of Biological Dynamics |
| spelling | doaj-art-e36660c6691346049344ee5a531beb872025-08-20T02:29:29ZengTaylor & Francis GroupJournal of Biological Dynamics1751-37581751-37662025-12-0119110.1080/17513758.2025.2508240Mathematical modeling and Hopf bifurcation analysis of tumor macrophage interaction with polarization delayJianping Li0Nan Liu1Danni Wang2Hongli Yang3School of Mathematical Sciences, Inner Mongolia University, Hohhot, People's Republic of ChinaSchool of Mathematical Sciences, Baotou Teachers' College, Baotou, People's Republic of ChinaSchool of Mathematical Sciences, Inner Mongolia University, Hohhot, People's Republic of ChinaSchool of Mathematical Sciences, Inner Mongolia University, Hohhot, People's Republic of ChinaMacrophages have both anti-tumor and pro-tumor effects. Time delay is commonly observed in real systems, yet its impact on tumor-macrophage dynamics remains unclear. This paper develops a new tumor-macrophage model with time delay. The model describes the interactions between tumor cells (T), the classically activated macrophages (M1), the alternatively activated macrophages (M2), and the inactive macrophages (M0). The system's solution is computed, and equilibrium stability is analyzed. The existence of Hopf bifurcation is subsequently established. There exists bifurcating periodic solutions near the internal equilibrium, showing tumor cells and macrophages can coexist in the long term, as well as the potential for tumor relapse. Furthermore, the normal form and center manifold theorem are utilized to study the nature of Hopf bifurcation. Sensitivity analysis highlights the effect of parameters on tumor population dynamics. Numerical simulations validate the theory, elaborating the model can serve as a useful tool for tumor system analysis.https://www.tandfonline.com/doi/10.1080/17513758.2025.2508240Tumor dynamicsmacrophagestime delaystabilityHopf bifurcation92D25 |
| spellingShingle | Jianping Li Nan Liu Danni Wang Hongli Yang Mathematical modeling and Hopf bifurcation analysis of tumor macrophage interaction with polarization delay Journal of Biological Dynamics Tumor dynamics macrophages time delay stability Hopf bifurcation 92D25 |
| title | Mathematical modeling and Hopf bifurcation analysis of tumor macrophage interaction with polarization delay |
| title_full | Mathematical modeling and Hopf bifurcation analysis of tumor macrophage interaction with polarization delay |
| title_fullStr | Mathematical modeling and Hopf bifurcation analysis of tumor macrophage interaction with polarization delay |
| title_full_unstemmed | Mathematical modeling and Hopf bifurcation analysis of tumor macrophage interaction with polarization delay |
| title_short | Mathematical modeling and Hopf bifurcation analysis of tumor macrophage interaction with polarization delay |
| title_sort | mathematical modeling and hopf bifurcation analysis of tumor macrophage interaction with polarization delay |
| topic | Tumor dynamics macrophages time delay stability Hopf bifurcation 92D25 |
| url | https://www.tandfonline.com/doi/10.1080/17513758.2025.2508240 |
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