Analysis of a free boundary problem for vascularized tumor growth with time delays and almost periodic nutrient supply

Shihe Xu; Zuxing Xuan; Fangwei Zhang; Shihe Xu; Zuxing Xuan; Fangwei Zhang

doi:10.3934/math.2024648

AIMS Mathematics

2024, Volume 9, Issue 5: 13291-13312. doi: 10.3934/math.2024648

Previous Article Next Article

Research article

Analysis of a free boundary problem for vascularized tumor growth with time delays and almost periodic nutrient supply

1.
School of Mathematics and Statistics, Zhaoqing University, Zhaoqing, Guangdong 526061, China
2.
Institute of Fundamental and Interdisciplinary Sciences, Beijing Union University, Beijing 100101, China
3.
School of International Business, Shandong Jiaotong University, Weihai, Shandong 264200, China

Received: 01 December 2023 Revised: 08 March 2024 Accepted: 19 March 2024 Published: 10 April 2024
MSC : 34K13, 35Q92

In this research, we have proposed and investigated a time-delayed free boundary problem concerning tumor growth in the presence of almost periodic nutrient supply with angiogenesis. This study primarily focused on examining the impact of almost periodic nutrient supply, angiogenesis, and time delay on tumor growth dynamics. We analyzed the existence, uniqueness, and exponential stability of almost periodic solutions. Furthermore, we established conditions for the disappearance of almost periodic oscillations in tumors. The existence and uniqueness of almost periodic solutions were proven, while sufficient conditions for the exponential stability of the unique solution were established. Finally, computer simulations were employed to illustrate our results.
- tumor growth,
- angiogenesis,
- almost periodic solution,
- existence and uniqueness,
- stability
Citation: Shihe Xu, Zuxing Xuan, Fangwei Zhang. Analysis of a free boundary problem for vascularized tumor growth with time delays and almost periodic nutrient supply[J]. AIMS Mathematics, 2024, 9(5): 13291-13312. doi: 10.3934/math.2024648

Related Papers:

Abstract

In this research, we have proposed and investigated a time-delayed free boundary problem concerning tumor growth in the presence of almost periodic nutrient supply with angiogenesis. This study primarily focused on examining the impact of almost periodic nutrient supply, angiogenesis, and time delay on tumor growth dynamics. We analyzed the existence, uniqueness, and exponential stability of almost periodic solutions. Furthermore, we established conditions for the disappearance of almost periodic oscillations in tumors. The existence and uniqueness of almost periodic solutions were proven, while sufficient conditions for the exponential stability of the unique solution were established. Finally, computer simulations were employed to illustrate our results.

References

[1]	M. Bodnar, The nonnegativity of solutions of delay differential equations, Appl. Math. Lett., 13 (2000), 91–95. https://doi.org/10.1016/S0893-9659(00)00061-6 doi: 10.1016/S0893-9659(00)00061-6
[2]	H. Bohr, Almost periodic functions, New York, 1947.
[3]	H. M. Byrne, The effect of time delays on the dynamics of avascular tumor growth, Math. Biosci., 144 (1997), 83–117. https://doi.org/10.1016/S0025-5564(97)00023-0 doi: 10.1016/S0025-5564(97)00023-0
[4]	C. Corduneanu, Almost periodic functions, New York, 1989.
[5]	S. B. Cui, A. Friedman, Analysis of a mathematical model of the effect of inhibitors on the growth of tumors, Math. Biosci., 164 (2000), 103–137. https://doi.org/10.1016/S0025-5564(99)00063-2 doi: 10.1016/S0025-5564(99)00063-2
[6]	S. B. Cui, S. H. Xu, Analysis of mathematical models for the growth of tumors with time delays in cell proliferation, J. Math. Anal. Appl., 336 (2007), 523–541. https://doi.org/10.1016/j.jmaa.2007.02.047 doi: 10.1016/j.jmaa.2007.02.047
[7]	H. S. Ding, J. Nieto, A new approach for positive almost periodic solutions to a class of Nicholson's blowflies model, J. Comput. Appl. Math., 253 (2013), 249–254. https://doi.org/10.1016/j.cam.2013.04.028 doi: 10.1016/j.cam.2013.04.028
[8]	H. S. Ding, T. J. Xiao, J. Liang, Existence of positive almost automorphic solutions to nonlinear delay integral equations, Nonlinear Anal. Theor., 70 (2009), 2216–2231. https://doi.org/10.1016/j.na.2008.03.001 doi: 10.1016/j.na.2008.03.001
[9]	A. M. Fink, Almost periodic differential equations, Berlin: Springer, 1974. https://doi.org/10.1007/BFb0070324
[10]	U. Forýs, M. Bodnar, Time delays in proliferation process for solid avascular tumour, Math. Comput. Model., 37 (2003), 1201–1209. https://doi.org/10.1016/S0895-7177(03)80019-5 doi: 10.1016/S0895-7177(03)80019-5
[11]	A. Friedman, F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol., 38 (1999), 262–284. https://doi.org/10.1007/s002850050149 doi: 10.1007/s002850050149
[12]	A. Friedman, K. Lam, Analysis of a free-boundary tumor model with angiogenesis, J. Differ. Equations, 259 (2015), 7636–7661. https://doi.org/10.1016/j.jde.2015.08.032 doi: 10.1016/j.jde.2015.08.032
[13]	D. Guo, V. Lakshmikantham, Nonlinear problems in abstract cones, New York: Academic Press, 2014.
[14]	G. Huang, A. P. Liu, U. Foryś, Global stability analysis of some nonlinear delay differential equations in population dynamics, J. Nonlinear Sci., 26 (2016), 27–41. https://doi.org/10.1007/s00332-015-9267-4 doi: 10.1007/s00332-015-9267-4
[15]	M. J. Piotrowska, Hopf bifurcation in a solid avascular tumor growth model with two discrete delays, Math. Comput. Model., 47 (2008), 597–603. https://doi.org/10.1016/j.mcm.2007.02.030 doi: 10.1016/j.mcm.2007.02.030
[16]	M. J. Piotrowska, A remark on the ODE with two discrete delays, J. Math. Anal. Appl., 329 (2007), 664–676. https://doi.org/10.1016/j.jmaa.2006.06.078 doi: 10.1016/j.jmaa.2006.06.078
[17]	F. A. Rihan, Delay differential equations and applications to biology, Singapore: Springer, 2021. https://doi.org/10.1007/978-981-16-0626-7
[18]	R. R. Sarkar, S. Banerjee, A time delay model for control of malignant tumor growth, National Conference on Nonlinear Systems and Dynamics, 1 (2006), 1–5.
[19]	D. R. Smart, Fixed points theorems, Cambridge: Cambridge University Press, 1980.
[20]	S. H. Xu, Analysis of a delayed free boundary problem for tumor growth, Discrete Cont. Dyn. B., 18 (2011), 293–308. https://doi.org/10.3934/dcdsb.2011.15.293 doi: 10.3934/dcdsb.2011.15.293
[21]	S. H. Xu, Analysis of a free boundary problem for tumor growth in a periodic external environment, Bound. Value Probl., 2015 (2015), 1–12. https://doi.org/10.1186/s13661-015-0399-0 doi: 10.1186/s13661-015-0399-0
[22]	S. H. Xu, M. Bai, X. Q. Zhao, Analysis of a solid avascular tumor growth model with time delays in proliferation process, J. Math. Anal. Appl., 391 (2012), 38–47. https://doi.org/10.1016/j.jmaa.2012.02.034 doi: 10.1016/j.jmaa.2012.02.034
[23]	S. H. Xu, Analysis of a free boundary problem for tumor growth with angiogenesis and time delays in proliferation, Nonlinear Anal. Real World Appl., 51 (2020), 103005. https://doi.org/10.1016/j.nonrwa.2019.103005 doi: 10.1016/j.nonrwa.2019.103005
[24]	S. H. Xu, Hopf bifurcation of a free boundary problem modeling tumor growth with two time delays, Chao Soliton. Fract., 41 (2009), 2491–2494. https://doi.org/10.1016/j.chaos.2008.09.029 doi: 10.1016/j.chaos.2008.09.029
[25]	Z. J. Ye, S. H. Xu, X. M. Wei, Analysis of a free boundary problem for vascularized tumor growth with a time delay in the process of tumor regulating apoptosis, AIMS Math., 7 (2022), 19440–19457. https://doi.org/10.3934/math.20221067 doi: 10.3934/math.20221067
[26]	H. H. Zhou, Z. J. Wang, D. M. Yuan, H. J. Song, Hopf bifurcation of a free boundary problem modeling tumor growth with angiogenesis and two time delays, Chaos Soliton. Fract., 153 (2021), 111578. https://doi.org/10.1016/j.chaos.2021.111578 doi: 10.1016/j.chaos.2021.111578

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)