
Mathematical Biosciences and Engineering, 2019, 16(6): 74337446. doi: 10.3934/mbe.2019372.
Research article
Export file:
Format
 RIS(for EndNote,Reference Manager,ProCite)
 BibTex
 Text
Content
 Citation Only
 Citation and Abstract
Qualitative analysis of a timedelayed free boundary problem for tumor growth with angiogenesis and GibbsThomson relation
1 School of Mathematics and Statistics, Zhaoqing University, Zhaoqing, Guangdong 526061, China
2 Department of Mathematics, Soochow University, Suzhou, Jiangsu 215006, China
Received: , Accepted: , Published:
Keywords: tumor growth; time delay; free boundary problem; GibbsThomson relation; stability
Citation: Shihe Xu, Junde Wu. Qualitative analysis of a timedelayed free boundary problem for tumor growth with angiogenesis and GibbsThomson relation. Mathematical Biosciences and Engineering, 2019, 16(6): 74337446. doi: 10.3934/mbe.2019372
References:
 1. H. Byrne, The effect of time delays on the dynamics of avascular tumor growth, Math. Biosci., 144 (1997), 83–117.
 2. H. Byrne and M. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151–181.
 3. H. Greenspan, Models for the growth of solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317–340.
 4. M. J. Piotrowska, Hopf bifurcation in a solid asascular tumor growth model with two discrete delays, Math. Comput. Modeling, 47 (2008), 597–603.
 5. K. Thompson and H. Byrne, Modelling the internalisation of labelled cells in tumor spheroids, Bull. Math. Biol., 61 (1999), 601–623.
 6. J. Ward and J. King, Mathematical modelling of avasculartumor growth II: Modelling growth saturation, IMA J. Math. Appl. Med. Biol., 15 (1998), 1–42.
 7. S. Cui and A. Friedman, Analysis of a mathematical model of the effect of inhibitors on the growth of tumors, Math. Biosci., 164 (2000), 103–137.
 8. S. Cui, Analysis of a free boundary problem modeling tumor growth, Acta. Math. Sinica., 21 (2005), 1071–1082.
 9. S. Cui and S. Xu, Analysis of mathematical models for the growth of tumors with time delays in cell proliferation, J. Math. Anal. Appl., 336 (2007), 523–541.
 10. A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol., 38 (1999), 262–284.
 11. J. Wu and F. Zhou, Asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with Gibbs–Thomson relation, J. Differential Eqs., 262 (2017),4907–4930.
 12. A.Friedman and K. Lam, Analysis of a freeboundary tumor model with angiogenesis, J. Differential Eqs., 259 (2015), 7636–7661.
 13. U. Fory´ s and M. Bodnar, Time delays in proliferation process for solid avascular tumour, Math. Comput. Modelling, 37 (2003), 1201–1209.
 14. S. Xu, M. Bai and X. Zhao, Analysis of a solid avascular tumor growth model with time delays in proliferation process, J. Math. Anal. Appl., 391 (2012), 38–47.
 15. S. Xu, M. Bai and F. Zhang, Analysis of a timedelayed mathematical model for tumour growth with an almost periodic supply of external nutrients, J. Biol. Dynam., 11 (2017), 504–520.
 16. S. Cui, Analysis of a mathematical model for the growth of tumors under the action of external inhibitors, J. Math. Biol. 44 (2002), 395–426.
 17. H. Byrne and M. Chaplain, Modelling the role of cellcell adhesion in the growth and development of carcinomas, Math. Comput. Modelling, 24 (1996), 1–17.
 18. J. Wu, Stationary solutions of a free boundary problem modeling the growth of tumors with Gibbs–Thomson relation, J. Differential Eqs., 260 (2016), 5875–5893.
 19. J. Wu, Analysis of a mathematical model for tumor growth with Gibbs–Thomson relation, J. Math. Anal. Appl., 450 (2017), 532–543.
 20. S. Xu, Analysis of tumor growth under direct effect of inhibitors with time delays in proliferation, Nonlinear Anal., 11 (2010), 401–406.
 21. S. Xu, M. Bai and F. Zhang, Analysis of a free boundary problem for tumor growth with GibbsThomson relation and time delays, Discrete Contin. Dyn. Syst. B., 23 (2018), 3535–3551.
 22. J. Hale, Theory of Functional Differential Equations, SpringerVerlag, New York, 1977.
 23. M. Bodnar, The nonnegativity of solutions of delay differential equations, Appl. Math. Lett., 13 (2000), 91–95.
This article has been cited by:
 1. Hasitha N. Weerasinghe, Pamela M. Burrage, Kevin Burrage, Dan V. Nicolau, Mathematical Models of Cancer Cell Plasticity, Journal of Oncology, 2019, 2019, 1, 10.1155/2019/2403483
Reader Comments
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)
Associated material
Metrics
Other articles by authors
Related pages
Tools
your name: * your email: *