Bifurcation from stability to instability for a free boundary tumor model with angiogenesis

Yaodan Huang; Zhengce Zhang; Bei Hu

doi:10.3934/dcds.2019105

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May 2019, 39(5): 2473-2510. doi: 10.3934/dcds.2019105

Bifurcation from stability to instability for a free boundary tumor model with angiogenesis

Yaodan Huang ^1, , Zhengce Zhang ^1,, and Bei Hu ^2,

1.	School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China
2.	Department of Applied Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA

^* Corresponding author: Zhengce Zhang

Received January 2018 Revised October 2018 Published January 2019

In this paper we consider a free boundary tumor model with angiogenesis. The model consists of a reaction-diffusion equation describing the concentration of nutrients $ \sigma $ and an elliptic equation describing the distribution of the internal pressure $ p $. The vasculature supplies nutrients to the tumor, so that $ \frac{\partial\sigma}{\partial \mathbf{n}}+\beta(\sigma-\bar{\sigma}) = 0 $ holds on the boundary, where a positive constant $ \beta $ is the rate of nutrient supply to the tumor and $ \bar{\sigma} $ is the nutrient concentration outside the tumor. The tumor cells proliferate at a rate $ \mu $. If $ 0<\widetilde{\sigma}<\overline{\sigma} $, where $ \widetilde{\sigma} $ is a threshold concentration for proliferating, then there exists a unique radially symmetric stationary solution $ (\sigma_S(r), p_S(r), R_S) $. In this paper, we found a function $ \mu^\ast = \mu^\ast(R_S) $ such that if $ \mu<\mu^\ast $ then the radially symmetric stationary solution is linearly stable with respect to non-radially symmetric perturbations, whereas if $ \mu>\mu^\ast $ then the radially symmetric stationary solution is linearly unstable.

Keywords: Free boundary problems, tumor growth, angiogenesis, stationary solution, stability, instability.

Mathematics Subject Classification: Primary: 35B35, 35R35, 35B40, 35C10, 92C37.

Citation: Yaodan Huang, Zhengce Zhang, Bei Hu. Bifurcation from stability to instability for a free boundary tumor model with angiogenesis. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2473-2510. doi: 10.3934/dcds.2019105

References:

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[2]	J. A. Adam, A mathematical model of tumor growth: Ⅲ. Comparison with experiment, Math. Biosci., 86 (1987), 213-227. doi: 10.1016/0025-5564(87)90011-3.
[3]	J. A. Adam and N. Bellomo, A Survey of Models for Tumor-Immune System Dynamics, 1^nd edition, Birkhäuser Basel, 1997. doi: 10.1007/978-0-8176-8119-7.
[4]	B. V. Bazaliy and A. Friedman, Global existence and asymptotic stability for an elliptic-parabolic free boundary problem: an application to a model of tumor growth, Indiana Univ. Math. J., 52 (2003), 1265-1304. doi: 10.1512/iumj.2003.52.2317.
[5]	N. F. Britton and M. A. Chaplain, A qualitative analysis of some models of tissue growth, Math. Biosci., 113 (1993), 77-89. doi: 10.1016/0025-5564(93)90009-Y.
[6]	H. M. Byrne, The importance of intercellular adhesion in the development of carcinomas, IMA J. Math. Appl. Med. Biol., 14 (1997), 305-323. doi: 10.1093/imammb/14.4.305.
[7]	H. M. Byrne and M. A. J. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151-181. doi: 10.1016/0025-5564(94)00117-3.
[8]	H. M. Byrne and M. A. J. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci., 135 (1996), 187-216. doi: 10.1016/0025-5564(96)00023-5.
[9]	H. M. Byrne and M. A. J. Chaplain, Modelling the role of cell-cell adhesion in the growth and development of carcinomas, Math. Comput. Modelling, 24 (1996), 1-17. doi: 10.1016/S0895-7177(96)00174-4.
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[12]	V. Cristini, J. Lowengrub and Q. Nie, Nonlinear simulation of tumor growth, J. Math. Biol., 46 (2003), 191-224. doi: 10.1007/s00285-002-0174-6.
[13]	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. doi: 10.1007/s002850100130.
[14]	S. Cui, Asymptotic stability of the stationary solution for a parabolic-hyperbolic free boundary problem modeling tumor growth, SIAM J. Math. Anal., 45 (2013), 2870-2893. doi: 10.1137/130906271.
[15]	S. Cui and J. Escher, Bifurcation analysis of an elliptic free boundary problem modelling the growth of avascular tumors, SIAM J. Math. Anal., 39 (2007), 210-235. doi: 10.1137/060657509.
[16]	S. Cui and J. Escher, Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth, Comm. Partial Differential Equations, 33 (2008), 636-655. doi: 10.1080/03605300701743848.
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[18]	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. doi: 10.1016/S0025-5564(99)00063-2.
[19]	S. Cui and A. Friedman, Analysis of a mathematical model of the growth of necrotic tumors, J. Math. Anal. Appl., 255 (2001), 636-677. doi: 10.1006/jmaa.2000.7306.
[20]	S. Cui and A. Friedman, A free boundary problem for a singular system of differential equations: an application to a model of tumor growth, Trans. Amer. Math. Soc., 355 (2003), 3537-3590. doi: 10.1090/S0002-9947-03-03137-4.
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[22]	J. Escher and A.-V. Matioc, Bifurcation analysis for a free boundary problem modeling tumor growth, Arch. Math., 97 (2011), 79-90. doi: 10.1007/s00013-011-0276-8.
[23]	J. Escher and A.-V. Matioc, Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 573-596. doi: 10.3934/dcdsb.2011.15.573.
[24]	J. Escher and A.-V. Matioc, Analysis of a two-phase model describing the growth of solid tumors, European J. Appl. Math., 24 (2013), 25-48. doi: 10.1017/S0956792512000290.
[25]	M. A. Fontelos and A. Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptotic Analysis, 35 (2003), 187-206.
[26]	A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem arising in a tumor model, Arch. Ration. Mech. Anal., 180 (2006), 293-330. doi: 10.1007/s00205-005-0408-z.
[27]	A. Friedman and B. Hu, Asymptotic stability for a free boundary problem arising in a tumor model, J. Differential Equations, 227 (2006), 598-639. doi: 10.1016/j.jde.2005.09.008.
[28]	A. Friedman and B. Hu, Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal., 39 (2007), 174-194. doi: 10.1137/060656292.
[29]	A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation, J. Math. Anal. Appl., 327 (2007), 643-664. doi: 10.1016/j.jmaa.2006.04.034.
[30]	A. Friedman and B. Hu, Stability and instability of Liapunov-Schmidt and Holf bifurcation for a free boundary problem arising in a tumor model, Trans. Amer. Math. Soc., 360 (2008), 5291-5342. doi: 10.1090/S0002-9947-08-04468-1.
[31]	A. Friedman and K.-Y. Lam, Analysis of a free-boundary tumor model with angiogenesis, J. Differential Equations, 259 (2015), 7636-7661. doi: 10.1016/j.jde.2015.08.032.
[32]	A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol., 38 (1999), 262-284. doi: 10.1007/s002850050149.
[33]	A. Friedman and F. Reitich, Nonlinear stability of a quasi-static Stefan problem with surface tension: A continuation approach, Ann. Sc. Norm. Super Pisa CI. Sci., 30 (2001), 341-403.
[34]	A. Friedman and F. Reitich, Symmetry-breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth, Trans. Amer. Math. Soc., 353 (2001), 1587-1634. doi: 10.1090/S0002-9947-00-02715-X.
[35]	H. P. Greenspan, Models for the growth of a solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340. doi: 10.1002/sapm1972514317.
[36]	H. P. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theoret. Biol., 56 (1976), 229-242. doi: 10.1016/S0022-5193(76)80054-9.
[37]	W. Hao, J. D. Hauenstein, B. Hu, Y. Liu, A. J. Sommese and Y. Zhang, Continuation along bifurcation branches for a tumor model with a necrotic core, J. Sci. Comput., 53 (2012), 395-413. doi: 10.1007/s10915-012-9575-x.
[38]	W. Hao, J. D. Hauenstein, B. Hu, T. McCoy and A. J. Sommese, Computing steady-state solutions for a free boundary problem modeling tumor growth by Stokes equation, J. Comput. Appl. Math., 237 (2013), 326-334. doi: 10.1016/j.cam.2012.06.001.
[39]	W. Hao, J. D. Hauenstein, B. Hu and A. J. Sommese, A three-dimensional steady-state tumor system, Appl. Math. Comput., 218 (2011), 2661-2669. doi: 10.1016/j.amc.2011.08.006.
[40]	Y. D. Huang, Z. C. Zhang and B. Hu, Bifurcation for a free-boundary tumor model with angiogenesis, Nonlinear. Anal. Real World Appl., 35 (2017), 483-502. doi: 10.1016/j.nonrwa.2016.12.003.
[41]	J. S. Lowengrub, H. B. Frieboes, F. Jin, Y.-L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modelling of cancer: Bridging the gap between cells and tumours, Nonlinearity, 23 (2010), 1-91. doi: 10.1088/0951-7715/23/1/R01.
[42]	Z. J. Wang, Bifurcation for a free boundary problem modeling tumor growth with inhibitors, Nonlinear Anal. Real World Appl., 19 (2014), 45-53. doi: 10.1016/j.nonrwa.2014.03.001.
[43]	J. P. Ward and J. R. King, Mathematical modelling of avascular-tumour growth, IMA J. Math. Appl. Med. Biol., 14 (1997), 39-69. doi: 10.1093/imammb/14.1.39.
[44]	J. P. Ward and J. R. King, Mathematical modelling of avascular-tumour growth Ⅱ: Modelling growth saturation, IMA J. Math. Appl. Med. Biol., 16 (1999), 171-211. doi: 10.1093/imammb/16.2.171.
[45]	J. Wu and S. Cui, Asymptotic behaviour of solutions of a free boundary problem modelling the growth of tumours in the presence of inhibitors, Nonlinearity, 20 (2007), 2389-2408. doi: 10.1088/0951-7715/20/10/007.
[46]	J. Wu and S. Cui, Asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with Stokes equations, Discrete Contin. Dyn. Syst., 24 (2009), 625-651. doi: 10.3934/dcds.2009.24.625.
[47]	J. Wu and S. Cui, Asymptotic stability of stationary solutions of a free boundary problem modeling the growth of tumors with fluid tissues, SIAM J. Math. Anal., 41 (2009), 391-414. doi: 10.1137/080726550.
[48]	J. Wu and S. Cui, Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth, Discrete Contin. Dyn. Syst., 26 (2010), 737-765. doi: 10.3934/dcds.2010.26.737.
[49]	F. Zhou, J. Escher and S. Cui, Well-posedness and stability of a free boundary problem modeling the growth of multi-layer tumors, J. Differential Equations, 244 (2008), 2909-2933. doi: 10.1016/j.jde.2008.02.038.
[50]	F. Zhou and J. Wu, Stability and bifurcation analysis of a free boundary problem modelling multi-layer tumours with Gibbs-Thomson relation, European J. Appl. Math., 26 (2015), 401-425. doi: 10.1017/S0956792515000108.
[51]	F. Zhou, J. Wu and S. Cui, Existence and asymptotic behavior of solutions to a moving boundary problem modeling the growth of multi-layer tumors, Comm. Pure Appl. Anal., 8 (2009), 1669-1688. doi: 10.3934/cpaa.2009.8.1669.

show all references

References:

[1]	J. A. Adam, A simplified mathematical model of tumor growth, Math. Biosci., 81 (1986), 229-244. doi: 10.1016/0025-5564(86)90119-7.
[2]	J. A. Adam, A mathematical model of tumor growth: Ⅲ. Comparison with experiment, Math. Biosci., 86 (1987), 213-227. doi: 10.1016/0025-5564(87)90011-3.
[3]	J. A. Adam and N. Bellomo, A Survey of Models for Tumor-Immune System Dynamics, 1^nd edition, Birkhäuser Basel, 1997. doi: 10.1007/978-0-8176-8119-7.
[4]	B. V. Bazaliy and A. Friedman, Global existence and asymptotic stability for an elliptic-parabolic free boundary problem: an application to a model of tumor growth, Indiana Univ. Math. J., 52 (2003), 1265-1304. doi: 10.1512/iumj.2003.52.2317.
[5]	N. F. Britton and M. A. Chaplain, A qualitative analysis of some models of tissue growth, Math. Biosci., 113 (1993), 77-89. doi: 10.1016/0025-5564(93)90009-Y.
[6]	H. M. Byrne, The importance of intercellular adhesion in the development of carcinomas, IMA J. Math. Appl. Med. Biol., 14 (1997), 305-323. doi: 10.1093/imammb/14.4.305.
[7]	H. M. Byrne and M. A. J. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151-181. doi: 10.1016/0025-5564(94)00117-3.
[8]	H. M. Byrne and M. A. J. Chaplain, Growth of necrotic tumors in the presence and absence of inhibitors, Math. Biosci., 135 (1996), 187-216. doi: 10.1016/0025-5564(96)00023-5.
[9]	H. M. Byrne and M. A. J. Chaplain, Modelling the role of cell-cell adhesion in the growth and development of carcinomas, Math. Comput. Modelling, 24 (1996), 1-17. doi: 10.1016/S0895-7177(96)00174-4.
[10]	H. M. Byrne and M. A. J. Chaplain, Free boundary value problems associated with the growth and development of multicellular spheroids, European J. Appl. Math., 8 (1997), 639-658. doi: 10.1017/S0956792597003264.
[11]	X. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: an application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986. doi: 10.1137/S0036141002418388.
[12]	V. Cristini, J. Lowengrub and Q. Nie, Nonlinear simulation of tumor growth, J. Math. Biol., 46 (2003), 191-224. doi: 10.1007/s00285-002-0174-6.
[13]	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. doi: 10.1007/s002850100130.
[14]	S. Cui, Asymptotic stability of the stationary solution for a parabolic-hyperbolic free boundary problem modeling tumor growth, SIAM J. Math. Anal., 45 (2013), 2870-2893. doi: 10.1137/130906271.
[15]	S. Cui and J. Escher, Bifurcation analysis of an elliptic free boundary problem modelling the growth of avascular tumors, SIAM J. Math. Anal., 39 (2007), 210-235. doi: 10.1137/060657509.
[16]	S. Cui and J. Escher, Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth, Comm. Partial Differential Equations, 33 (2008), 636-655. doi: 10.1080/03605300701743848.
[17]	S. Cui and J. Escher, Well-posedness and stability of a multi-dimensional tumor growth model, Arch. Ration. Mech. Anal., 191 (2009), 173-193. doi: 10.1007/s00205-008-0158-9.
[18]	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. doi: 10.1016/S0025-5564(99)00063-2.
[19]	S. Cui and A. Friedman, Analysis of a mathematical model of the growth of necrotic tumors, J. Math. Anal. Appl., 255 (2001), 636-677. doi: 10.1006/jmaa.2000.7306.
[20]	S. Cui and A. Friedman, A free boundary problem for a singular system of differential equations: an application to a model of tumor growth, Trans. Amer. Math. Soc., 355 (2003), 3537-3590. doi: 10.1090/S0002-9947-03-03137-4.
[21]	J. Escher and A.-V. Matioc, Radially symmetric growth of nonnecrotic tumors, Nonlinear Differ. Equ. Appl., 17 (2010), 1-20. doi: 10.1007/s00030-009-0037-6.
[22]	J. Escher and A.-V. Matioc, Bifurcation analysis for a free boundary problem modeling tumor growth, Arch. Math., 97 (2011), 79-90. doi: 10.1007/s00013-011-0276-8.
[23]	J. Escher and A.-V. Matioc, Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 573-596. doi: 10.3934/dcdsb.2011.15.573.
[24]	J. Escher and A.-V. Matioc, Analysis of a two-phase model describing the growth of solid tumors, European J. Appl. Math., 24 (2013), 25-48. doi: 10.1017/S0956792512000290.
[25]	M. A. Fontelos and A. Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptotic Analysis, 35 (2003), 187-206.
[26]	A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem arising in a tumor model, Arch. Ration. Mech. Anal., 180 (2006), 293-330. doi: 10.1007/s00205-005-0408-z.
[27]	A. Friedman and B. Hu, Asymptotic stability for a free boundary problem arising in a tumor model, J. Differential Equations, 227 (2006), 598-639. doi: 10.1016/j.jde.2005.09.008.
[28]	A. Friedman and B. Hu, Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal., 39 (2007), 174-194. doi: 10.1137/060656292.
[29]	A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation, J. Math. Anal. Appl., 327 (2007), 643-664. doi: 10.1016/j.jmaa.2006.04.034.
[30]	A. Friedman and B. Hu, Stability and instability of Liapunov-Schmidt and Holf bifurcation for a free boundary problem arising in a tumor model, Trans. Amer. Math. Soc., 360 (2008), 5291-5342. doi: 10.1090/S0002-9947-08-04468-1.
[31]	A. Friedman and K.-Y. Lam, Analysis of a free-boundary tumor model with angiogenesis, J. Differential Equations, 259 (2015), 7636-7661. doi: 10.1016/j.jde.2015.08.032.
[32]	A. Friedman and F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol., 38 (1999), 262-284. doi: 10.1007/s002850050149.
[33]	A. Friedman and F. Reitich, Nonlinear stability of a quasi-static Stefan problem with surface tension: A continuation approach, Ann. Sc. Norm. Super Pisa CI. Sci., 30 (2001), 341-403.
[34]	A. Friedman and F. Reitich, Symmetry-breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth, Trans. Amer. Math. Soc., 353 (2001), 1587-1634. doi: 10.1090/S0002-9947-00-02715-X.
[35]	H. P. Greenspan, Models for the growth of a solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340. doi: 10.1002/sapm1972514317.
[36]	H. P. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theoret. Biol., 56 (1976), 229-242. doi: 10.1016/S0022-5193(76)80054-9.
[37]	W. Hao, J. D. Hauenstein, B. Hu, Y. Liu, A. J. Sommese and Y. Zhang, Continuation along bifurcation branches for a tumor model with a necrotic core, J. Sci. Comput., 53 (2012), 395-413. doi: 10.1007/s10915-012-9575-x.
[38]	W. Hao, J. D. Hauenstein, B. Hu, T. McCoy and A. J. Sommese, Computing steady-state solutions for a free boundary problem modeling tumor growth by Stokes equation, J. Comput. Appl. Math., 237 (2013), 326-334. doi: 10.1016/j.cam.2012.06.001.
[39]	W. Hao, J. D. Hauenstein, B. Hu and A. J. Sommese, A three-dimensional steady-state tumor system, Appl. Math. Comput., 218 (2011), 2661-2669. doi: 10.1016/j.amc.2011.08.006.
[40]	Y. D. Huang, Z. C. Zhang and B. Hu, Bifurcation for a free-boundary tumor model with angiogenesis, Nonlinear. Anal. Real World Appl., 35 (2017), 483-502. doi: 10.1016/j.nonrwa.2016.12.003.
[41]	J. S. Lowengrub, H. B. Frieboes, F. Jin, Y.-L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modelling of cancer: Bridging the gap between cells and tumours, Nonlinearity, 23 (2010), 1-91. doi: 10.1088/0951-7715/23/1/R01.
[42]	Z. J. Wang, Bifurcation for a free boundary problem modeling tumor growth with inhibitors, Nonlinear Anal. Real World Appl., 19 (2014), 45-53. doi: 10.1016/j.nonrwa.2014.03.001.
[43]	J. P. Ward and J. R. King, Mathematical modelling of avascular-tumour growth, IMA J. Math. Appl. Med. Biol., 14 (1997), 39-69. doi: 10.1093/imammb/14.1.39.
[44]	J. P. Ward and J. R. King, Mathematical modelling of avascular-tumour growth Ⅱ: Modelling growth saturation, IMA J. Math. Appl. Med. Biol., 16 (1999), 171-211. doi: 10.1093/imammb/16.2.171.
[45]	J. Wu and S. Cui, Asymptotic behaviour of solutions of a free boundary problem modelling the growth of tumours in the presence of inhibitors, Nonlinearity, 20 (2007), 2389-2408. doi: 10.1088/0951-7715/20/10/007.
[46]	J. Wu and S. Cui, Asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with Stokes equations, Discrete Contin. Dyn. Syst., 24 (2009), 625-651. doi: 10.3934/dcds.2009.24.625.
[47]	J. Wu and S. Cui, Asymptotic stability of stationary solutions of a free boundary problem modeling the growth of tumors with fluid tissues, SIAM J. Math. Anal., 41 (2009), 391-414. doi: 10.1137/080726550.
[48]	J. Wu and S. Cui, Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth, Discrete Contin. Dyn. Syst., 26 (2010), 737-765. doi: 10.3934/dcds.2010.26.737.
[49]	F. Zhou, J. Escher and S. Cui, Well-posedness and stability of a free boundary problem modeling the growth of multi-layer tumors, J. Differential Equations, 244 (2008), 2909-2933. doi: 10.1016/j.jde.2008.02.038.
[50]	F. Zhou and J. Wu, Stability and bifurcation analysis of a free boundary problem modelling multi-layer tumours with Gibbs-Thomson relation, European J. Appl. Math., 26 (2015), 401-425. doi: 10.1017/S0956792515000108.
[51]	F. Zhou, J. Wu and S. Cui, Existence and asymptotic behavior of solutions to a moving boundary problem modeling the growth of multi-layer tumors, Comm. Pure Appl. Anal., 8 (2009), 1669-1688. doi: 10.3934/cpaa.2009.8.1669.

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2021 Impact Factor: 1.588

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American Institute of Mathematical Sciences

Bifurcation from stability to instability for a free boundary tumor model with angiogenesis

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American Institute of Mathematical Sciences

Bifurcation from stability to instability for a free boundary tumor model with angiogenesis

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