December 2018, 15(6): 1345-1385. doi: 10.3934/mbe.2018062

Early and late stage profiles for a chemotaxis model with density-dependent jump probability

1. 

School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, China

2. 

School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510641, China

3. 

Department of Mathematics, Champlain College Saint-Lambert, Quebec, J4P 3P2, Canada

4. 

Department of Mathematics and Statistics, McGill University, Montreal, Quebec, H3A 2K6, Canada

* Corresponding author: jism@scut.edu.cn

Received  October 19, 2017 Accepted  July 24, 2018 Published  September 2018

In this paper, we derive a chemotaxis model with degenerate diffusion and density-dependent chemotactic sensitivity, and we provide a more realistic description of cell migration process for its early and late stages. Different from the existing studies focusing on the case of non-degenerate diffusion, this model with degenerate diffusion causes us some essential difficulty on the boundedness estimates and the propagation behavior of its compact support. In the presence of logistic damping, for the early stage before tumour cells spread to the whole domain, we first estimate the expanding speed of tumour region as $O(t^{β})$ for $ 0 < β < \frac{1}{2}$. Then, for the late stage of cell migration, we further prove that the asymptotic profile of the original system is just its corresponding steady state. The global convergence of the original weak solution to the steady state with exponential rate $O(e^{-ct})$ for some $c>0$ is also obtained.

Citation: Tianyuan Xu, Shanming Ji, Chunhua Jin, Ming Mei, Jingxue Yin. Early and late stage profiles for a chemotaxis model with density-dependent jump probability. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1345-1385. doi: 10.3934/mbe.2018062
References:
[1]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X.

[2]

A. A. Blumberg, Logistic growth rate functions, J. Theor. Biol., 21 (1968), 42-44.

[3]

H. M. Byrne, Dissecting cancer through mathematics: from the cell to the animal model, Nature Reviews Cancer, 10 (2010), 221-230.

[4]

M. A. Chaplain and A. R. Anderson, Mathematical modelling of tissue invasion, in Cancer Model. Simul., Chapman and Hall/CRC, London, (2003), 269-297.

[5]

Y. H. Du, F. Quirós and M. L. Zhou, Logarithmic corrections in Fisher-KPP type Porous Medium Equations, preprint, arXiv: 1806.02022.

[6]

P. Friedl and K. Wolf, Tumour-cell invasion and migration: Diversity and escape mechanisms, Nature Reviews Cancer, 3 (2003), 362-374.

[7]

K. FujieA. ItoM. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Disc. Cont. Dyn. Syst., 36 (2016), 151-169. doi: 10.3934/dcds.2016.36.151.

[8]

R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753.

[9]

W. S. C. Gurney and R. M. Nisbet, The regulation of inhomogeneous population, J. Theors. Biol., 52 (1975), 441-457.

[10]

M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49. doi: 10.1016/0025-5564(77)90062-1.

[11]

T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301. doi: 10.1006/aama.2001.0721.

[12]

T. Hillen and K. J. Painter, A user's guide to pde models in a chemotaxis, J. Math. Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

[13]

C. H. Jin, Boundedness and global solvability to a chemotaxis model with nonlinear diffusion, J. Differential Equations, 263 (2017), 5759-5772. doi: 10.1016/j.jde.2017.06.034.

[14]

C. H. Jin, Boundedness and global solvability to a chemotaxis-haptotaixs model with slow and fast diffusion, Disc. Cont. Dyn. Syst., 23 (2018), 1675-1688. doi: 10.3934/dcdsb.2018069.

[15]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biology, 30 (1971), 225-234.

[16]

R. D. Leek, The role of tumour associated macrophages in breast cancer angiogenesis, Ph.D thesis, Oxford Brookes University, Oxford, 1999.

[17]

D. LiC. Mu and P. Zheng, Boundedness and large time behavior in a quasilinear chemotaxis model for tumor invasion, Math. Models Methods Appl. Sci., 28 (2018), 1413-1451. doi: 10.1142/S0218202518500380.

[18]

J. Liu and Y. F. Wang, A quasilinear chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, Math. Methods Appl. Sci., 40 (2017), 2107-2121. doi: 10.1002/mma.4126.

[19]

Y. Lou and M. Winkler, Global existence and uniform boundedness of smooth solutions to a cross-diffusion system with equal diffusion rates, Comm. Part. Diff. Equ., 40 (2015), 1905-1941. doi: 10.1080/03605302.2015.1052882.

[20]

P. LuV. Weaver and Z. Werb, The extracellular matrix: a dynamic niche in cancer progression, J. Cell Biol., 196 (2012), 395-406.

[21]

H. McAneney and S. F. C. O'Rourke, Investigation of various growth mechanisms of solid tumour growth within the linear quadratic model for radiotherapy, Phys. Med. Biol., 52 (2007), 1039-1054.

[22]

M. MeiH. Y. Peng and Z. A. Wang, Asymptotic profile of a parabolic-hyperbolic system with boundary effect arising from tumor angiogenesis, J. Differential Equations, 259 (2015), 5168-5191. doi: 10.1016/j.jde.2015.06.022.

[23]

Y. Mimura, The variational formulation of the fully parabolic Keller-Segel system with degenerate diffusion, J. Differential Equations, 263 (2017), 1477-1521. doi: 10.1016/j.jde.2017.03.020.

[24]

J. D. Murry, Mathematical Biology I: An Introduction, Springer, New York, USA, 2002.

[25]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Springer Science Business Media, 2013. doi: 10.1007/978-1-4757-4978-6.

[26]

M. E. Orme and M. A. J. Chaplain, A mathematical model of vascular tumour growth and invasion, Math. Comput. Modelling, 23 (1996), 43-60.

[27]

M. R. OwenH. M. Byrne and C. E. Lewis, Mathematical modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumour sites, J. Theor. Biol., 226 (2004), 377-391. doi: 10.1016/j.jtbi.2003.09.004.

[28]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canadian Appl. Math. Quart., 10 (2002), 501-543.

[29]

K. J. Painter and J. A. Sherratt, Modelling the movement of interacting cell populations, J. Theor. Biol., 225 (2003), 327-339. doi: 10.1016/S0022-5193(03)00258-3.

[30]

B. G. SengersC. P. Please and R. O. C. Oreffo, Experimental characterization and computational modelling of two-dimensional cell spreading for skeletal regeneration, J. R. Soc. Interface, 4 (2007), 1107-1117.

[31]

J. A. Sherratt, On the form of smooth-front travelling waves in a reaction-diffusion equation with degenerate nonlinear diffusion, Math. Model. Nat. Phenom, 5 (2010), 64-79. doi: 10.1051/mmnp/20105505.

[32]

J. A. Sherratt and M. A. J. Chaplain, A new mathematical model for avascular tumour growth, J. Math. Biol., 43 (2001), 291-312. doi: 10.1007/s002850100088.

[33]

J. A. Sherratt and J. D. Murray, Models of epidermal wound healing, Proc. R. Soc. London B, 241 (1990), 29-35.

[34]

M. J. Simpson, R. E. Baker and S. W. McCue, Models of collective cell spreading with variable cell aspect ratio: A motivation for degenerate diffusion models, Phys. Rev. E, 83 (2011), 021901.

[35]

M. J. SimpsonD. C. ZhangM. MarianiK. A. Landman and D. F. Newgreen, Cell proliferation drives neural crest cell invasion of the intestine, Dev. Biol., 302 (2007), 553-568.

[36]

A. Stevens and H. G. Othmer, Aggregation, blowup, and collapse: The abc's of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (2001), 1044-1081. doi: 10.1137/S0036139995288976.

[37]

Z. SzymańskaC. M. RodrigoM. Lachowicz and M. A. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models Methods Appl. Sci., 19 (2009), 257-281. doi: 10.1142/S0218202509003425.

[38]

Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-705. doi: 10.1137/100802943.

[39]

Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 1901-1914. doi: 10.3934/dcds.2012.32.1901.

[40]

Y. Tao and M. Winkler, Dominance of chemotaxis in a chemotaxis-haptotaxis model, Nonlinearity, 27 (2014), 1225-1239. doi: 10.1088/0951-7715/27/6/1225.

[41]

M. E. TurnerB. A. Blumenstein and J. L. Sebaugh, A generalization of the logistic law of growth, Biometrics, 25 (1969), 577-580.

[42]

H. A. S. Van den Brenk, Studies in restorative growth processes in mammalian wound healing, Br. J. Surg., 43 (1956), 525-550.

[43]

J. L. Vàzquez, The Porous Medium Equation: Mathematical Theory, Oxford Univ. Press, Oxford, 2007.

[44]

L. Von Bertalanffy, Quantitative laws in metabolism and growth, Q. Rev. Biol., 32 (1957), 217-231.

[45]

Y. F. Wang, Boundedness in a multi-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Appl. Math. Lett., 59 (2016), 122-126. doi: 10.1016/j.aml.2016.03.019.

[46]

Y. F. Wang, Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Differential Equations, 260 (2016), 1975-1989. doi: 10.1016/j.jde.2015.09.051.

[47]

Y. F. Wang and Y. Y. Ke, Large time behavior of solution to a fully parabolic chemotaxis-haptotaxis model in higher demensions, J. Differential Equations, 260 (2016), 6960-6988. doi: 10.1016/j.jde.2016.01.017.

[48]

Z. A. WangM. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect, Nonlinearity, 24 (2011), 3279-3297. doi: 10.1088/0951-7715/24/12/001.

[49]

Z. A. WangZ. Y. Xiang and P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 2225-2258. doi: 10.1016/j.jde.2015.09.063.

[50]

Z. A. WangM. Winkler and D. Wrzosek, Global regularity vs. infinite-time singularity formation in chemotaxis model with volume-filling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 3502-3525. doi: 10.1137/110853972.

[51]

M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Analysis, 72 (2010), 1044-1064. doi: 10.1016/j.na.2009.07.045.

[52]

M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differential Equations, 264 (2018), 6109-6151, arXiv: 1704.05648. doi: 10.1016/j.jde.2018.01.027.

[53]

M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828. doi: 10.1007/s00526-015-0922-2.

[54]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.

[55]

T. P. Witelski, Segregation and mixing in degenerate diffusion in population dynamics, J. Math. Biol., 35 (1997), 695-712. doi: 10.1007/s002850050072.

[56]

Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations, World Scientific Publishing Co. Pvt. Ltd., 2001. doi: 10.1142/9789812799791.

[57]

T. Y. XuS. M. JiM. Mei and J. X. Yin, Global existence of solutions to a chemotaxis-haptotaxis model with density-dependent jump probability and quorum-sensing mechanisms, Math. Meth. Appl. Sci., 41 (2018), 4208-4226.

[58]

T. Y. XuS. M. JiM. Mei and J. X. Yin, Traveling waves for time-delayed reaction diffusion equations with degenerate diffusion, J. Differential Equations, 265 (2018), 4442-4485. doi: 10.1016/j.jde.2018.06.008.

[59]

A. ZhigunC. Surulescu and A. Uatay, Global existence for a degenerate haptotaxis model of cancer invasion, Z. Angew. Math. Phys., 67 (2016), 1-29. doi: 10.1007/s00033-016-0741-0.

show all references

References:
[1]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X.

[2]

A. A. Blumberg, Logistic growth rate functions, J. Theor. Biol., 21 (1968), 42-44.

[3]

H. M. Byrne, Dissecting cancer through mathematics: from the cell to the animal model, Nature Reviews Cancer, 10 (2010), 221-230.

[4]

M. A. Chaplain and A. R. Anderson, Mathematical modelling of tissue invasion, in Cancer Model. Simul., Chapman and Hall/CRC, London, (2003), 269-297.

[5]

Y. H. Du, F. Quirós and M. L. Zhou, Logarithmic corrections in Fisher-KPP type Porous Medium Equations, preprint, arXiv: 1806.02022.

[6]

P. Friedl and K. Wolf, Tumour-cell invasion and migration: Diversity and escape mechanisms, Nature Reviews Cancer, 3 (2003), 362-374.

[7]

K. FujieA. ItoM. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Disc. Cont. Dyn. Syst., 36 (2016), 151-169. doi: 10.3934/dcds.2016.36.151.

[8]

R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753.

[9]

W. S. C. Gurney and R. M. Nisbet, The regulation of inhomogeneous population, J. Theors. Biol., 52 (1975), 441-457.

[10]

M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49. doi: 10.1016/0025-5564(77)90062-1.

[11]

T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301. doi: 10.1006/aama.2001.0721.

[12]

T. Hillen and K. J. Painter, A user's guide to pde models in a chemotaxis, J. Math. Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

[13]

C. H. Jin, Boundedness and global solvability to a chemotaxis model with nonlinear diffusion, J. Differential Equations, 263 (2017), 5759-5772. doi: 10.1016/j.jde.2017.06.034.

[14]

C. H. Jin, Boundedness and global solvability to a chemotaxis-haptotaixs model with slow and fast diffusion, Disc. Cont. Dyn. Syst., 23 (2018), 1675-1688. doi: 10.3934/dcdsb.2018069.

[15]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biology, 30 (1971), 225-234.

[16]

R. D. Leek, The role of tumour associated macrophages in breast cancer angiogenesis, Ph.D thesis, Oxford Brookes University, Oxford, 1999.

[17]

D. LiC. Mu and P. Zheng, Boundedness and large time behavior in a quasilinear chemotaxis model for tumor invasion, Math. Models Methods Appl. Sci., 28 (2018), 1413-1451. doi: 10.1142/S0218202518500380.

[18]

J. Liu and Y. F. Wang, A quasilinear chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, Math. Methods Appl. Sci., 40 (2017), 2107-2121. doi: 10.1002/mma.4126.

[19]

Y. Lou and M. Winkler, Global existence and uniform boundedness of smooth solutions to a cross-diffusion system with equal diffusion rates, Comm. Part. Diff. Equ., 40 (2015), 1905-1941. doi: 10.1080/03605302.2015.1052882.

[20]

P. LuV. Weaver and Z. Werb, The extracellular matrix: a dynamic niche in cancer progression, J. Cell Biol., 196 (2012), 395-406.

[21]

H. McAneney and S. F. C. O'Rourke, Investigation of various growth mechanisms of solid tumour growth within the linear quadratic model for radiotherapy, Phys. Med. Biol., 52 (2007), 1039-1054.

[22]

M. MeiH. Y. Peng and Z. A. Wang, Asymptotic profile of a parabolic-hyperbolic system with boundary effect arising from tumor angiogenesis, J. Differential Equations, 259 (2015), 5168-5191. doi: 10.1016/j.jde.2015.06.022.

[23]

Y. Mimura, The variational formulation of the fully parabolic Keller-Segel system with degenerate diffusion, J. Differential Equations, 263 (2017), 1477-1521. doi: 10.1016/j.jde.2017.03.020.

[24]

J. D. Murry, Mathematical Biology I: An Introduction, Springer, New York, USA, 2002.

[25]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Springer Science Business Media, 2013. doi: 10.1007/978-1-4757-4978-6.

[26]

M. E. Orme and M. A. J. Chaplain, A mathematical model of vascular tumour growth and invasion, Math. Comput. Modelling, 23 (1996), 43-60.

[27]

M. R. OwenH. M. Byrne and C. E. Lewis, Mathematical modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumour sites, J. Theor. Biol., 226 (2004), 377-391. doi: 10.1016/j.jtbi.2003.09.004.

[28]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canadian Appl. Math. Quart., 10 (2002), 501-543.

[29]

K. J. Painter and J. A. Sherratt, Modelling the movement of interacting cell populations, J. Theor. Biol., 225 (2003), 327-339. doi: 10.1016/S0022-5193(03)00258-3.

[30]

B. G. SengersC. P. Please and R. O. C. Oreffo, Experimental characterization and computational modelling of two-dimensional cell spreading for skeletal regeneration, J. R. Soc. Interface, 4 (2007), 1107-1117.

[31]

J. A. Sherratt, On the form of smooth-front travelling waves in a reaction-diffusion equation with degenerate nonlinear diffusion, Math. Model. Nat. Phenom, 5 (2010), 64-79. doi: 10.1051/mmnp/20105505.

[32]

J. A. Sherratt and M. A. J. Chaplain, A new mathematical model for avascular tumour growth, J. Math. Biol., 43 (2001), 291-312. doi: 10.1007/s002850100088.

[33]

J. A. Sherratt and J. D. Murray, Models of epidermal wound healing, Proc. R. Soc. London B, 241 (1990), 29-35.

[34]

M. J. Simpson, R. E. Baker and S. W. McCue, Models of collective cell spreading with variable cell aspect ratio: A motivation for degenerate diffusion models, Phys. Rev. E, 83 (2011), 021901.

[35]

M. J. SimpsonD. C. ZhangM. MarianiK. A. Landman and D. F. Newgreen, Cell proliferation drives neural crest cell invasion of the intestine, Dev. Biol., 302 (2007), 553-568.

[36]

A. Stevens and H. G. Othmer, Aggregation, blowup, and collapse: The abc's of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (2001), 1044-1081. doi: 10.1137/S0036139995288976.

[37]

Z. SzymańskaC. M. RodrigoM. Lachowicz and M. A. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models Methods Appl. Sci., 19 (2009), 257-281. doi: 10.1142/S0218202509003425.

[38]

Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-705. doi: 10.1137/100802943.

[39]

Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 1901-1914. doi: 10.3934/dcds.2012.32.1901.

[40]

Y. Tao and M. Winkler, Dominance of chemotaxis in a chemotaxis-haptotaxis model, Nonlinearity, 27 (2014), 1225-1239. doi: 10.1088/0951-7715/27/6/1225.

[41]

M. E. TurnerB. A. Blumenstein and J. L. Sebaugh, A generalization of the logistic law of growth, Biometrics, 25 (1969), 577-580.

[42]

H. A. S. Van den Brenk, Studies in restorative growth processes in mammalian wound healing, Br. J. Surg., 43 (1956), 525-550.

[43]

J. L. Vàzquez, The Porous Medium Equation: Mathematical Theory, Oxford Univ. Press, Oxford, 2007.

[44]

L. Von Bertalanffy, Quantitative laws in metabolism and growth, Q. Rev. Biol., 32 (1957), 217-231.

[45]

Y. F. Wang, Boundedness in a multi-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Appl. Math. Lett., 59 (2016), 122-126. doi: 10.1016/j.aml.2016.03.019.

[46]

Y. F. Wang, Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Differential Equations, 260 (2016), 1975-1989. doi: 10.1016/j.jde.2015.09.051.

[47]

Y. F. Wang and Y. Y. Ke, Large time behavior of solution to a fully parabolic chemotaxis-haptotaxis model in higher demensions, J. Differential Equations, 260 (2016), 6960-6988. doi: 10.1016/j.jde.2016.01.017.

[48]

Z. A. WangM. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect, Nonlinearity, 24 (2011), 3279-3297. doi: 10.1088/0951-7715/24/12/001.

[49]

Z. A. WangZ. Y. Xiang and P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 2225-2258. doi: 10.1016/j.jde.2015.09.063.

[50]

Z. A. WangM. Winkler and D. Wrzosek, Global regularity vs. infinite-time singularity formation in chemotaxis model with volume-filling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 3502-3525. doi: 10.1137/110853972.

[51]

M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Analysis, 72 (2010), 1044-1064. doi: 10.1016/j.na.2009.07.045.

[52]

M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differential Equations, 264 (2018), 6109-6151, arXiv: 1704.05648. doi: 10.1016/j.jde.2018.01.027.

[53]

M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828. doi: 10.1007/s00526-015-0922-2.

[54]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.

[55]

T. P. Witelski, Segregation and mixing in degenerate diffusion in population dynamics, J. Math. Biol., 35 (1997), 695-712. doi: 10.1007/s002850050072.

[56]

Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations, World Scientific Publishing Co. Pvt. Ltd., 2001. doi: 10.1142/9789812799791.

[57]

T. Y. XuS. M. JiM. Mei and J. X. Yin, Global existence of solutions to a chemotaxis-haptotaxis model with density-dependent jump probability and quorum-sensing mechanisms, Math. Meth. Appl. Sci., 41 (2018), 4208-4226.

[58]

T. Y. XuS. M. JiM. Mei and J. X. Yin, Traveling waves for time-delayed reaction diffusion equations with degenerate diffusion, J. Differential Equations, 265 (2018), 4442-4485. doi: 10.1016/j.jde.2018.06.008.

[59]

A. ZhigunC. Surulescu and A. Uatay, Global existence for a degenerate haptotaxis model of cancer invasion, Z. Angew. Math. Phys., 67 (2016), 1-29. doi: 10.1007/s00033-016-0741-0.

Figure 1.  (a) Comparison of experimental and simulated cell distribution for MG63 cells. The measured cell density (gray histogram) are fitted using the solution of degenerate nonlinear diffusion model (gray lines). (b) Comparison of experimental and simulated cell distribution for HBMSC cells. The measured cell density (gray histogram) are matched with the solution of linear diffusion model (gray lines). This diagram was redrawn from the one in Ref. [30]
Figure 2.  The growth curve of HEPA-1 spheroids. The solid line represents the position of the outer tumour boundary. Dimensional diameters are shown in $\mu m$. This diagram was redrawn from the one in Ref. [27]
[1]

S. Bonafede, G. R. Cirmi, A.F. Tedeev. Finite speed of propagation for the porous media equation with lower order terms. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 305-314. doi: 10.3934/dcds.2000.6.305

[2]

Tracy L. Stepien, Erica M. Rutter, Yang Kuang. A data-motivated density-dependent diffusion model of in vitro glioblastoma growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1157-1172. doi: 10.3934/mbe.2015.12.1157

[3]

Kaigang Huang, Yongli Cai, Feng Rao, Shengmao Fu, Weiming Wang. Positive steady states of a density-dependent predator-prey model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3087-3107. doi: 10.3934/dcdsb.2017209

[4]

Jishan Fan, Tohru Ozawa. An approximation model for the density-dependent magnetohydrodynamic equations. Conference Publications, 2013, 2013 (special) : 207-216. doi: 10.3934/proc.2013.2013.207

[5]

Shangbing Ai, Wenzhang Huang, Zhi-An Wang. Reaction, diffusion and chemotaxis in wave propagation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 1-21. doi: 10.3934/dcdsb.2015.20.1

[6]

Francisco Guillén-González, Mamadou Sy. Iterative method for mass diffusion model with density dependent viscosity. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 823-841. doi: 10.3934/dcdsb.2008.10.823

[7]

Messoud Efendiev, Anna Zhigun. On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 651-673. doi: 10.3934/dcds.2018028

[8]

Matthieu Alfaro, Thomas Giletti. Varying the direction of propagation in reaction-diffusion equations in periodic media. Networks & Heterogeneous Media, 2016, 11 (3) : 369-393. doi: 10.3934/nhm.2016001

[9]

Azmy S. Ackleh, Linda J. S. Allen. Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 175-188. doi: 10.3934/dcdsb.2005.5.175

[10]

Jacques A. L. Silva, Flávia T. Giordani. Density-dependent dispersal in multiple species metapopulations. Mathematical Biosciences & Engineering, 2008, 5 (4) : 843-857. doi: 10.3934/mbe.2008.5.843

[11]

Jishan Fan, Tohru Ozawa. Global Cauchy problem of an ideal density-dependent MHD-$\alpha$ model. Conference Publications, 2011, 2011 (Special) : 400-409. doi: 10.3934/proc.2011.2011.400

[12]

Zhuo Jin, Linyi Qian. Lookback option pricing for regime-switching jump diffusion models. Mathematical Control & Related Fields, 2015, 5 (2) : 237-258. doi: 10.3934/mcrf.2015.5.237

[13]

Xin Zhang, Hui Meng, Jie Xiong, Yang Shen. Robust optimal investment and reinsurance of an insurer under Jump-diffusion models. Mathematical Control & Related Fields, 2018, 8 (0) : 1-18. doi: 10.3934/mcrf.2019003

[14]

Antoine Benoit. Finite speed of propagation for mixed problems in the $WR$ class. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2351-2358. doi: 10.3934/cpaa.2014.13.2351

[15]

Lianzhang Bao, Wenjie Gao. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony with volume filling. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2813-2829. doi: 10.3934/dcdsb.2017152

[16]

Peng Feng, Zhengfang Zhou. Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1145-1165. doi: 10.3934/cpaa.2007.6.1145

[17]

Sungrim Seirin Lee. Dependence of propagation speed on invader species: The effect of the predatory commensalism in two-prey, one-predator system with diffusion. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 797-825. doi: 10.3934/dcdsb.2009.12.797

[18]

Philip Gerlee, Alexander R. A. Anderson. Diffusion-limited tumour growth: Simulations and analysis. Mathematical Biosciences & Engineering, 2010, 7 (2) : 385-400. doi: 10.3934/mbe.2010.7.385

[19]

Mikhail Kuzmin, Stefano Ruggerini. Front propagation in diffusion-aggregation models with bi-stable reaction. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 819-833. doi: 10.3934/dcdsb.2011.16.819

[20]

Roberta Sirovich, Laura Sacerdote, Alessandro E. P. Villa. Cooperative behavior in a jump diffusion model for a simple network of spiking neurons. Mathematical Biosciences & Engineering, 2014, 11 (2) : 385-401. doi: 10.3934/mbe.2014.11.385

2017 Impact Factor: 1.23

Metrics

  • PDF downloads (46)
  • HTML views (81)
  • Cited by (0)

[Back to Top]