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:
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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.  Google Scholar

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[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.  Google Scholar

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

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[8]

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

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[42]

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

[43]

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

[44]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[55]

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

[56]

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

[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.   Google Scholar

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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]
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