
Mathematical Biosciences and Engineering, 2018, 15(6): 13451385. doi: 10.3934/mbe.2018062
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EARLY AND LATE STAGE PROFILES FOR A CHEMOTAXIS MODEL WITH DENSITYDEPENDENT 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 SaintLambert, Quebec, J4P 3P2, Canada
4. Department of Mathematics and Statistics, McGill University, Montreal, Quebec, H3A 2K6, Canada
Received: , Accepted: , Published:
References
[1] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of KellerSegel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 16631763.
[2] A. A. Blumberg, Logistic growth rate functions, J. Theor. Biol., 21 (1968), 4244.
[3] H. M. Byrne, Dissecting cancer through mathematics: from the cell to the animal model, Nature Reviews Cancer, 10 (2010), 221230.
[4] M. A. Chaplain and A. R. Anderson, Mathematical modelling of tissue invasion, in Cancer Model. Simul., Chapman and Hall/CRC, London, (2003), 269297.
[5] Y. H. Du, F. Quirós and M. L. Zhou, Logarithmic corrections in FisherKPP type Porous Medium Equations, preprint, arXiv: 1806.02022.
[6] P. Friedl and K. Wolf, Tumourcell invasion and migration: Diversity and escape mechanisms, Nature Reviews Cancer, 3 (2003), 362374.
[7] K. Fujie, A. Ito, M. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Disc. Cont. Dyn. Syst., 36 (2016), 151169. doi:
[8] R. A. Gatenby and E. T. Gawlinski, A reactiondiffusion model of cancer invasion, Cancer Res., 56 (1996), 57455753.
[9] W. S. C. Gurney and R. M. Nisbet, The regulation of inhomogeneous population, J. Theors. Biol., 52 (1975), 441457.
[10] M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 3549.
[11] T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280301.
[12] T. Hillen and K. J. Painter, A user's guide to pde models in a chemotaxis, J. Math. Biology, 58 (2009), 183217.
[13] C. H. Jin, Boundedness and global solvability to a chemotaxis model with nonlinear diffusion, J. Differential Equations, 263 (2017), 57595772.
[14] C. H. Jin, Boundedness and global solvability to a chemotaxishaptotaixs model with slow and fast diffusion, Disc. Cont. Dyn. Syst., 23 (2018), 16751688.
[15] E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biology, 30 (1971), 225234.
[16] R. D. Leek, The role of tumour associated macrophages in breast cancer angiogenesis, Ph.D thesis, Oxford Brookes University, Oxford, 1999.
[17] D. Li, C. Mu and P. Zheng, Boundedness and large time behavior in a quasilinear chemotaxis model for tumor invasion, Math. Models Methods Appl. Sci., 28 (2018), 14131451.
[18] J. Liu and Y. F. Wang, A quasilinear chemotaxishaptotaxis model: The roles of nonlinear diffusion and logistic source, Math. Methods Appl. Sci., 40 (2017), 21072121.
[19] Y. Lou and M. Winkler, Global existence and uniform boundedness of smooth solutions to a crossdiffusion system with equal diffusion rates, Comm. Part. Diff. Equ., 40 (2015), 19051941.
[20] P. Lu, V. Weaver and Z. Werb, The extracellular matrix: a dynamic niche in cancer progression, J. Cell Biol., 196 (2012), 395406.
[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), 10391054.
[22] M. Mei, H. Y. Peng and Z. A. Wang, Asymptotic profile of a parabolichyperbolic system with boundary effect arising from tumor angiogenesis, J. Differential Equations, 259 (2015), 51685191.
[23] Y. Mimura, The variational formulation of the fully parabolic KellerSegel system with degenerate diffusion, J. Differential Equations, 263 (2017), 14771521.
[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.
[26] M. E. Orme and M. A. J. Chaplain, A mathematical model of vascular tumour growth and invasion, Math. Comput. Modelling, 23 (1996), 4360.
[27] M. R. Owen, H. 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), 377391.
[28] K. J. Painter and T. Hillen, Volumefilling and quorumsensing in models for chemosensitive movement, Canadian Appl. Math. Quart., 10 (2002), 501543.
[29] K. J. Painter and J. A. Sherratt, Modelling the movement of interacting cell populations, J. Theor. Biol., 225 (2003), 327339.
[30] B. G. Sengers, C. P. Please and R. O. C. Oreffo, Experimental characterization and computational modelling of twodimensional cell spreading for skeletal regeneration, J. R. Soc. Interface, 4 (2007), 11071117.
[31] J. A. Sherratt, On the form of smoothfront travelling waves in a reactiondiffusion equation with degenerate nonlinear diffusion, Math. Model. Nat. Phenom, 5 (2010), 6479.
[32] J. A. Sherratt and M. A. J. Chaplain, A new mathematical model for avascular tumour growth, J. Math. Biol., 43 (2001), 291312.
[33] J. A. Sherratt and J. D. Murray, Models of epidermal wound healing, Proc. R. Soc. London B, 241 (1990), 2935.
[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. Simpson, D. C. Zhang, M. Mariani, K. A. Landman and D. F. Newgreen, Cell proliferation drives neural crest cell invasion of the intestine, Dev. Biol., 302 (2007), 553568.
[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), 10441081.
[37] Z. Szymańska, C. M. Rodrigo, M. 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), 257281.
[38] Y. Tao and M. Winkler, A chemotaxishaptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685705.
[39] Y. Tao and M. Winkler, Global existence and boundedness in a KellerSegelStokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 19011914.
[40] Y. Tao and M. Winkler, Dominance of chemotaxis in a chemotaxishaptotaxis model, Nonlinearity, 27 (2014), 12251239.
[41] M. E. Turner, B. A. Blumenstein and J. L. Sebaugh, A generalization of the logistic law of growth, Biometrics, 25 (1969), 577580.
[42] H. A. S. Van den Brenk, Studies in restorative growth processes in mammalian wound healing, Br. J. Surg., 43 (1956), 525550.
[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), 217231.
[45] Y. F. Wang, Boundedness in a multidimensional chemotaxishaptotaxis model with nonlinear diffusion, Appl. Math. Lett., 59 (2016), 122126.
[46] Y. F. Wang, Boundedness in the higherdimensional chemotaxishaptotaxis model with nonlinear diffusion, J. Differential Equations, 260 (2016), 19751989.
[47] Y. F. Wang and Y. Y. Ke, Large time behavior of solution to a fully parabolic chemotaxishaptotaxis model in higher demensions, J. Differential Equations, 260 (2016), 69606988.
[48] Z. A. Wang, M. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volumefilling effect, Nonlinearity, 24 (2011), 32793297.
[49] Z. A. Wang, Z. Y. Xiang and P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 22252258.
[50] Z. A. Wang, M. Winkler and D. Wrzosek, Global regularity vs. infinitetime singularity formation in chemotaxis model with volumefilling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 35023525.
[51] M. Winkler and K. C. Djie, Boundedness and finitetime collapse in a chemotaxis system with volumefilling effect, Nonlinear Analysis, 72 (2010), 10441064.
[52] M. Winkler, Global existence and stabilization in a degenerate chemotaxisStokes system with mildly strong diffusion enhancement, J. Differential Equations, 264 (2018), 61096151, arXiv: 1704.05648.
[53] M. Winkler, Boundedness and large time behavior in a threedimensional chemotaxisStokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 37893828.
[54] M. Winkler, Finitetime blowup in the higherdimensional parabolicparabolic KellerSegel system, J. Math. Pures Appl., 100 (2013), 748767.
[55] T. P. Witelski, Segregation and mixing in degenerate diffusion in population dynamics, J. Math. Biol., 35 (1997), 695712.
[56] Z. Wu, J. Zhao, J. Yin and H. Li, Nonlinear Diffusion Equations, World Scientific Publishing Co. Pvt. Ltd., 2001.
[57] T. Y. Xu, S. M. Ji, M. Mei and J. X. Yin, Global existence of solutions to a chemotaxishaptotaxis model with densitydependent jump probability and quorumsensing mechanisms, Math. Meth. Appl. Sci., 41 (2018), 42084226.
[58] T. Y. Xu, S. M. Ji, M. Mei and J. X. Yin, Traveling waves for timedelayed reaction diffusion equations with degenerate diffusion, J. Differential Equations, 265 (2018), 44424485.
[59] A. Zhigun, C. Surulescu and A. Uatay, Global existence for a degenerate haptotaxis model of cancer invasion, Z. Angew. Math. Phys., 67 (2016), 129.
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