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On Chemotaxis Models with Cell Population Interactions

Published online by Cambridge University Press:  28 April 2010

Z. A. Wang
Z. A. Wang*
Affiliation:
Department of Mathematics, University of Vanderbilt, Nashville, TN 37240
*
* E-mail:zhian.wang@vanderbilt.edu
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Abstract

This paper extends the volume filling chemotaxis model [18, 26] by taking into account the cell population interactions. Theextended chemotaxis models have nonlinear diffusion and chemotactic sensitivity dependingon cell population density, which is a modification of the classical Keller-Segel model inwhich the diffusion and chemotactic sensitivity are constants (linear). The existence andboundedness of global solutions of these models are discussed and the numerical patternformations are shown. The further improvement is proposed in the end.

Keywords

chemotaxisKeller-Segel modelcell interactionsnonlinear diffusionblow upvolume fillingchemotactic sensitivitypattern formation
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 3: Mathematical modeling in the medical sciences , 2010 , pp. 173 - 190
Copyright
© EDP Sciences, 2010

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References

Chavanis, P.H.. A stochastic keller-segel model of chemotaxis . Commun. Nonlinear Sci Numer Simulat . 15 (2010), 60-70. CrossRefGoogle Scholar
Choi, Y.Z., Wang, Z.A.. Prevention of blow up by fast diffusion in chemotaxis. , J. Math. Anal. Appl., 362 (2010), 553-564. CrossRefGoogle Scholar
Kaiser, D.. Cell-cell interactions . Prokaryotes, 1 (2006), 221-245. CrossRefGoogle Scholar
M. Eisenbach. Chemotaxis. Imperial College Press, London, 2004.
Hillen, T., Painter, K.. A users guide to PDE models for chemotaxis . J. Math. Biol. , 57 (2009), 183-217. CrossRefGoogle Scholar
Hillen, T., Painter, K.. Global existence for a parabolic chemotaxis model with prevention of overcrowding . Adv. Appl. Math., 26 (2001), 280-301. CrossRefGoogle Scholar
Höfer, T., Sherratt, J.A., Maini, P.K.. Dictyostelium discoideum: cellular self-organisation in an excitable biological medium . Proc. R. Soc. Lond. B., 259 (1995), 249-257.CrossRefGoogle Scholar
Hortsmann, D.. From 1970 until present: the keller-segel model in chemotaxis and its consequences: I . Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.Google Scholar
Hortsmann, D.. From 1970 until present: the keller-segel model in chemotaxis and its consequences: II . Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69.Google Scholar
Keller, E.F., Segel, L.A.. Initiation of slime mold aggregation viewd as an instability . J. Theor. Biol., 26 (1970), 399-415.CrossRefGoogle ScholarPubMed
Kuiper, H., Dung, L.. Global attractors for cross-diffusion systems on domains of arbitrary dimensions . Rocky Mountain J. Math., 37 (2007), No 5, 1645-1668. CrossRefGoogle Scholar
Laurençot, P., Wrzosek, D.. A chemotaxis model with threshold density and degenerate diffusion . In: Progress in Nonlinear Diffusion Equations and Their Application., 64 (2005): 273-290. Google Scholar
Lushnikov, P.M., Chen, N., Alber, M.. Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact . Phys. Rev. E., 78 (2008), 061904. CrossRefGoogle ScholarPubMed
J. Murray, Mathematical biology: an introduction. Third edition, Springer, 2002.
Childress, S., Percus, J.K. Nonlinear aspects of chemotaxis , Math. Biosci. , 56 (1981), 217-237. CrossRefGoogle Scholar
Kowalczyk, T.. Preventing blow-up in a chemotaxis model . J. Math. Anal. Appl., 305 (2005), 566-588.CrossRefGoogle Scholar
Neuman, W.I.. The long-time behavior of the solution to a non-linear diffusion problem in population genetics and combustion . J. Theor. Biol., 104 (1985), 472-484.Google Scholar
Painter, P., Hillen, T.. Volume-filling and quorum-sensing in models for chemosensitive move- ment . Can. Appl. Math. Quart., 10 (2002), No 4, 501-543. Google Scholar
Painter, K., Sherratt, J. A.. Modelling the movement of interacting cell populations . J. Theor. Biol., 225 (2003), 327-339.CrossRefGoogle ScholarPubMed
B. Perthame. Transport equations in biology. Birkhäuser, Basel, 2007.
Peter Pivonka.Personal communication. 2009.
A. Okubo, Diffusion and Ecological problems: Mathematical Models. Springer-Verlag, Berlin-Heidelberg-New York, 1980.
Okubo, A.. Dynamical aspects of animal grouping: swarms, schools, flocks and herds . Adv. Biophys., 22 (1986), 1-94.CrossRefGoogle ScholarPubMed
N. Shigesada, K. Kawasaki. Biological Invasions: Theory and Practice. Oxford University Press, Oxford, 1997.
Othmer, H.G., Stevens, A.. Aggregation, blowup and collapse: The ABC of taxis in reinforced random walks . SIAM J. Appl. Math., 57 (1997), 1044-1081.Google Scholar
Wang, Z.A., Hillen, T.. Classical solutions and pattern formation for a volume filling chemotaxis model . Chaos., 17 (2007), 037108, 13 pages. CrossRefGoogle ScholarPubMed
Willard, S.S., Devreotes, P.N.. Signalling pathways mediating chemotaxis in the social amoeba, dictyostelium discoideum . Euro. J. Cell. Biol., 85 (2006), 897-904.CrossRefGoogle Scholar
Wrzosek, D.. Global attractor for a chemotaxis model with prevention of overcrowding . Nonlinear Analysis., 59 (2004), 1293-1310.CrossRefGoogle Scholar
Wrzosek, D.. Long time behavior of solutions to a chemotaxis model with volume filling effects . Proc. R. Soc. Edinburgh A: Math., 136 (2006), 431-444.CrossRefGoogle Scholar
D. Wrzosek. Model of chemotaxis with threshold density and singular diffusion. Nonlinear Anal. TMA, DOI:10.1016/j.na.2010.02.047, 2010. CrossRef