American Institute of Mathematical Sciences

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March  2012, 5(1): 51-95. doi: 10.3934/krm.2012.5.51

On a chemotaxis model with saturated chemotactic flux

Alina Chertock 1, , Alexander Kurganov 2, , Xuefeng Wang 2, and  Yaping Wu 3,

1. 

Department of Mathematics, North Carolina State University, Raleigh, NC 27695, United States
2. 

Mathematics Department, Tulane University, New Orleans, LA 70118
3. 

Department of Mathematics, Capital Normal University, Beijing 100048, China

Received  May 2011 Revised  August 2011 Published  January 2012

We propose a PDE chemotaxis model, which can be viewed as a regularization of the Patlak-Keller-Segel (PKS) system. Our modification is based on a fundamental physical property of the chemotactic flux function---its boundedness. This means that the cell velocity is proportional to the magnitude of the chemoattractant gradient only when the latter is small, while when the chemoattractant gradient tends to infinity the cell velocity saturates. Unlike the original PKS system, the solutions of the modified model do not blow up in either finite or infinite time in any number of spatial dimensions, thus making it possible to use bounded spiky steady states to model cell aggregation. After obtaining local and global existence results, we use the local and global bifurcation theories to show the existence of one-dimensional spiky steady states; we also study the stability of bifurcating steady states. Finally, we numerically verify these analytical results, and then demonstrate that solutions of the two-dimensional model with nonlinear saturated chemotactic flux function typically develop very complicated spiky structures.
Keywords: spiky steady states, local and global existence, Chemotaxis, bifurcation, saturation of flux, second-order positivity preserving upwind scheme., hybrid finite-volume-finite-difference method.
Mathematics Subject Classification: Primary: 92C17, 35K50, 35B36, 76M12, 76M20; Secondary: 35B3.
Citation: Alina Chertock, Alexander Kurganov, Xuefeng Wang, Yaping Wu. On a chemotaxis model with saturated chemotactic flux. Kinetic & Related Models, 2012, 5 (1) : 51-95. doi: 10.3934/krm.2012.5.51
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show all references

References:
[1]

Ann. Rev. Biochem., 44 (1975), 341-356. doi: 10.1146/annurev.bi.44.070175.002013.  Google Scholar

[2]

J. Math. Biol., 9 (1980), 147-177. doi: 10.1007/BF00275919.  Google Scholar

[3]

Differential Integral Equations, 3 (1990), 13-75.  Google Scholar

[4]

in "Function Spaces, Differential Operators and Nonlinear Analysis" (Friedrichroda, 1992), Teubner-Texte Math., 133, Teubner, Stuttgart, (1993), 9-126.  Google Scholar

[5]

2nd ed., Princeton University Press, Princeton, New Jersey, 1967. Google Scholar

[6]

Nature, 349 (1991), 630-633. doi: 10.1038/349630a0.  Google Scholar

[7]

Nature, 376 (1995), 49-53. doi: 10.1038/376049a0.  Google Scholar

[8]

A. Chertock, Y. Epshteyn and A. Kurganov, High-order finite-difference and finite-volume methods for chemotaxis models,, in preparartion., ().   Google Scholar

[9]

Math. Biosc., 56 (1981), 217-237. doi: 10.1016/0025-5564(81)90055-9.  Google Scholar

[10]

J. Theor. Biol., 31 (1971), 101-118. doi: 10.1016/0022-5193(71)90124-X.  Google Scholar

[11]

J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[12]

Arch. Rational Mech. Anal., 52 (1973), 161-180. doi: 10.1007/BF00282325.  Google Scholar

[13]

SIAM Rev., 43 (2001), 89-112. doi: 10.1137/S003614450036757X.  Google Scholar

[14]

Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[15]

Ann. Scuola Normale Superiore Pisa Cl. Sci. (4), 24 (1997), 633-683.  Google Scholar

[16]

J. Sci. Comput., 39 (2009), 115-128. doi: 10.1007/s10915-008-9252-2.  Google Scholar

[17]

Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 125-144 (electronic). doi: 10.3934/dcdsb.2007.7.125.  Google Scholar

[18]

J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

[19]

Jahresber. DMV, 105 (2003), 103-165.  Google Scholar

[20]

Jahresber. DMV, 106 (2004), 51-69.  Google Scholar

[21]

J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[22]

J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[23]

J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[24]

Translated from the Russian by S. Smith, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967.  Google Scholar

[25]

World Scientific Publishing Co., Inc., River Edge, NJ, 1996.  Google Scholar

[26]

J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.  Google Scholar

[27]

Funkcial. Ekvac., 40 (1997), 411-433.  Google Scholar

[28]

Notices Amer. Math. Soc., 45 (1998), 9-18.  Google Scholar

[29]

J. Math. Biol., 26 (1988), 263-298. doi: 10.1007/BF00277392.  Google Scholar

[30]

Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407.  Google Scholar

[31]

Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.  Google Scholar

[32]

J. Anal. Math., 76 (1998), 289-319. doi: 10.1007/BF02786939.  Google Scholar

[33]

Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[34]

3rd ed., Wm. C. Brown Publishers, Chicago-London, 1996. Google Scholar

[35]

Chem. Eng. Sci., 44 (1989), 1-17. doi: 10.1016/0009-2509(89)85098-5.  Google Scholar

[36]

J. Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009.  Google Scholar

[37]

SIAM J. Appl. Math., 65 (2005), 790-817 (electronic). doi: 10.1137/S0036139902415117.  Google Scholar

[38]

SIAM J. Math. Anal., 31 (2000), 535-560 (electronic). doi: 10.1137/S0036141098339897.  Google Scholar

[39]

Biophys. J., 68 (1995), 2181-2189. doi: 10.1016/S0006-3495(95)80400-5.  Google Scholar

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