American Institute of Mathematical Sciences

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May  2013, 18(3): 601-641. doi: 10.3934/dcdsb.2013.18.601

Mathematics of traveling waves in chemotaxis --Review paper--

Zhi-An Wang 1,

1. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received  November 2012 Revised  November 2012 Published  December 2012

This article surveys the mathematical aspects of traveling waves of a class of chemotaxis models with logarithmic sensitivity, which describe a variety of biological or medical phenomena including bacterial chemotactic motion, initiation of angiogenesis and reinforced random walks. The survey is focused on the existence, wave speed, asymptotic decay rates, stability and chemical diffusion limits of traveling wave solutions. The main approaches are reviewed and related analytical results are given with sketchy proofs. We also develop some new results with detailed proofs to fill the gap existing in the literature. The numerical simulations of steadily propagating waves will be presented along the study. Open problems are proposed for interested readers to pursue.
Keywords: traveling waves, stability, bacteria, wave speed, angiogenesis, Fisher equation, Chemotaxis, conservation laws., chemical diffusion, transformation.
Mathematics Subject Classification: 35C07, 35K55, 46N60, 62P10, 92C1.
Citation: Zhi-An Wang. Mathematics of traveling waves in chemotaxis --Review paper--. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 601-641. doi: 10.3934/dcdsb.2013.18.601
References:
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show all references

References:
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[2]

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[16]

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[17]

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[18]

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[19]

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[20]

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[21]

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[22]

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[23]

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[24]

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[28]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.  Google Scholar

[29]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II. Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69.  Google Scholar

[30]

D. Horstmann and A. Stevens, A constructive approach to traveling waves in chemotaxis, J. Nonlin. Sci., 14 (2004), 1-25. doi: 10.1007/s00332-003-0548-y.  Google Scholar

[31]

C. K. R. T. Jones, Geometric singular perturbation theory, in "Dynamical Systems" (ed. J. Russell) (Montecatini Terme, 1994), Lecture Notes in Math., 1609, Springer, Berlin, 1995. doi: 10.1007/BFb0095239.  Google Scholar

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[33]

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[34]

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

[35]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis, J. Theor. Biol., 26 (1971), 235-248. Google Scholar

[36]

C. R. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death, Bull. Math. Biol., 42 (1980), 397-429. doi: 10.1016/S0092-8240(80)80057-7.  Google Scholar

[37]

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[38]

A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Byul. Moskovskogo Gos. Univ., 1 (1937), 1-25. Google Scholar

[39]

M. Kot, "Elementals of Mathematical Biology," Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511608520.  Google Scholar

[40]

K. A. Landman, M. J. Simpson, J. L Slater and D. F. Newgreen, Diffusive and chemotactic celluar migration: Smooth and disconcinuous traveling wave solutions, SIAM J. Appl. Math., 65 (2005), 1420-1442. doi: 10.1137/040604066.  Google Scholar

[41]

I. R. Lapidus and R. Schiller, A model for traveling bands of chemotactic bacteria, Biophy. J., 22 (1978), 1-13. Google Scholar

[42]

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