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

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

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.
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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[3]

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[11]

E. Budrene and H. Berg, Complex patterns formed by motile cells of Escherichia coli,, Nature, 349 (1991), 630.   Google Scholar

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E. Budrene and H. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria,, Nature, 376 (1995), 49.   Google Scholar

[13]

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[14]

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[15]

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[16]

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[17]

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[18]

Y. Ebihara, Y. Furusho and T. Nagai, Singular solution of traveling waves in a chemotactic model,, Bull. Kyushu Inst. Tech. Math. Natur. Sci., 39 (1992), 29.   Google Scholar

[19]

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[20]

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[21]

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[22]

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[23]

R. E. Goldstein, Traveling-wave chemotaxis,, Phys. Rev. Lett., 77 (1996), 775.   Google Scholar

[24]

J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws,, Arch. Rat. Mech. Anal., 95 (1986), 325.  doi: 10.1007/BF00276840.  Google Scholar

[25]

S. Gueron and N. Liron, A model of herd grazing as a traveling wave: Chemotaxis and stability,, J. Math. Biol., 27 (1989), 595.  doi: 10.1007/BF00288436.  Google Scholar

[26]

J. Guo, J. X. Xiao, H. J. Zhao and C. J. Zhu, Global solutions to a hyperbolic-parabolic coupled system with large initial data,, Acta Math. Sci. Ser. B Engl. Ed, 29 (2009), 629.  doi: 10.1016/S0252-9602(09)60059-X.  Google Scholar

[27]

M. Holz and S. H. Chen, Spatio-temporal structure of migrating chemotactic band of escherichia coli,, Biophys. J., 26 (1979), 243.   Google Scholar

[28]

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

[30]

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

[31]

C. K. R. T. Jones, Geometric singular perturbation theory,, in, 1609 (1994).  doi: 10.1007/BFb0095239.  Google Scholar

[32]

E. F. Keller and G. M. Odell, Necessary and sufficient conditions for chemotactic bands,, Math. Biosci., 27 (1975), 309.   Google Scholar

[33]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.   Google Scholar

[34]

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

[35]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis,, J. Theor. Biol., 26 (1971), 235.   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.  doi: 10.1016/S0092-8240(80)80057-7.  Google Scholar

[37]

I. T. Kiguradse, On the non-negative non-increasing solutions of non-linear second order differential equations,, Ann. Mat. Pura ed Appl. (4), 81 (1969), 169.   Google Scholar

[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.   Google Scholar

[39]

M. Kot, "Elementals of Mathematical Biology,", Cambridge University Press, (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.  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.   Google Scholar

[42]

D. Lauffenburger, C. R. Kennedy and R. Aris, Traveling bands of chemotactic bacteria in the context of population growth,, Bull. Math. Biol., 46 (1984), 19.   Google Scholar

[43]

K. J. Lee, E. C. Cox and R. E. Goldstein, Competing patterns of signaling activity in dictyostelium discoideum,, Phys. Rev. Lett., 76 (1996), 1174.   Google Scholar

[44]

H. A. Levine and B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks,, SIAM J. Appl. Math., 57 (1997), 683.  doi: 10.1137/S0036139995291106.  Google Scholar

[45]

D. Li, T. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis,, Math. Models Methods Appl. Sci., 21 (2011), 1631.  doi: 10.1142/S0218202511005519.  Google Scholar

[46]

T. Li, R. H. Pan and K. Zhao, Global dynamics of a hyperbolic-parabolic model arising from chemotaxis,, SIAM. J. Appl. Math., 72 (2012), 417.  doi: 10.1137/110829453.  Google Scholar

[47]

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