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

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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--

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

Received November 2012 Revised November 2012 Published December 2012

Abstract
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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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[2]	J. Adler, Chemoreceptors in bacteria,, Science, 166 (1969), 1588.
[3]	S. B. Ai, W. Z. Huang and Z. A. Wang, Traveling wave solutions to a chemotaxis system with logistic growth,, preprint, (2012).
[4]	W. Alt and D. A. Lauffenburger, Transient behavior of a chemotaxis system modelling certain types of tissue inflammation,, J. Math. Biol., 24 (1987), 691. doi: 10.1007/BF00275511.
[5]	S. Balasuriya and G. A. Gottwald, Wavespeed in reaction diffusion systems, with applications to chemotaxis and population pressure,, J. Math. Biol., 61 (2010), 377. doi: 10.1007/s00285-009-0305-4.
[6]	D. Balding and D. L. McElwain, A mathematical model of tumour-induced capillary growth,, J. Theor. Biol., 114 (1985), 53.
[7]	R. Bellman, "Stability Theory of Differential Equations,", McGraw-Hill Book Company, (1953).
[8]	R. D. Benguria, M. C. Depassier and V. Mendez, Minimum speed of fronts of reaction-convenction-diffusion equations,, Phys. Rev. E (3), 69 (2004). doi: 10.1103/PhysRevE.69.031106.
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[11]	E. Budrene and H. Berg, Complex patterns formed by motile cells of Escherichia coli,, Nature, 349 (1991), 630.
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show all references

References:

[1]	J. Adler, Chemotaxis in bacteria,, Science, 153 (1966), 708.
[2]	J. Adler, Chemoreceptors in bacteria,, Science, 166 (1969), 1588.
[3]	S. B. Ai, W. Z. Huang and Z. A. Wang, Traveling wave solutions to a chemotaxis system with logistic growth,, preprint, (2012).
[4]	W. Alt and D. A. Lauffenburger, Transient behavior of a chemotaxis system modelling certain types of tissue inflammation,, J. Math. Biol., 24 (1987), 691. doi: 10.1007/BF00275511.
[5]	S. Balasuriya and G. A. Gottwald, Wavespeed in reaction diffusion systems, with applications to chemotaxis and population pressure,, J. Math. Biol., 61 (2010), 377. doi: 10.1007/s00285-009-0305-4.
[6]	D. Balding and D. L. McElwain, A mathematical model of tumour-induced capillary growth,, J. Theor. Biol., 114 (1985), 53.
[7]	R. Bellman, "Stability Theory of Differential Equations,", McGraw-Hill Book Company, (1953).
[8]	R. D. Benguria, M. C. Depassier and V. Mendez, Minimum speed of fronts of reaction-convenction-diffusion equations,, Phys. Rev. E (3), 69 (2004). doi: 10.1103/PhysRevE.69.031106.
[9]	F. S. Berezovskaya, A. S. Novozhilov and G. P. Karev, Families of traveling impulses and fronts in some models with cross-diffusion,, Nonlinear Analysis: Real World Applications, 9 (2008), 1866. doi: 10.1016/j.nonrwa.2007.06.001.
[10]	M. P. Brenner, L. S. Levitor and E. O. Budrene, Physical mechanisms for chemotactic pattern formation by bacterial,, Biophys. J., 74 (1988), 1677.
[11]	E. Budrene and H. Berg, Complex patterns formed by motile cells of Escherichia coli,, Nature, 349 (1991), 630.
[12]	E. Budrene and H. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria,, Nature, 376 (1995), 49.
[13]	S. Childress, Chemotactic collapse in two dimensions,, in, 55 (1984), 61. doi: 10.1007/978-3-642-45589-66.
[14]	W. A. Coppel, "Stability and Asymptotic Behavior of Differential Equations,", D. C. Heath and Co., (1965).
[15]	L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis,, C. R. Acad. Sci. Paris. Ser. I., 336 (2003), 141. doi: 10.1016/S1631-073X(02)00008-0.
[16]	L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis system in high space dimensions,, Milan j. Math., 72 (2004), 1. doi: 10.1007/s00032-003-0026-x.
[17]	F. W. Dahlquist, P. Lovely and D. E. Jr Koshland, Qualitative analysis of bacterial migration in chemotaxis,, Nature, 236 (1972), 120.
[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.
[19]	N. Fenichel, Geometric singular perturbation theory of ordinary differential equations,, J. Differential Equations, 31 (1979), 53. doi: 10.1016/0022-0396(79)90152-9.
[20]	R. A. Fisher, The advance of advantageous genes,, Ann. Eugenics, 7 (1937), 355.
[21]	M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis,, SIAM J. Math. Anal., 33 (2002), 1330. doi: 10.1137/S0036141001385046.
[22]	M. Funaki, M. Mimura and T. Tsujikawa, Travelling front solutions arising in the chemotaxis-growth model,, Interfaces Free Bound., 8 (2006), 223. doi: 10.4171/IFB/141.
[23]	R. E. Goldstein, Traveling-wave chemotaxis,, Phys. Rev. Lett., 77 (1996), 775.
[24]	J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws,, Arch. Rat. Mech. Anal., 95 (1986), 325. doi: 10.1007/BF00276840.
[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.
[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.
[27]	M. Holz and S. H. Chen, Spatio-temporal structure of migrating chemotactic band of escherichia coli,, Biophys. J., 26 (1979), 243.
[28]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I., Jahresber. Deutsch. Math.-Verein., 105 (2003), 103.
[29]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II., Jahresber. Deutsch. Math.-Verein., 106 (2004), 51.
[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.
[31]	C. K. R. T. Jones, Geometric singular perturbation theory,, in, 1609 (1994). doi: 10.1007/BFb0095239.
[32]	E. F. Keller and G. M. Odell, Necessary and sufficient conditions for chemotactic bands,, Math. Biosci., 27 (1975), 309.
[33]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.
[34]	E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225.
[35]	E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis,, J. Theor. Biol., 26 (1971), 235.
[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.
[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.
[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.
[39]	M. Kot, "Elementals of Mathematical Biology,", Cambridge University Press, (2001). doi: 10.1017/CBO9780511608520.
[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.
[41]	I. R. Lapidus and R. Schiller, A model for traveling bands of chemotactic bacteria,, Biophy. J., 22 (1978), 1.
[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.
[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.
[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.
[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.
[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.
[47]	T. Li and Z.-A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,, SIAM J. Appl. Math., 70 (): 1522. doi: 10.1137/09075161X.
[48]	T. Li and Z.-A. Wang, Nonlinear stability of large amplitude viscous shock waves of a hyperbolic-parabolic system arising in chemotaxis,, Math. Models Methods Appl. Sci., 20 (2010), 1967. doi: 10.1142/S0218202510004830.
[49]	T. Li and Z.-A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis,, J. Differential Equations, 250 (2011), 1310. doi: 10.1016/j.jde.2010.09.020.
[50]	T. Li and Z.-A. Wang, Steadily propogating waves of a chemotaxis model,, Math. Biosci., 240 (2012), 161.
[51]	Y. Li, The existence of traveling waves in a biological model for chemotaxis,, Acta Mathematicae Applicatae Sinica (in Chinese), 27 (2004), 123.
[52]	C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, J. Differential Equations, 72 (1988), 1. doi: 10.1016/0022-0396(88)90147-7.
[53]	T.-P. Liu and Y. Zeng, Time-asymptotic behavior of wave propagation around a viscous shock profile,, Comm. Math. Phy., 290 (2009), 23. doi: 10.1007/s00220-009-0820-6.
[54]	R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models,, J. Math. Biol., 61 (2010), 739. doi: 10.1007/s00285-009-0317-0.
[55]	A. Matsumura and K. Nishihara, On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas,, Japan J. Appl. Math., 2 (1985), 17. doi: 10.1007/BF03167036.
[56]	P. M. McCabe, J. A. Leach and D. J. Needham, A note on the non-existence of permanent form traveling wave solutions in a class of singular reaction-diffusion problems,, Dynamical Systems, 17 (2002), 131. doi: 10.1080/14689360110116498.
[57]	M. Meyries, Local well posedness and instability of travelling waves in a chemotaxis model,, Adv. Differential Equations, 16 (2011), 31.
[58]	J. D. Murray, "Mathematical Biology. I. An Introduction,", Third edition, 17 (2002).
[59]	G. Nadin, B. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with fisher birth terms,, Interfaces Free Bound., 10 (2008), 517. doi: 10.4171/IFB/200.
[60]	T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model,, J. Math. Biol., 30 (1991), 169. doi: 10.1007/BF00160334.
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