# American Institute of Mathematical Sciences

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:
 [1] J. Adler, Chemotaxis in bacteria,, Science, 153 (1966), 708. Google Scholar [2] J. Adler, Chemoreceptors in bacteria,, Science, 166 (1969), 1588. Google Scholar [3] S. B. Ai, W. Z. Huang and Z. A. Wang, Traveling wave solutions to a chemotaxis system with logistic growth,, preprint, (2012). Google Scholar [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. Google Scholar [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. Google Scholar [6] D. Balding and D. L. McElwain, A mathematical model of tumour-induced capillary growth,, J. Theor. Biol., 114 (1985), 53. Google Scholar [7] R. Bellman, "Stability Theory of Differential Equations,", McGraw-Hill Book Company, (1953). Google Scholar [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. Google Scholar [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. Google Scholar [10] M. P. Brenner, L. S. Levitor and E. O. Budrene, Physical mechanisms for chemotactic pattern formation by bacterial,, Biophys. J., 74 (1988), 1677. Google Scholar [11] E. Budrene and H. Berg, Complex patterns formed by motile cells of Escherichia coli,, Nature, 349 (1991), 630. Google Scholar [12] E. Budrene and H. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria,, Nature, 376 (1995), 49. Google Scholar [13] S. Childress, Chemotactic collapse in two dimensions,, in, 55 (1984), 61. doi: 10.1007/978-3-642-45589-66. Google Scholar [14] W. A. Coppel, "Stability and Asymptotic Behavior of Differential Equations,", D. C. Heath and Co., (1965). Google Scholar [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. Google Scholar [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. Google Scholar [17] F. W. Dahlquist, P. Lovely and D. E. Jr Koshland, Qualitative analysis of bacterial migration in chemotaxis,, Nature, 236 (1972), 120. Google Scholar [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] N. Fenichel, Geometric singular perturbation theory of ordinary differential equations,, J. Differential Equations, 31 (1979), 53. doi: 10.1016/0022-0396(79)90152-9. Google Scholar [20] R. A. Fisher, The advance of advantageous genes,, Ann. Eugenics, 7 (1937), 355. Google Scholar [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. Google Scholar [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. Google Scholar [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] 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. Google Scholar [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. Google Scholar [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. Google Scholar [50] T. Li and Z.-A. Wang, Steadily propogating waves of a chemotaxis model,, Math. Biosci., 240 (2012), 161. Google Scholar [51] Y. Li, The existence of traveling waves in a biological model for chemotaxis,, Acta Mathematicae Applicatae Sinica (in Chinese), 27 (2004), 123. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [57] M. Meyries, Local well posedness and instability of travelling waves in a chemotaxis model,, Adv. Differential Equations, 16 (2011), 31. Google Scholar [58] J. D. Murray, "Mathematical Biology. I. An Introduction,", Third edition, 17 (2002). Google Scholar [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. Google Scholar [60] T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model,, J. Math. Biol., 30 (1991), 169. doi: 10.1007/BF00160334. Google Scholar [61] W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices Amer. Math. Soc., 1 (1998), 9. Google Scholar [62] R. Nossal, Boundary movement of chemotactic bacterial populations,, Math. Biosci., 13 (1972), 397. Google Scholar [63] A. Novick-Cohen and L. A. Segel, A gradually slowing traveling band of chemotactic bacteria,, J. Math. Biol., 19 (1984), 125. Google Scholar [64] G. M. Odell and E. F. Keller, Traveling bands of chemotactic bacteria revisited,, J. Theor. Biol., 56 (1976), 243. Google Scholar [65] H. Othmer and A. Stevens, Aggregation, blowup and collapse: The ABC's of taxis in reinforced random walks,, SIAM J. Appl. Math., 57 (1997), 1044. doi: 10.1137/S0036139995288976. Google Scholar [66] C. H. Ou and W. Yuan, Traveling wavefronts in a volume-filling chemotaxis model,, SIAM Appl. Dyn. Sys., 8 (2009), 390. doi: 10.1137/08072797X. Google Scholar [67] M. Rascle and C. Ziti, Finite time blow-up in some models of chemotaxis,, J. Math. Biol., 33 (1995), 388. doi: 10.1007/BF00176379. Google Scholar [68] G. Rosen, On the propogation theory for bands of chemotactic bacteria,, Math. Biosci., 20 (1974), 185. Google Scholar [69] G. Rosen, Analytical solution to the initial value problem for traveling bands of chemotactic bacteria,, J. Theor. Biol., 49 (1975), 311. Google Scholar [70] G. Rosen, Existence and nature of band solutions to generic chemotactic transport equations,, J. Theor. Biol., 59 (1976), 243. Google Scholar [71] G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen,, Bull. Math. Biol., 40 (1978), 671. doi: 10.1016/S0092-8240(78)80025-1. Google Scholar [72] G. Rosen, Theoretical significance of the condition $\delta=2\mu$ in bacterial chemotaxis,, Bull. Math. Biol., 45 (1983), 151. Google Scholar [73] G. Rosen and S. Baloga, On the stability of steadily propogating bands of chemotactic bacteria,, Math. Biosci., 24 (1975), 273. Google Scholar [74] J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses,, PLoS Computational Biology, 6 (2010). doi: 10.1371/journal.pcbi.1000890. Google Scholar [75] H. Schwetlick, Traveling waves for chemotaxis systems,, Proc. Appl. Math. Mech., 3 (2003), 476. Google Scholar [76] T. L. Scribner, L. A. Segel and E. H. Rogers, A numerical study of the formation and propogation of traveling bands of chemotactic bacteria,, J. Theor. Biol., 46 (1974), 189. Google Scholar [77] L. A. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis,, SIAM J. Appl. Math., 32 (1977), 653. Google Scholar [78] J. A. Sherratt, E. H. Sage and J. D. Murray, Chemical control of eukaryotic cell movement: A new model,, J. Theor. Biol., 162 (1993), 23. Google Scholar [79] Y. S. Tao, L. H. Wang and Z. A. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension,, Discrete Contin. Dyn. Syst.-Series B, (2012). Google Scholar [80] M. J. Tindall, P. K. Maini, S. L. Porter and J. P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis. II. Bacterial populations,, Bull. Math. Biol., 70 (2008), 1570. doi: 10.1007/s11538-008-9322-5. Google Scholar [81] C. Walker and G. F. Webb, Global existence of classical solutions for a haptoaxis model,, SIAM J. Math. Anal., 38 (): 1694. doi: 10.1137/060655122. Google Scholar [82] Z.-A. Wang, Wavefront of an angiogenesis model,, Discrete Contin. Dyn. Syst.-Series B, 17 (2012), 2849. Google Scholar [83] Z.-A. Wang and T. Hillen, Shock formation in a chemotaxis model,, Math. Methods. Appl. Sci., 31 (2008), 45. doi: 10.1002/mma.898. Google Scholar [84] Z.-A. Wang and K. Zhao, Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model,, Comm. Pure Appl. Anal., (). Google Scholar [85] R. Welch and D. Kaiser, Cell behavior in traveling wave patterns of myxobacteria,, Proc. Natl. Acad. Sci., 98 (2001), 14907. Google Scholar [86] M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity,, Math. Methods Appl. Sci., 34 (2011), 176. doi: 10.1002/mma.1346. Google Scholar [87] Y. P. Wu, Stability of travelling waves for a cross-diffusion model,, J. Math. Anal. Appl, 215 (1997), 388. doi: 10.1006/jmaa.1997.5636. Google Scholar [88] J. Xin, Front propagation in heterogeneous media,, SIAM Review, 42 (2000), 161. doi: 10.1137/S0036144599364296. Google Scholar [89] C. Xue, H. J. Hwang, K. J. Painter and R. Erban, Travelling waves in hyperbolic chemotaxis equations,, Bull. Math. Biol., 73 (2011), 1695. doi: 10.1007/s11538-010-9586-4. Google Scholar [90] M. Zhang and C. J. Zhu, Global existence of solutions to a hyperbolic-parabolic system,, Proc. Amer. Math. Soc., 135 (2006), 1017. doi: 10.1090/S0002-9939-06-08773-9. Google Scholar

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##### References:
 [1] J. Adler, Chemotaxis in bacteria,, Science, 153 (1966), 708. Google Scholar [2] J. Adler, Chemoreceptors in bacteria,, Science, 166 (1969), 1588. Google Scholar [3] S. B. Ai, W. Z. Huang and Z. A. Wang, Traveling wave solutions to a chemotaxis system with logistic growth,, preprint, (2012). Google Scholar [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. Google Scholar [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. Google Scholar [6] D. Balding and D. L. McElwain, A mathematical model of tumour-induced capillary growth,, J. Theor. Biol., 114 (1985), 53. Google Scholar [7] R. Bellman, "Stability Theory of Differential Equations,", McGraw-Hill Book Company, (1953). Google Scholar [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. Google Scholar [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. Google Scholar [10] M. P. Brenner, L. S. Levitor and E. O. Budrene, Physical mechanisms for chemotactic pattern formation by bacterial,, Biophys. J., 74 (1988), 1677. Google Scholar [11] E. Budrene and H. Berg, Complex patterns formed by motile cells of Escherichia coli,, Nature, 349 (1991), 630. Google Scholar [12] E. Budrene and H. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria,, Nature, 376 (1995), 49. Google Scholar [13] S. Childress, Chemotactic collapse in two dimensions,, in, 55 (1984), 61. doi: 10.1007/978-3-642-45589-66. Google Scholar [14] W. A. Coppel, "Stability and Asymptotic Behavior of Differential Equations,", D. C. Heath and Co., (1965). Google Scholar [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. Google Scholar [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. Google Scholar [17] F. W. Dahlquist, P. Lovely and D. E. Jr Koshland, Qualitative analysis of bacterial migration in chemotaxis,, Nature, 236 (1972), 120. Google Scholar [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] N. Fenichel, Geometric singular perturbation theory of ordinary differential equations,, J. Differential Equations, 31 (1979), 53. doi: 10.1016/0022-0396(79)90152-9. Google Scholar [20] R. A. Fisher, The advance of advantageous genes,, Ann. Eugenics, 7 (1937), 355. Google Scholar [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. Google Scholar [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. Google Scholar [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] 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. Google Scholar [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. Google Scholar [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. Google Scholar [50] T. Li and Z.-A. Wang, Steadily propogating waves of a chemotaxis model,, Math. Biosci., 240 (2012), 161. Google Scholar [51] Y. Li, The existence of traveling waves in a biological model for chemotaxis,, Acta Mathematicae Applicatae Sinica (in Chinese), 27 (2004), 123. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [57] M. Meyries, Local well posedness and instability of travelling waves in a chemotaxis model,, Adv. Differential Equations, 16 (2011), 31. Google Scholar [58] J. D. Murray, "Mathematical Biology. I. An Introduction,", Third edition, 17 (2002). Google Scholar [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. Google Scholar [60] T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model,, J. Math. Biol., 30 (1991), 169. doi: 10.1007/BF00160334. Google Scholar [61] W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices Amer. Math. Soc., 1 (1998), 9. Google Scholar [62] R. Nossal, Boundary movement of chemotactic bacterial populations,, Math. Biosci., 13 (1972), 397. Google Scholar [63] A. Novick-Cohen and L. A. Segel, A gradually slowing traveling band of chemotactic bacteria,, J. Math. Biol., 19 (1984), 125. Google Scholar [64] G. M. Odell and E. F. Keller, Traveling bands of chemotactic bacteria revisited,, J. Theor. Biol., 56 (1976), 243. Google Scholar [65] H. Othmer and A. Stevens, Aggregation, blowup and collapse: The ABC's of taxis in reinforced random walks,, SIAM J. Appl. Math., 57 (1997), 1044. doi: 10.1137/S0036139995288976. Google Scholar [66] C. H. Ou and W. Yuan, Traveling wavefronts in a volume-filling chemotaxis model,, SIAM Appl. Dyn. Sys., 8 (2009), 390. doi: 10.1137/08072797X. Google Scholar [67] M. Rascle and C. Ziti, Finite time blow-up in some models of chemotaxis,, J. Math. Biol., 33 (1995), 388. doi: 10.1007/BF00176379. Google Scholar [68] G. Rosen, On the propogation theory for bands of chemotactic bacteria,, Math. Biosci., 20 (1974), 185. Google Scholar [69] G. Rosen, Analytical solution to the initial value problem for traveling bands of chemotactic bacteria,, J. Theor. Biol., 49 (1975), 311. Google Scholar [70] G. Rosen, Existence and nature of band solutions to generic chemotactic transport equations,, J. Theor. Biol., 59 (1976), 243. Google Scholar [71] G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen,, Bull. Math. Biol., 40 (1978), 671. doi: 10.1016/S0092-8240(78)80025-1. Google Scholar [72] G. Rosen, Theoretical significance of the condition $\delta=2\mu$ in bacterial chemotaxis,, Bull. Math. Biol., 45 (1983), 151. Google Scholar [73] G. Rosen and S. Baloga, On the stability of steadily propogating bands of chemotactic bacteria,, Math. Biosci., 24 (1975), 273. Google Scholar [74] J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses,, PLoS Computational Biology, 6 (2010). doi: 10.1371/journal.pcbi.1000890. Google Scholar [75] H. Schwetlick, Traveling waves for chemotaxis systems,, Proc. Appl. Math. Mech., 3 (2003), 476. Google Scholar [76] T. L. Scribner, L. A. Segel and E. H. Rogers, A numerical study of the formation and propogation of traveling bands of chemotactic bacteria,, J. Theor. Biol., 46 (1974), 189. Google Scholar [77] L. A. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis,, SIAM J. Appl. Math., 32 (1977), 653. Google Scholar [78] J. A. Sherratt, E. H. Sage and J. D. Murray, Chemical control of eukaryotic cell movement: A new model,, J. Theor. Biol., 162 (1993), 23. Google Scholar [79] Y. S. Tao, L. H. Wang and Z. A. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension,, Discrete Contin. Dyn. Syst.-Series B, (2012). Google Scholar [80] M. J. Tindall, P. K. Maini, S. L. Porter and J. P. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis. II. Bacterial populations,, Bull. Math. Biol., 70 (2008), 1570. doi: 10.1007/s11538-008-9322-5. Google Scholar [81] C. Walker and G. F. Webb, Global existence of classical solutions for a haptoaxis model,, SIAM J. Math. Anal., 38 (): 1694. doi: 10.1137/060655122. Google Scholar [82] Z.-A. Wang, Wavefront of an angiogenesis model,, Discrete Contin. Dyn. Syst.-Series B, 17 (2012), 2849. Google Scholar [83] Z.-A. Wang and T. Hillen, Shock formation in a chemotaxis model,, Math. Methods. Appl. Sci., 31 (2008), 45. doi: 10.1002/mma.898. Google Scholar [84] Z.-A. Wang and K. Zhao, Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model,, Comm. Pure Appl. Anal., (). Google Scholar [85] R. Welch and D. Kaiser, Cell behavior in traveling wave patterns of myxobacteria,, Proc. Natl. Acad. Sci., 98 (2001), 14907. Google Scholar [86] M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity,, Math. Methods Appl. Sci., 34 (2011), 176. doi: 10.1002/mma.1346. Google Scholar [87] Y. P. Wu, Stability of travelling waves for a cross-diffusion model,, J. Math. Anal. Appl, 215 (1997), 388. doi: 10.1006/jmaa.1997.5636. Google Scholar [88] J. Xin, Front propagation in heterogeneous media,, SIAM Review, 42 (2000), 161. doi: 10.1137/S0036144599364296. Google Scholar [89] C. Xue, H. J. Hwang, K. J. Painter and R. Erban, Travelling waves in hyperbolic chemotaxis equations,, Bull. Math. Biol., 73 (2011), 1695. doi: 10.1007/s11538-010-9586-4. Google Scholar [90] M. Zhang and C. J. Zhu, Global existence of solutions to a hyperbolic-parabolic system,, Proc. Amer. Math. Soc., 135 (2006), 1017. doi: 10.1090/S0002-9939-06-08773-9. Google Scholar
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