January  2015, 20(1): 1-21. doi: 10.3934/dcdsb.2015.20.1

Reaction, diffusion and chemotaxis in wave propagation

1. 

Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899, United States

2. 

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

Received  June 2014 Published  November 2014

By constructing an invariant set in the three dimensional space, we establish the existence of traveling wave solutions to a reaction-diffusion-chemotaxis model describing biological processes such as the bacterial chemotactic movement in response to oxygen and the initiation of angiogenesis. The minimal wave speed is shown to exist and the role of each process of reaction, diffusion and chemotaxis in the wave propagation is investigated. Our results reveal three essential biological implications: (1) the cell growth increases the wave speed; (2) the chemotaxis must be strong enough to make a contribution to the increment of the wave speed; (3) the diffusion rate plays a role in increasing the wave speed only when the cell growth is present.
Citation: Shangbing Ai, Wenzhang Huang, Zhi-An Wang. Reaction, diffusion and chemotaxis in wave propagation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 1-21. doi: 10.3934/dcdsb.2015.20.1
References:
[1]

J. Adler, Chemotaxis in bacteria,, Annual Review of Biochemistry, 44 (1975), 341.  doi: 10.1146/annurev.bi.44.070175.002013.  Google Scholar

[2]

J. Adler, Chemoreceptors in bacteria,, Science, 166 (1969), 1588.  doi: 10.1126/science.166.3913.1588.  Google Scholar

[3]

F. S. Berezovskaya, A. S. Novozhilov and G. P. Karev, Families of traveling impulse 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

[4]

M. A. J. Chaplain, Avascular growth, angiogenesis and vascular growth in solid tumors: The mathamatical modeling of the stages of tumor development,, Math. Comput. Modeling, 23 (1996), 47.   Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis,, J. Theor. Biol., 30 (1971), 235.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

[11]

H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton, Mathematical modeling of the onset of capillary formation initiating angiogenesis,, J. Math. Biol., 42 (2001), 195.  doi: 10.1007/s002850000037.  Google Scholar

[12]

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

[13]

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

[14]

T. Li, R. H. Pan and K. Zhao, Global dynamics of a chemotaxis model on bounded domains with large data,, SIAM J. Appl. Math., 72 (2012), 417.  doi: 10.1137/110829453.  Google Scholar

[15]

T. Li and Z. A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,, SIAM J. Appl. Math., 70 (2009), 1522.  doi: 10.1137/09075161X.  Google Scholar

[16]

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

[17]

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

[18]

M. Meyries, Local well posedness and instability of travelling waves in a chemotaxis model,, Adv. Differential Equations, 16 (2011), 31.   Google Scholar

[19]

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

[20]

T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model,, J. Math. Biol., 30 (1991), 169.  doi: 10.1007/BF00160334.  Google Scholar

[21]

, National Cancer Institute,, , ().   Google Scholar

[22]

R. Nossal, Boundary movement of chemotactic bacterial population,, Math. Biosci., 13 (1972), 397.  doi: 10.1016/0025-5564(72)90058-2.  Google Scholar

[23]

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

[24]

K. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model,, Physica D: Nonlinear Phenomena, 240 (2011), 363.  doi: 10.1016/j.physd.2010.09.011.  Google Scholar

[25]

G. Rosen, Analytically solution to the initial-value problem for traveling bands of chemotaxis bacteria,, J. Theor. Biol., 49 (1975), 311.   Google Scholar

[26]

G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen,, Bull. Math. Biol., 40 (1978), 671.  doi: 10.1007/BF02460738.  Google Scholar

[27]

G. Rosen, Theoretical significance of the condition $\delta=2 \mu$ in bacterical chemotaxis,, Bull. Math. Biol., 45 (1983), 151.   Google Scholar

[28]

G. Rosen and S. Baloga, On the stability of steadily propogating bands of chemotactic bacteria,, Math. Biosci., 24 (1975), 273.  doi: 10.1016/0025-5564(75)90080-2.  Google Scholar

[29]

H. Schwetlick, Traveling waves for chemotaxis systems,, Proc. Appl. Math. Mech., 3 (2003), 476.  doi: 10.1002/pamm.200310508.  Google Scholar

[30]

Y. S. Tao, L. H. Wang and Z. A. Wang, Long-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension,, Discrete Cont. Dyn. Syst.-Seris B, 18 (2013), 821.  doi: 10.3934/dcdsb.2013.18.821.  Google Scholar

[31]

C. Walker and G. F. Webb, Global existence of classical solutions for a haptoaxis model,, SIAM J. Math. Anal., 38 (2006), 1694.  doi: 10.1137/060655122.  Google Scholar

[32]

Z. A. Wang, Wavefront of an angiogenesis model,, Discrete Cont. Dyn. Syst.-Series B, 17 (2012), 2849.  doi: 10.3934/dcdsb.2012.17.2849.  Google Scholar

[33]

Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model,, Chaos, 17 (2007).  doi: 10.1063/1.2766864.  Google Scholar

[34]

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

[35]

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

show all references

References:
[1]

J. Adler, Chemotaxis in bacteria,, Annual Review of Biochemistry, 44 (1975), 341.  doi: 10.1146/annurev.bi.44.070175.002013.  Google Scholar

[2]

J. Adler, Chemoreceptors in bacteria,, Science, 166 (1969), 1588.  doi: 10.1126/science.166.3913.1588.  Google Scholar

[3]

F. S. Berezovskaya, A. S. Novozhilov and G. P. Karev, Families of traveling impulse 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

[4]

M. A. J. Chaplain, Avascular growth, angiogenesis and vascular growth in solid tumors: The mathamatical modeling of the stages of tumor development,, Math. Comput. Modeling, 23 (1996), 47.   Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis,, J. Theor. Biol., 30 (1971), 235.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

[11]

H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton, Mathematical modeling of the onset of capillary formation initiating angiogenesis,, J. Math. Biol., 42 (2001), 195.  doi: 10.1007/s002850000037.  Google Scholar

[12]

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

[13]

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

[14]

T. Li, R. H. Pan and K. Zhao, Global dynamics of a chemotaxis model on bounded domains with large data,, SIAM J. Appl. Math., 72 (2012), 417.  doi: 10.1137/110829453.  Google Scholar

[15]

T. Li and Z. A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,, SIAM J. Appl. Math., 70 (2009), 1522.  doi: 10.1137/09075161X.  Google Scholar

[16]

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

[17]

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

[18]

M. Meyries, Local well posedness and instability of travelling waves in a chemotaxis model,, Adv. Differential Equations, 16 (2011), 31.   Google Scholar

[19]

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

[20]

T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model,, J. Math. Biol., 30 (1991), 169.  doi: 10.1007/BF00160334.  Google Scholar

[21]

, National Cancer Institute,, , ().   Google Scholar

[22]

R. Nossal, Boundary movement of chemotactic bacterial population,, Math. Biosci., 13 (1972), 397.  doi: 10.1016/0025-5564(72)90058-2.  Google Scholar

[23]

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

[24]

K. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model,, Physica D: Nonlinear Phenomena, 240 (2011), 363.  doi: 10.1016/j.physd.2010.09.011.  Google Scholar

[25]

G. Rosen, Analytically solution to the initial-value problem for traveling bands of chemotaxis bacteria,, J. Theor. Biol., 49 (1975), 311.   Google Scholar

[26]

G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen,, Bull. Math. Biol., 40 (1978), 671.  doi: 10.1007/BF02460738.  Google Scholar

[27]

G. Rosen, Theoretical significance of the condition $\delta=2 \mu$ in bacterical chemotaxis,, Bull. Math. Biol., 45 (1983), 151.   Google Scholar

[28]

G. Rosen and S. Baloga, On the stability of steadily propogating bands of chemotactic bacteria,, Math. Biosci., 24 (1975), 273.  doi: 10.1016/0025-5564(75)90080-2.  Google Scholar

[29]

H. Schwetlick, Traveling waves for chemotaxis systems,, Proc. Appl. Math. Mech., 3 (2003), 476.  doi: 10.1002/pamm.200310508.  Google Scholar

[30]

Y. S. Tao, L. H. Wang and Z. A. Wang, Long-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension,, Discrete Cont. Dyn. Syst.-Seris B, 18 (2013), 821.  doi: 10.3934/dcdsb.2013.18.821.  Google Scholar

[31]

C. Walker and G. F. Webb, Global existence of classical solutions for a haptoaxis model,, SIAM J. Math. Anal., 38 (2006), 1694.  doi: 10.1137/060655122.  Google Scholar

[32]

Z. A. Wang, Wavefront of an angiogenesis model,, Discrete Cont. Dyn. Syst.-Series B, 17 (2012), 2849.  doi: 10.3934/dcdsb.2012.17.2849.  Google Scholar

[33]

Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model,, Chaos, 17 (2007).  doi: 10.1063/1.2766864.  Google Scholar

[34]

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

[35]

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

[1]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[2]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[3]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[4]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[5]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[6]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[7]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215

[8]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466

[9]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[10]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[11]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218

[12]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[13]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[14]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426

[15]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

[16]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273

[17]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075

[18]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[19]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252

[20]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (66)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]