December  2019, 24(12): 6465-6480. doi: 10.3934/dcdsb.2019147

Traveling waves in a chemotaxis model with logistic growth

Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA

* Corresponding author: Jeungeun Park

Received  November 2018 Revised  January 2019 Published  December 2019 Early access  July 2019

Traveling wave solutions of a chemotaxis model with a reaction term are studied. We investigate the existence and non-existence of traveling wave solutions in certain ranges of parameters. Particularly for a positive rate of chemical growth, we prove the existence of a heteroclinic orbit by constructing a positively invariant set in the three dimensional space. The monotonicity of traveling waves is also analyzed in terms of chemotaxis, reaction and diffusion parameters. Finally, the traveling wave solutions are shown to be linearly unstable.

Citation: Tong Li, Jeungeun Park. Traveling waves in a chemotaxis model with logistic growth. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6465-6480. doi: 10.3934/dcdsb.2019147
References:
[1]

J. Adler, Chemotaxis in bacteria, Science, 40 (1975), 341-356. 

[2]

J. Adler, Chemoreceptors in bacteria, Science, 166 (1969), 1588-1597.  doi: 10.1126/science.166.3913.1588.

[3]

S. AiW. Huang and Z. A. Wang, Reaction, diffusion and chemotaxis in wave propagation, Discrete Contin. Dyn. Syst. Series B, 20 (2015), 1-21.  doi: 10.3934/dcdsb.2015.20.1.

[4]

S. Ai and Z. A. Wang, Traveling bands for the Keller-Segel model with population growth, Math. Biosci. Eng., 12 (2015), 717-737.  doi: 10.3934/mbe.2015.12.717.

[5]

F. R. Gantmacher, The Theory of Matrices, Vols. 1, 2. Translated by K. A. Hirsch Chelsea Publishing Co., New York, 1959.

[6]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Vol. 180, Springer-Verlag, New York/Berlin, 1981. doi: 10.1007/BFb0089647.

[7]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[8]

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

[9]

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.

[10]

H.-Y. Jin, J. Li and Z.-A. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differential Equations, 255 (2013), 193–219. doi: 10.1016/j.jde.2013.04.002.

[11]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.

[12]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theoret. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.

[13]

A. KolmogorovI. Petrovskii and N. Piskunov, Study of a diffusion equation that is related to the growth of a quality of matter and its application to a biological problem, Moscow University Mathematics Bulletin, 1 (1937), 1-26. 

[14]

T. LiH. Liu and L. Wang, Oscillatory traveling wave solutions to an attractive chemotaxis system, J. Differential Equations, 261 (2016), 7080-7098.  doi: 10.1016/j.jde.2016.09.012.

[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), 1522-1541.  doi: 10.1137/09075161X.

[16]

R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, J. Math. Biol., 61 (2010), 739-361.  doi: 10.1007/s00285-009-0317-0.

[17]

G. NadinB. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with Fisher birth terms, Interfaces Free Bound, 10 (2008), 517-538.  doi: 10.4171/IFB/200.

[18]

T. Nagai and T. Ikeda, Traveling waves in a chemotactic model, J. Math. Biol., 30 (1991), 169-184.  doi: 10.1007/BF00160334.

[19]

R. B. Salako and W. Shen, Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^{N}$, Discrete Contin. Dyn. Syst. Series A, 37 (2017), 6189-6225.  doi: 10.3934/dcds.2017268.

[20]

R. B. Salako and W. Shen, Existence of traveling wave solution of parabolic-parabolic chemotaxis systems, Nonlinear Anal.: Real World Appl., 42 (2018), 93-119.  doi: 10.1016/j.nonrwa.2017.12.004.

[21]

D. G. Schaeffer and J. W. Cain, Ordinary Differential Equations: Basics and Beyond, Volume 65 of Texts in Applied Mathematics, Springer, 2016. doi: 10.1007/978-1-4939-6389-8.

[22]

H. Schwetlick, Traveling waves for chemotaxis systems, Proc. Appl. Math. Mech., 3 (2003), 476-478.  doi: 10.1002/pamm.200310508.

[23]

Z. A. Wang, Mathematics of traveling waves in chemotaxis -Review paper, Discrete Contin. Dyn. Syst. Series B, 18 (2013), 601-641.  doi: 10.3934/dcdsb.2013.18.601.

[24]

C. XueH. J. HwangK. J. Painter and R. Erban, Travelling waves in hyperbolic chemotaxis equations, Bull. Math. Biol., 73 (2011), 1695-1733.  doi: 10.1007/s11538-010-9586-4.

show all references

References:
[1]

J. Adler, Chemotaxis in bacteria, Science, 40 (1975), 341-356. 

[2]

J. Adler, Chemoreceptors in bacteria, Science, 166 (1969), 1588-1597.  doi: 10.1126/science.166.3913.1588.

[3]

S. AiW. Huang and Z. A. Wang, Reaction, diffusion and chemotaxis in wave propagation, Discrete Contin. Dyn. Syst. Series B, 20 (2015), 1-21.  doi: 10.3934/dcdsb.2015.20.1.

[4]

S. Ai and Z. A. Wang, Traveling bands for the Keller-Segel model with population growth, Math. Biosci. Eng., 12 (2015), 717-737.  doi: 10.3934/mbe.2015.12.717.

[5]

F. R. Gantmacher, The Theory of Matrices, Vols. 1, 2. Translated by K. A. Hirsch Chelsea Publishing Co., New York, 1959.

[6]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Vol. 180, Springer-Verlag, New York/Berlin, 1981. doi: 10.1007/BFb0089647.

[7]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[8]

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

[9]

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.

[10]

H.-Y. Jin, J. Li and Z.-A. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differential Equations, 255 (2013), 193–219. doi: 10.1016/j.jde.2013.04.002.

[11]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.

[12]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theoret. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.

[13]

A. KolmogorovI. Petrovskii and N. Piskunov, Study of a diffusion equation that is related to the growth of a quality of matter and its application to a biological problem, Moscow University Mathematics Bulletin, 1 (1937), 1-26. 

[14]

T. LiH. Liu and L. Wang, Oscillatory traveling wave solutions to an attractive chemotaxis system, J. Differential Equations, 261 (2016), 7080-7098.  doi: 10.1016/j.jde.2016.09.012.

[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), 1522-1541.  doi: 10.1137/09075161X.

[16]

R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, J. Math. Biol., 61 (2010), 739-361.  doi: 10.1007/s00285-009-0317-0.

[17]

G. NadinB. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with Fisher birth terms, Interfaces Free Bound, 10 (2008), 517-538.  doi: 10.4171/IFB/200.

[18]

T. Nagai and T. Ikeda, Traveling waves in a chemotactic model, J. Math. Biol., 30 (1991), 169-184.  doi: 10.1007/BF00160334.

[19]

R. B. Salako and W. Shen, Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^{N}$, Discrete Contin. Dyn. Syst. Series A, 37 (2017), 6189-6225.  doi: 10.3934/dcds.2017268.

[20]

R. B. Salako and W. Shen, Existence of traveling wave solution of parabolic-parabolic chemotaxis systems, Nonlinear Anal.: Real World Appl., 42 (2018), 93-119.  doi: 10.1016/j.nonrwa.2017.12.004.

[21]

D. G. Schaeffer and J. W. Cain, Ordinary Differential Equations: Basics and Beyond, Volume 65 of Texts in Applied Mathematics, Springer, 2016. doi: 10.1007/978-1-4939-6389-8.

[22]

H. Schwetlick, Traveling waves for chemotaxis systems, Proc. Appl. Math. Mech., 3 (2003), 476-478.  doi: 10.1002/pamm.200310508.

[23]

Z. A. Wang, Mathematics of traveling waves in chemotaxis -Review paper, Discrete Contin. Dyn. Syst. Series B, 18 (2013), 601-641.  doi: 10.3934/dcdsb.2013.18.601.

[24]

C. XueH. J. HwangK. J. Painter and R. Erban, Travelling waves in hyperbolic chemotaxis equations, Bull. Math. Biol., 73 (2011), 1695-1733.  doi: 10.1007/s11538-010-9586-4.

Figure 1.  Sketch of the compact sets: $ \mathcal{P} $ in (A) and $ \mathcal{T} $ in (B)
Figure 2.  Numerical simulations of traveling wave solutions $ (U(\xi), V(\xi)) $ of the system (1). For (A), $ D = 1, s = 2, \beta = 0.2, \mu = 1 $ and $ \chi(v) = \cos(10 v) - \sin(20 v) +2 $. For (B), $ D = 1, s = 2, \beta = 1, \mu = \frac{1}{2} $ and $ \chi(v) = \frac{1}{(1+v)^{2}}. $
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