Advertisement

BIT Numerical Mathematics

, Volume 46, Supplement 1, pp 85–97 | Cite as

A Lyapunov function for a two-chemical species version of the chemotaxis model

  • V. Calvez
  • B. PerthameEmail author
Article

Abstract

We answer partially the question of global existence for a chemotaxis model involving two chemical species: a chemo-attractant and a stimulant. We introduce a Lyapunov function for this system and we show that it is non-decreasing assuming a family of threshold conditions.

Key words

chemotaxis bacterial motility global existence Lyapunov function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. R. A. Anderson and M. A. J. Chaplain, A mathematical model for capillary network formation in the absence of endothelial cell proliferation, Appl. Math. Lett., 11 (1998), pp. 109–114.CrossRefzbMATHGoogle Scholar
  2. 2.
    W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), pp. 147–177.CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    E. O. Budrene and H. C. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), pp. 49–53.CrossRefGoogle Scholar
  4. 4.
    P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), pp. 347–359.MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. Blanchet, J. Dolbeault, and B. Perthame, Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differ. Equ., Vol. 2006 (2006), No. 44, pp. 1–32.Google Scholar
  6. 6.
    M. P. Brenner, L. S. Levitov, and E. O. Budrene, Physical mechanisms for chemotactic pattern formation by bacteria, Biophys. J., 74 (1998), pp. 1677–1693.CrossRefGoogle Scholar
  7. 7.
    V. Calvez and J. A. Carrillo, Volume effects in the Keller–Segel model: energy estimates preventing blow-up, J. Math. Pures Appl., to appear.Google Scholar
  8. 8.
    L. Corrias, B. Perthame, and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R., Math., Acad. Sci. Paris, 336 (2003), pp. 141–146.MathSciNetzbMATHGoogle Scholar
  9. 9.
    L. Corrias, B. Perthame, and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), pp. 1–28.CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    L. Corrias and B. Perthame, Critical space for the parabolic-parabolic Keller–Segel model C. R. Acad. Sci., Paris, Sér. I, 342 (2006), pp. 747–750.Google Scholar
  11. 11.
    J. Dolbeault and B. Perthame, Optimal critical mass in the two dimensional Keller–Segel model in \(\mathbb{R}^2\), C. R., Math., Acad. Sci. Paris, 339 (2004), pp. 611–616.MathSciNetzbMATHGoogle Scholar
  12. 12.
    F. Filbet, P. Laurencot, and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2004), pp. 189–207.CrossRefMathSciNetGoogle Scholar
  13. 13.
    E. Gagliardo, Ulteriori proprietà di alcune classi di funzioni in più variabili, Ric. Mat., 8 (1959), pp. 24–51.MathSciNetzbMATHGoogle Scholar
  14. 14.
    H. Gajewski and K. Zacharias, Global behavior of a reaction–diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), pp. 77–114.MathSciNetzbMATHGoogle Scholar
  15. 15.
    A. Gamba, D. Ambrosi, A. Coniglio, A. de Candia, S. Di Talia, E. Giraudo, G. Serini, L. Preziosi, and F. Bussolino, Percolation, morphogenesis, and Burgers dynamics in blood vessels formation, Phys. Rev. Lett., 90 (2003) 118101.Google Scholar
  16. 16.
    D. Horstmann, From 1970 until present: the Keller–Segel model in chemotaxis and its consequences, I, Jahresber. Dtsch. Math.-Ver., 105 (2003), pp. 103–165.MathSciNetzbMATHGoogle Scholar
  17. 17.
    T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), pp. 280–301.CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    T. Höfer, J. A. Sherratt, and P. K. Maini, Cellular pattern formation in a model of Dictyostelium aggregation, Physica D, 85 (1995), pp. 425–444.CrossRefzbMATHGoogle Scholar
  19. 19.
    W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), pp. 819–824.CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), pp. 566–588.CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), pp. 225–234.CrossRefGoogle Scholar
  22. 22.
    E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: a theoretical analysis, J. Theor. Biol., 30 (1971), pp. 235–248.CrossRefGoogle Scholar
  23. 23.
    H. A. Levine, S. Pamuk, B. D. Sleeman, and M. Nilsen-Hamilton, Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma, Bull. Math. Biol., 63 (2001), pp. 801–863.CrossRefGoogle Scholar
  24. 24.
    A. Marrocco, 2D simulation of chemotactic bacteria aggregation, ESAIM, Math. Model. Numer. Anal., 37 (2003), pp. 617–630.CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    J. D. Murray, Mathematical Biology, 3rd edn., Springer, New York, 2003.Google Scholar
  26. 26.
    N. V. Mantzaris, S. Webb, and H. G. Othmer, Mathematical modeling of tumor-induced angiogenesis, J. Math. Biol., 49 (2004), pp. 111–187.CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), pp. 115–162.MathSciNetGoogle Scholar
  28. 28.
    C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), pp. 311–338.CrossRefMathSciNetGoogle Scholar
  29. 29.
    K. J. Painter, P. K. Maini, and H. G. Othmer, Chemotactic response to multiple signalling cues, J. Math. Biol., 41 (2000), pp. 285–314.CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    G. Serini, D. Ambrosi, E. Giraudo, A. Gamba, L. Preziosi, and F. Bussolino, Modeling the early stages of vascular network assembly, EMBO J., 22 (2003), pp. 1771–1779.CrossRefGoogle Scholar
  31. 31.
    R. Tyson, S. R. Lubkin, and J. D. Murray, A minimal mechanism for bacterial pattern formation, Proc. R. Soc. Lond. B, 266 (1999), pp. 299–304.CrossRefGoogle Scholar
  32. 32.
    J. J. L. Velázquez, Point dynamics in a singular limit of the Keller–Segel model, I., Motion of the concentration regions, SIAM J. Appl. Math., 64 (2004), pp. 1198–1223.CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Département de Mathématiques et Applications, CNRS UMR 8553Ecole Normale SupérieureParisFrance

Personalised recommendations