Global Attractor of a Coupled Two-Cell Brusselator Model

Chapter
Part of the Fields Institute Communications book series (FIC, volume 64)

Abstract

In this work the existence of a global attractor for the solution semiflow of the coupled two-cell Brusselator model equations is proved. A grouping estimation method and a new decomposition approach are introduced to deal with the challenges in proving the absorbing property and the asymptotic compactness of this type of four-variable reaction-diffusion systems with cubic autocatalytic nonlinearity and with linear coupling. It is also proved that the Hausdorff dimension and the fractal dimension of the global attractor are finite.

References

  1. 1.
    M. Ashkenazi, H.G. Othmer, Spatial patterns in coupled biochemical oscillators. J. Math. Biology 5, 305–350 (1978)MathSciNetMATHGoogle Scholar
  2. 2.
    J.F.G. Auchmuty, G. Nicolis, Bifurcation analysis of nonlinear reaction-diffusion equations I: Evolution equations and the steady state solutions. Bull. Math. Biol. 37, 323–365 (1975)MathSciNetMATHGoogle Scholar
  3. 3.
    A.Yu. Berezin, A. Gainoval, Yu.G. Matushkin, V.A. Likhoshval, S.I. Fadeev, Numerical study of mathematical models described dynamics of gene nets functioning: software package STEP, BGRS 2000, 243–245Google Scholar
  4. 4.
    K.J. Brown, F.A. Davidson, Global bifurcation in the Brusselator system. Nonlinear Anal. 24, 1713–1725 (1995)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    X. Chen, Y. Qi, Sharp estimates on minimum traveling wave speed of reaction-diffusion systems modeling autocatalysis. SIAM J. Math. Anal. 39, 437–448 (2007)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    V.V. Chepyzhov, M.I. Vishik, Attractors for Equations of Mathematical Physics, vol. 49 (AMS Colloquium Publications, Providence, 2002)MATHGoogle Scholar
  7. 7.
    G. Dangelmayr, Degenerate bifurcation near a double eigenvalue in the Brusselator. J. Austral. Math. Soc. Ser. B 28, 486–535 (1987)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    A. Doelman, T.J. Kaper, P.A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model. Nonlinearity 10, 523–563 (1997)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    I.R. Epstein, Complex dynamical behavior in simple chemical systems. J. Phys. Chemistry 88, 187–198 (1984)CrossRefGoogle Scholar
  10. 10.
    T. Erneux, E. Reiss, Brusselator isolas. SIAM J. Appl. Math. 43, 1240–1246 (1983)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Z. Fu, X. Xu, H. Wang, Q. Ouyang, Stochastic simulation of Turing patterns. Chin. Phys. Lett. 25, 1220–1223 (2008)CrossRefGoogle Scholar
  12. 12.
    P. Gormley, K. Li, G.W. Irwin, Modeling molecular interaction pathways using a two-stage identification algorithm. Syst. Synthetic Biol. 1, 145–160 (2007)CrossRefGoogle Scholar
  13. 13.
    P. Gray, S.K. Scott, Autocatalytic reactions in the isothermal continuous stirred tank reactor: isolas and other forms of multistability. Chem. Eng. Sci. 38, 29–43 (1983)CrossRefGoogle Scholar
  14. 14.
    P. Gray, S.K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilities in the system A + 2B → 3B, B → C. Chem. Eng. Sci. 39, 1087–1097 (1984)Google Scholar
  15. 15.
    J.K. Hale, Asymptotic Behavior of Dissipative Systems. (Amer. Math. Soc., Providence, 1988)Google Scholar
  16. 16.
    S. Ishihara, K. Kanedo, Turing pattern with proportion preservation. J. Theor. Biol. 238, 683–693 (2006)CrossRefGoogle Scholar
  17. 17.
    I. Karafyllis, P.D. Christofides, P. Daoutidis, Dynamical analysis of a reaction-diffusion system with Brusselator kinetics under feedback control. Proc. Amer. Control Conference, Albuquerque, NM, June 1997, pp. 2213–2217Google Scholar
  18. 18.
    M. Kawato, R. Suzuki, Two coupled neural oscillators as a model of the circadian pacemaker. J. Theor. Biol. 86, 547–575 (1980)MathSciNetCrossRefGoogle Scholar
  19. 19.
    H. Kitano, Systems biology: a brief overview. Science 295, 1662–1664 (2002)Google Scholar
  20. 20.
    A. Klic̆, Period doubling bifurcations in a two-box model of the Brusselator. Aplikace Matematiky 28, 335–343 (1983)Google Scholar
  21. 21.
    T. Kolokolnikov, T. Erneux, J. Wei, Mesa-type patterns in one-dimensional Brusselator and their stability. Physica D 214(1), 63–77 (2006)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    K.J. Lee, W.D. McCormick, Q. Ouyang, H. Swinney, Pattern formation by interacting chemical fronts. Science 261, 192–194 (1993)CrossRefGoogle Scholar
  23. 23.
    J.E. Pearson, Complex patterns in a simple system. Science 261, 189–192 (1993)CrossRefGoogle Scholar
  24. 24.
    R. Peng, M. Wang, Pattern formation in the Brusselator system. J. Math. Anal. Appl. 309, 151–166 (2005)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    I. Prigogine, R. Lefever, Symmetry-breaking instabilities in dissipative systems. J. Chem. Phys. 48, 1665–1700 (1968)CrossRefGoogle Scholar
  26. 26.
    B. Peña, C. Pérez-García, Stability of Turing patterns in the Brusselator model. Phys. Review E 64(5), (2001)Google Scholar
  27. 27.
    Y. Qi, The development of traveling waves in cubic auto-catalysis with different rates of diffusion. Physica D 226, 129–135 (2007)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    T. Rauber, G. Runger, Aspects of a distributed solution of the Brusselator equation. Proc. of the First Aizu International Symposium on Parallel Algorithms and Architecture Syntheses, 114–120, 1995Google Scholar
  29. 29.
    W. Reynolds, J.E. Pearson, S. Ponce-Dawson, Dynamics of self-replicating patterns in reaction-diffusion systems. Phys. Rev. E 56, 185–198 (1997)MathSciNetCrossRefGoogle Scholar
  30. 30.
    J. Schnackenberg, Simple chemical reaction systems with limit cycle behavior. J. Theor. Biol. 81, 389–400 (1979)CrossRefGoogle Scholar
  31. 31.
    I. Schreiber, M. Marek, Strange attractors in coupled reaction-diffusion cells. Physica D 5, 258–272 (1982)MathSciNetCrossRefGoogle Scholar
  32. 32.
    S.K. Scott, K. Showalter, Simple and complex reaction-diffusion fronts, in Chemical Waves and Patterns, ed. by R. Kapral, K. Showalter (Kluwer Acad. Publ., Dordrecht, 1995), pp. 485–516CrossRefGoogle Scholar
  33. 33.
    E.E. Sel’kov, Self-oscillations in glycolysis: a simple kinetic model. Euro. J. Biochem. 4, 79–86 (1968)Google Scholar
  34. 34.
    G.R. Sell, Y. You, Dynamics of Evolutionary Equations (Springer, New York, 2002)MATHGoogle Scholar
  35. 35.
    S.H. Strogatz, Nonlinear Dynamics and Chaos (Westview Press, 1994)Google Scholar
  36. 36.
    R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics (Springer, New York, 1988)MATHCrossRefGoogle Scholar
  37. 37.
    J.J. Tyson, K. Chen, B. Novak, Network dynamics and cell physiology. Nature Rev. Mol. Cell Biol. 2, 908–916 (2001)CrossRefGoogle Scholar
  38. 38.
    J. Wei, M. Winter, Asymmetric spotty patterns for the Gray-Scott model in 2. Stud. Appl. math. 110, 63–102 (2003)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    A. De Wit, D. Lima, G. Dewel, P. Borckmans, Spatiotemporal dynamics near a codimension-two point. Phys. Review E 54(1), (1996)Google Scholar
  40. 40.
    L. Yang, A.M. Zhabotinsky, I.R. Epstein, Stable squares and other oscillatory Turing patterns in a reaction-diffusion model. Phys. Rev. Lett. 92, 198303:1–4 (2004)Google Scholar
  41. 41.
    Y. You, Global dynamics of the Brusselator equations. Dynamics of PDE 4, 167–196 (2007)MATHGoogle Scholar
  42. 42.
    Y. You, Global attractor of the Gray-Scott equations. Comm. Pure Appl. Anal. 7, 947–970 (2008)MATHCrossRefGoogle Scholar
  43. 43.
    Y. You, Asymptotical dynamics of Selkov equations. Discrete Continuous Dyn. Syst. Ser. S 2, 193–219 (2009)MATHCrossRefGoogle Scholar
  44. 44.
    Y. You, Asymptotical dynamics of the modified Schnackenberg equations. Discrete and Continuous Dynamical Systems–Supplement, 857–868 (2009)Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA

Personalised recommendations