Acta Mathematica Sinica, English Series

, Volume 34, Issue 6, pp 975–991 | Cite as

Qualitative analysis of a Belousov–Zhabotinskii reaction model

  • Aung Zaw Myint
  • Li Li
  • Ming Xin Wang


This paper deals with one kind of Belousov–Zhabotinskii reaction model. Linear stability is discussed for the spatially homogeneous problem firstly. Then we focus on the stationary problem with diffusion. Non-existence and existence of non-constant positive solutions are obtained by using implicit function theorem and Leray–Schauder degree theory, respectively.

MR(2010) Subject Classification

35J57 35B09 35B35 92E20 


Belousov–Zhabotinskii reaction stability positive stationary solutions 


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  1. [1]
    Belousov, B. P.: An oscillating reaction and its mechanism, In Sborn. Referat. Radiat. Med. (Collection of Abstracts on Radiation Medicine), page 145. Medgiz, Moscow, 1959Google Scholar
  2. [2]
    Brown, K. J., Davidson, F. A.: Global bifurcation in the Brusselator system. Nonlinear Anal. TMA, 24, 1713–1725 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Capasso, V., Diekmann, O.: Mathematics Inspired by Biology, Lecture Notes in Mathematics, Vol. 1714, Springer, Berlin, CIME, Florence, 1999Google Scholar
  4. [4]
    Chen, X. F., Qi, Y. W., Wang, M. X.: A strongly coupled predator-prey system with non-monotonic functional response. Nonlinear Anal. TMA, 67, 1966–1979 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Davidson, F. A., Rynne, B. P.: A priori bounds and global existence of solutions of the steady-state Sel’kov model. Proc. R. Soc. Edinburgh A 130, 507–516 (2000)CrossRefzbMATHGoogle Scholar
  6. [6]
    Field, R. J., Burger, M.: Oscillations and Travelling Waves in Chemical Systems, Wiley, New York, 1985Google Scholar
  7. [7]
    Field, R. J., Noyes, R. M.: Oscillations in chemical systems. IV. limit cycle behaviour in a model of a real chemical reaction. J. Chem. Phys., 60, 1877–1884 (1974)Google Scholar
  8. [8]
    Li, J. J., Gao, W. J.: Analysis of a prey-predator model with disease in prey. Appl. Math. Comput., 217(8), 4024–4035 (2010)MathSciNetzbMATHGoogle Scholar
  9. [9]
    Lieberman, G. M.: Bounds for the steady-state Sel’kov model for arbitrary p in any number of dimensions. SIAM J. Math. Anal., 36(5), 1400–1406 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Lou, Y., Martinez, S., Ni, W. M.: On 3×3 Lotka-Volterra competition systems with cross-diffusion. Discrete Contin. Dynamic Systems, 6(1), 175–190 (2000)MathSciNetzbMATHGoogle Scholar
  11. [11]
    Lou, Y., Ni, W. M.: Diffusion, self-diffusion and cross-diffusion. J. Differential Equations, 131, 79–131 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Lou, Y., Ni, W. M.: Diffusion vs cross-diffusion: an elliptic approach. J. Differential Equations, 154, 157–190 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Murray, J. D.: Mathematical Biology, 2nd Edition, Springer, Berlin, 1993CrossRefzbMATHGoogle Scholar
  14. [14]
    Nirenberg, L.: Topics in Nonlinear Functional Analysis, American Mathematical Society, Providence, RI, 2001CrossRefzbMATHGoogle Scholar
  15. [15]
    Pang, P. Y. H., Wang, M. X.: Qualitative analysis of a ratio-dependent predator-prey system with diffusion. Proc. R. Soc. Edinburgh A, 133(4), 919–942 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Pang, P. Y. H., Wang, M. X.: Non-constant positive steady states of a predator-prey system with nonmonotonic functional response and diffusion. Proc. London Math. Soc., 88, 135–157 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Pang, P. Y. H., Wang, M. X.: Strategy and stationary pattern in a three-species predator-prey model. J. Differential Equations, 200(2), 245–273 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Peng, R.: Qualitative analysis of steady states to the Sel’kov model. J. Differential Equations, 241(2), 386–398 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Peng, R., Wang, M. X.: Positive steady state solutions of the Noyes–Field model for Belousov–Zhabotinskii reaction. Nonlinear Anal. TMA, 56(3), 451–464 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Peng, R., Wang, M. X.: Pattern formation in the Brusselator system. J. Math. Anal. Appl., 309(1), 151–166 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Peng, R., Shi, J. P., Wang, M. X.: Station pattern of ratio-dependent food chain model with diffusion. SIAM J. Appl. Math., 67, 1479–1503 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Shi, H. P., Li, W. T., Lin, G.: Positive steady states of a diffusive predator-prey system with modified Holling–Tanner functional response. Nonlinear Anal. Real World Appl., 11(5), 3711–3721 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Tian, C. R., Ling, Z., Lin, Z. G.: Turing pattern formation in a predator-prey-mutualist system. Nonlinear Anal. Real World Appl., 12(6), 3224–3237 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Turing, A.: The chemical basis of morphogenesis. Philos. Trans. Roy. Soc., B(237), 37–72 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Tyson, J. J.: The Belousov–Zhabotinskii Reaction, Lecture Notes in Biomathematics, Springer-Verlag, Berlin, 1976zbMATHGoogle Scholar
  26. [26]
    Wang, M. X.: Non-constant positive steady states of the Sel’kov model. J. Differential Equations, 190(2), 600–620 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Wang, M. X.: Nonlinear Equations of Parabolic Type (in Chinese), Science Press, Beijing, 1993Google Scholar
  28. [28]
    Wang, X. F.: Qualitative behavior of solutions of chemotactic diffusion systems: effects of motility and chemotaxis and dynamics. SIAM J. Math. Anal., 31(3), 535–560 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Zhabotinskii, A. M.: Periodic processes of the oxidation of malonic acid in solution (Study of the kinetics of Belousov-reaction). Biofizika, 9, 306–311 (1964)Google Scholar
  30. [30]
    Zhou, J., Kim, C. G., Shi, J. P.: Positive steady state solutions of a diffusive Leslie–Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete Contin. Dyn. Syst, 34(9), 3875–3899 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    Zhou, J., Mu, C. L.: Pattern formation of a coupled two-cell Brusselator model. J. Math. Anal. Appl., 366(2), 679–693 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinP. R. China

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