Rocky Mountain Journal of Mathematics

Bifurcations of patterned solutions in the diffusive Lengyel-Epstein system of Cima chemical reactions

Jiayin Jin, Junping Shi, Junjie Wei, and Fengqi Yi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Rocky Mountain J. Math. Volume 43, Number 5 (2013), 1637-1674.

First available in Project Euclid: 25 October 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K57: Reaction-diffusion equations 35B32: Bifurcation [See also 37Gxx, 37K50] 37G15: Bifurcations of limit cycles and periodic orbits 35B10: Periodic solutions 92E20: Classical flows, reactions, etc. [See also 80A30, 80A32] 80A32: Chemically reacting flows [See also 92C45, 92E20] 92B05: General biology and biomathematics

Lengyel-Epstein chemical reaction reaction-diffusion system Hopf bifurcation steady state bifurcation spatially non-homogeneous periodic orbits global bifurcation


Jin, Jiayin; Shi, Junping; Wei, Junjie; Yi, Fengqi. Bifurcations of patterned solutions in the diffusive Lengyel-Epstein system of Cima chemical reactions. Rocky Mountain J. Math. 43 (2013), no. 5, 1637--1674. doi:10.1216/RMJ-2013-43-5-1637.

Export citation


  • P. De Kepper, V. Castets, E. Dulos and J. Boissonade, Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction, Phys. D 49 (1991), 161-169.
  • I.R. Epstein and J.A. Pojman, An introduction to nonlinear chemical dynamics, Oxford University Press, Oxford, 1998.
  • P. Gray and S.K. Scott, Chemical oscillations and instabilities, Oxford Univ. Press, New York-Oxford, 1990.
  • B.D. Hassard, N.D. Kazarinoff and Y.-H. Wan, Theory and application of Hopf bifurcation, Cambridge University Press, Cambridge, 1981.
  • S. Hsu and J. Shi, Relaxation oscillator profile of limit cycle in predator-prey system, Disc. Cont. Dyn. Syst. 11 (2009), 893-911.
  • J. Jang, W. Ni and M. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model, J. Dyn. Diff. Eqs. 16 (2005), 297-320.
  • S.L. Judd and M. Silber, Simple and superlattice Turing patterns in reaction-diffusion systems: Bifurcation, bistability, and parameter collapse, Phys. D 136 (2000), 45-65.
  • I. Lengyel and I.R. Epstein, A chemical approach to designing Turing patterns in reaction-diffusion system, Proc. Natl. Acad. Sci. 89 (1992), 3977-3979.
  • –––, Modeling of Turing structure in the Chlorite-iodide-malonic acid-starch reaction system, Science 251 (1991), 650-652.
  • S.G. Lenhoff and H.M. Lenhoff, Hydra and the birth of experimental biology-1744, Boxwood, Pacific Grove, CA, 1986.
  • P.K. Maini, K.J. Painter and H.N.P. Chau, Spatial pattern formation in chemical and biological systems, Faraday Transactions 20 (1997), 3601-3610.
  • J.D. Murray, Mathematical biology, Third edition. I. An introduction, Interdisc. Appl. Math. 17, Springer-Verlag, New York, 2002; II. Spatial models and biomedical applications, Interdisc. Appl. Math. 18, Springer-Verlag, New York, 2003.
  • W. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Trans. Amer. Math. Soc. 357 (2005), 3953-3969.
  • Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal. 13 (1982), 555-593.
  • J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal. 169 (1999), 494-531.
  • J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Diff. Equat. 246 (2009), 2788-2812.
  • L.E. Stephenson and D.J. Wollkind, Weakly nonlinear stability analyses of one-dimensional Turing pattern formation in activator-inhibitor/immobilizer model systems, J. Math. Biol. 33 (1995), 771-815.
  • I. Takagi, Point-condensation for a reaction-diffusion system, J. Diff. Equat. 61 (1986), 208-249.
  • A. Trembley, Mémoires, Pour Servir à l'Histoire d'un Genre de Polypes d'Eau Douce, à Bras en Forme de Cornes, Verbeek, Leiden, The Netherlands, 1744.
  • A. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. Lond. 237 (1952), 37-72.
  • D.J. Wollkind and L.E. Stephenson, Chemical Turing pattern formation analyses: Comparison of theory with experiment, SIAM J. Appl. Math. 61 (2000), 387-431.
  • J. Wu, Theory and applications of partial functional-differential equations, Appl. Math. Sci. 119, Springer-Verlag, New York, 1996.
  • –––, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998), 4799-4838.
  • F. Yi, J. Wei and J. Shi, Diffusion-driven instability and bifurcation in the Lengyel-Epstein model, Nonl. Anal. Real World Appl. 9 (2008), 1038-1051.
  • –––, Global asymptotical behavior of the Lengyel-Epstein reaction-diffusion system, Appl. Math. Lett. 22 (2009), 52-55.
  • –––, Bifurcation and spatio-temporal patterns in a diffusive homogenous predator-prey system, J. Diff. Equat. 246 (2009), 1944-1977. \noindentstyle