Bulletin of Mathematical Biology

, Volume 68, Issue 5, pp 981–995 | Cite as

Mode Transitions in a Model Reaction–Diffusion System Driven by Domain Growth and Noise

  • Iain Barrass
  • Edmund J. Crampin
  • Philip K. Maini
Original Paper


Pattern formation in many biological systems takes place during growth of the underlying domain. We study a specific example of a reaction–diffusion (Turing) model in which peak splitting, driven by domain growth, generates a sequence of patterns. We have previously shown that the pattern sequences which are presented when the domain growth rate is sufficiently rapid exhibit a mode-doubling phenomenon. Such pattern sequences afford reliable selection of certain final patterns, thus addressing the robustness problem inherent of the Turing mechanism. At slower domain growth rates this regular mode doubling breaks down in the presence of small perturbations to the dynamics. In this paper we examine the breaking down of the mode doubling sequence and consider the implications of this behaviour in increasing the range of reliably selectable final patterns.


Turing model Schnakenberg Spatial pattern Robustness Mode-doubling failure Mode selection 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bard, J., Lauder, I., 1974. How well does Turing's theory of morphogenesis work? J. Theor. Biol. 45, 501–531CrossRefGoogle Scholar
  2. Crampin, E.J., 2000. Reaction–diffusion patterns on growing domains. DPhil thesis, University of Oxford, UK.Google Scholar
  3. Crampin, E.J., Gaffney, E.A., Maini, P.K., 1999. Reaction and diffusion on growing domains: Scenarios for robust pattern formation. Bull. Math. Biol. 61(6), 1093–1120.CrossRefGoogle Scholar
  4. Crampin, E.J., Gaffney, E.A., Maini, P.K., 2002. Mode doubling and tripling in reaction–diffusion patterns on growing domains: A piecewise linear model. J. Math. Biol. 44(2), 107–128.zbMATHCrossRefMathSciNetGoogle Scholar
  5. DeKepper, D., Castets, V., Dulos, E., Boissonade, J., 1991. Turing-type chemical patterns in the chlorite–iodide–malonic acid reaction. Physica D 61, 161–169.CrossRefGoogle Scholar
  6. Dillon, R., Maini, P.K., Othmer, H.G., 1994. Pattern formation in generalised Turing systems I: Steady state patterns in systems with mixed boundary conditions. J. Math. Biol. 32, 345–393.zbMATHCrossRefMathSciNetGoogle Scholar
  7. Doedel, E.J., Paffenroth, R.C., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang X., 1997. AUTO 2000: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont). Technical Report, Caltech, February 2001.Google Scholar
  8. Hayes, C., Brown, J.M., Lyon, M.F., Morriss-Kay, G.M., 1998. Sonic hedgehog is not required for polarising activity in the Doublefoot mutant mouse limb bud. Development 125, 351–357.Google Scholar
  9. Iron, D., Wei, J., Winter, M., 2004. Stability analysis of Turing patterns generated by the Schnakenberg Model. J. Math. Biol. 49(4), 358–390zbMATHCrossRefMathSciNetGoogle Scholar
  10. Keller, E.F., Segel, L.A., 1970. The initiation of slime mold aggregation viewed as an instability. J. theor. Biol. 26, 399–415.CrossRefGoogle Scholar
  11. Kolokolnikov, T., Ward, M.J., Wei. J., 2005. The existence and stability of spike equilibria in the one-dimensional Gray–Scott model on a finite domain. Appl. Math. Lett. 18(8), 951–956.zbMATHCrossRefMathSciNetGoogle Scholar
  12. Kondo, S., Asai, R., 1995. A reaction–diffusion wave on the skin of the marine angelfish Pomacanthus. Nature 376, 765–768CrossRefGoogle Scholar
  13. Meinhardt, H., 1995. The Algorithmic Beauty of Sea Shells. Springer, Heidelberg.Google Scholar
  14. Murray, J.D., 2003. Mathematical Biology. 3rd edition in 2 volumes: Mathematical Biology: II. Spatial Models and Biomedical Applications, Springer, New York.Google Scholar
  15. Ouyang, Q., Swinney, H.L., 1991. Transition from a uniform state to hexagonal and striped Turing patterns. Nature 353(6336), 610–612.CrossRefGoogle Scholar
  16. Segel, L.A., Jackson, J.L., 1972. Dissipative structure: An explanation and an ecological example. J. Theor. Biol. 37, 545–559.CrossRefGoogle Scholar
  17. Turing, A.M., 1952. The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37–72.CrossRefGoogle Scholar
  18. Yoon, H.-S., Golden, J.W., 1998. Heterocyst pattern formation controlled by a diffusible peptide. Science 282, 935–938.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Iain Barrass
    • 1
  • Edmund J. Crampin
    • 1
    • 2
  • Philip K. Maini
    • 1
  1. 1.Centre for Mathematical Biology, Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.Bioengineering Institute and Department of Engineering ScienceThe University of AucklandAucklandNew Zealand

Personalised recommendations