Journal of Mathematical Biology

, Volume 11, Issue 3, pp 295–310 | Cite as

Positional differentiation as pattern formation in reaction-diffusion systems with permeable boundaries. Bifurcation analysis

  • M. A. Livshits
  • G. T. Gurija
  • B. N. Belintsev
  • M. V. Volkenstein
Article

Abstract

Pattern formation in a unicomponent reaction-diffusion system with trigger type dynamics and combined boundary conditions is considered. The boundary permeabilities and reservoir concentrations as well as the dimension of the system are the control parameters. The whole assemblage of steady states, their bifurcations and changes under the variation of these parameters is described. Among all steady distributions possible for given values of the parameters, only the simplest ones prove to be asymptotically stable. The relation to catastrophe theory is discussed.

Key words

Morphogenesis Pattern formation Bifurcation 

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References

  1. 1.
    Arnold, V. I.: Theory of ordinary differential equations (Advanced course). M. Nauka, 1978 (Russian)Google Scholar
  2. 2.
    Bass, F. G., Bochkov, V. S., Gurevich, Yu. G.: Influence of sample size on the volt-ampere characteristic in media with an ambiguous dependence of electron temperature on field strength. JETP 58, 1814–1824 (1970)Google Scholar
  3. 3.
    Belintsev, B. N., Livshits, M. A., Volkenstein, M. V.: The study of stability of inhomogeneous states in spatially distributed systems. Biofizika 23, 1056–1062 (1978) (Russian)Google Scholar
  4. 4.
    Belintsev, B. N., Livshits, M. A., Volkenstein, M. V.: Spatially inhomogeneous stationary forms in kinetic systems with reactions and diffusion. Positional differentiation. Biofizika 24, 117–123 (1979) (Russian)Google Scholar
  5. 5.
    Belintsev, B. N., Livshits, M. A., Volkenstein, M. V.: On the multi-stationary state transitions in the spatial kinetic systems. Z. Physik B 30, 211–218 (1978)Google Scholar
  6. 6.
    Belintsev, B. N., Livshits, M. A., Gurija, G. T., Volkenstein, M. V.: Transient processes in the development of multicellular organisms. Dokl. Akad. Nauk., USSR 244, 1239–1243 (1979) (Russian)Google Scholar
  7. 7.
    Crick, F.: Diffusion in embryogenesis. Nature 225, 420–422 (1970)Google Scholar
  8. 8.
    Chafee, N.: Asymptotic behaviour for solutions of a one-dimensional parabolic equation with homogeneous Neumann boundary conditions. J. Differential Equations 18, 111–134 (1975)Google Scholar
  9. 9.
    Edelstein, B. B.: The dynamics of cellular differentiation and associated pattern formation. J. Theoret. Biol. 37, 221–243 (1972)Google Scholar
  10. 10.
    Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kibernetika 12, 30–39 (1972)Google Scholar
  11. 11.
    Grigorov, L. N., Poljakova, Chernavskii, D. S.: Model investigation of the trigger schemes and of the differentiation process. Mol. Biol. (Russ.) 1, 410–418 (1967)Google Scholar
  12. 12.
    Hanson, M. P.: Dissipative structures in one dimensional pseudo-one component systems. J. Chem. Phys. 66, 5551–5556 (1977)Google Scholar
  13. 13.
    Jacob, F., Monod, J.: Genetic regulatory mechanisms in the synthesis of proteins. J. Mol. Biol. 3, 318–356 (1961)Google Scholar
  14. 14.
    Lavenda, B. H.: The theory of multi-stationary state transitions and biosynthetic control processes. Quart. Rev. Biophys. 5, 429–479 (1972)Google Scholar
  15. 15.
    Lemarchand, H., Nicolis, G.: Long range correlations and the onset of chemical instabilities. Physica A 82, 521–542 (1976)Google Scholar
  16. 16.
    Maginu, K.: Stability of stationary solutions of a semilinear parabolic partial differential equation, J. Math. Anal. Appl. 63, 224–243 (1978)Google Scholar
  17. 17.
    Nitzan, A., Ortoleva, P., Deutch, J., Ross, J.: Fluctuations and transitions at chemical instabilities. The analogy phase transitions. J. Chem. Phys. 61, 1056–1074 (1974)Google Scholar
  18. 18.
    Robertson, A., Cohen, M. H.: Control of developing fields. Ann. Rev. Biophys. Bioeng. 1, 409–464 (1972)Google Scholar
  19. 19.
    Sattinger, D. H.: Stability of bifurcating solutions by Leray-Schauder degree. Arch. Rational Mech. Anal. 43, 154–166(1971)Google Scholar
  20. 20.
    Thom, R.: Stabilité structurelle et morphogenese. New York: Benjamin 1972Google Scholar
  21. 21.
    Wolpert, L.: Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol. 25, 1–47 (1969)Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • M. A. Livshits
    • 1
  • G. T. Gurija
    • 1
  • B. N. Belintsev
    • 1
  • M. V. Volkenstein
    • 1
  1. 1.Institute of Molecular BiologyUSSR Academy of SciencesMoscowUSSR

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