Skip to main content
Log in

Pattern sensitivity to boundary and initial conditions in reaction-diffusion models

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript


We consider Turing-type reaction-diffusion equations and study (via computer simulations) how the relationship between initial conditions and the asymptotic steady state solutions varies as a function of the boundary conditions. The results indicate that boundary conditions which are non-homogeneous with respect to the kinetic steady state give rise to spatial patterns which are much less sensitive to variations in the initial conditions than those obtained with homogeneous boundary conditions, such as zero flux conditions. We also compare linear pattern predictions with the numerical solutions of the full nonlinear problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  • Bard, J. B. L., Lauder, I.: How well does Turing's theory of morphogenesis work? J. Theor. Biol. 45, 501 (1974)

    Google Scholar 

  • Bunow, B., Kernevez, J. P.: Turing and the physico-chemical basis for biological pattern formation. (preprint) (1980)

  • Bunow, B., Kernevez, J. P., Joly, G., Thomas, D.: Pattern formation by reaction-diffusion instabilities: application to morphogenesis in Drosophila. J. Theor. Biol. 84, 629 (1980)

    Google Scholar 

  • Catalano, G., Eilbeck, J. C., Monroy, A., Parisi, E.: A mathematical model for pattern formation in biological systems. Physica (3D), 439 (1981)

  • Edelstein, B. B.: The dynamics of cellular differentiation and associated pattern formation. J. Theor. Biol. 37, 221 (1972)

    Google Scholar 

  • Gear, C. W.: Numerical initial value problems in ordinary differential equations. Prentice Hall (1971)

  • Gierer, A.: Generation of biological patterns and form: some physical, mathematical, and logical aspects. Prog. Biophys. Molec. Biol. 37, 1 (1981)

    Google Scholar 

  • Hadeler, K. P., an der Heiden, Rothe, F.: Nonhomogeneous spatial distributions of populations. J. Math. Biol. 1, 165–176 (1974)

    Google Scholar 

  • Hunding, A.: Dissipative structures in reaction-diffusion systems: numerical determination of bifurcations in the sphere. J. Chem. Phys. 72(9), 5241 (1980)

    Google Scholar 

  • Jones, D. S., Sleeman, B. D.: Differential equations and mathematical biology. George Allen and Unwin, London (1983)

    Google Scholar 

  • Kauffman, S. A., Shymko, R. M., Trabert, K.: Control of sequential compartment formation in Drosophila. Science 199, 259 (1978)

    Google Scholar 

  • Lewis, J., Slack, J. M. W., Wolpert, L.: Thresholds in development. J. Theor. Biol. 65, 579 (1977)

    Google Scholar 

  • Meinhardt, H.: Models of biological pattern formation. Academic Press, London (1982)

    Google Scholar 

  • Murray, J. D.: Nonlinear differential equation models in biology. Clarendon Press, Oxford (1977)

    Google Scholar 

  • Murray, J. D.: A prepattern formation mechanism for animal coat markings. J. Theor. Biol. 88, 161 (1981a)

    Google Scholar 

  • Murray, J. D.: On pattern formation mechanisms for lepidoptera wing patterns and mammalian coat markings. Phil. Trans. Roy. Soc. B 295, 473 (1981b)

    Google Scholar 

  • Murray, J. D.: Parameter space for Turing instability in reaction-difiusion mechanisms: a comparison of models. J. Theor. Biol. 98, 143 (1982)

    Google Scholar 

  • Schnakenberg, J.: Simple chemical reaction systems with limit cycle behaviour. J. Theor. Biol. 81, 389 (1979)

    Google Scholar 

  • Seelig, F. F.: Chemical oscillations by substrate inhibition. Z. Naturforsch 31A.2, 731 (1976)

    Google Scholar 

  • Thomas, D.: In: Thomas, D., Kernevez, J. P. (eds.) Analysis and control of immobilized enzyme systems, p. 115. Springer, Berlin, Heidelberg, New York (1976)

    Google Scholar 

  • Turing, A. M.: The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. B 27, 37 (1952)

    Google Scholar 

  • Wolpert, L.: Positional information and pattern formation. Curr. Top. Dev. Biol. 6, 183 (1971)

    Google Scholar 

Download references

Author information

Authors and Affiliations


Additional information

This work supported in part by U.S. Army Grant DAJA 37-81-C-0220 and the Science and Engineering Research Council of Great Britain Grant GR/c/63595

Rights and permissions

Reprints and permissions

About this article

Cite this article

Arcuri, P., Murray, J.D. Pattern sensitivity to boundary and initial conditions in reaction-diffusion models. J. Math. Biology 24, 141–165 (1986).

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI:

Key words