Summary
We first perform a linear stability analysis of the Gierer-Meinhardt model to determine the critical parameters where the homogeneous distribution of activator and inhibitor concentrations becomes unstable. There are two kinds of instabilities, namely, one leading to spatial patterns and another one leading to temporal oscillations. Focussing our attention on spatial pattern formation we solve the corresponding nonlinear equations by means of our previously introduced method of generalized Ginzburg-Landau equations. We explicitly consider the two-dimensional case and find both rolls and hexagon-like structures. The impact of different boundary conditions on the resulting patterns is also discussed. The occurrence of the new patterns has all the features of nonequilibrium phase transitions.
Similar content being viewed by others
References
Babloyantz, A., Hiernaux, J.: Models for cell differentiation. Bull. Math. Biology37, 637–657 (1975)
Busse, F. H.: Patterns of convection in spherical shells. J. Fluid Mech.72, 67–85 (1975)
Busse, F. H.: The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech.30, 625–649 (1967)
Busse, F. H.: Thermal Instabilities in rapidly rotating systems. J. Fluid Mech.44, 441 (1970)
Di Prima, R. C., Eckhaus, W., Segel, L. A.: Nonlinear wave-number interaction in nearcritical two-dimensional flows. J. Fluid Mech.,49, 705–744 (1971)
Gierer, A., Meinhardt, H.: Biological Pattern Formation Involving Lateral Inhibition. Lectures on Mathematics in Life Science.7, 163–183 (1974)
Gierer, A., Meinhardt, H.: Theory of Biological Pattern Formation. Kybernetik12, 30–39 (1972)
Granero, M. I., Porati, A., Zanacca, D.: A Bifurcation Analysis of Pattern Formation in a Diffusion Governed Morphogenetic Field. J. Math. Biology4, 21–27 (1977)
Haken, H.: Generalized Ginsburg-Landau Equations for Phase Transition-like Phenomena in Lasers, Nonlinear Optics, Hydrodynamics and Chemical Reactions. Z. Phys.B21, 105 (1975)
Haken, H.: Higher Order Corrections to Generalized Ginsburg-Landau-Equations of Non-Equilibrium Systems. Z. PhysikB22, 69–72 (1975)
Haken, H.: Synergetics. An Introduction. Nonequilibrium Phase Transitions and Selforganization in Physics, Chemistry and Biology. Berlin, Heidelberg, New York: Springer, 1977
Levin, S. A.: Uniqueness Theorems for the Compressible Flow Equation. Appl. Anal.5, 207–215 (1976)
Newell, A. C., Whitehead, L. A.: Finite bandwidth, finite amplitude convection. J. Fluid Mech.38, 279–303 (1969)
Mahar, T. J., Matkowsky, B. J.: A Model Biochemical Reaction Exhibiting Secondary Bifurcation. SIAM J. Appl. Math.32, 394 (1977)
Marsden, J. E.: The Hopf-Bifurcation and its Applications. Berlin, Heidelberg, New York: Springer, 1976
Matkowsky, B. J.: On boundary layer problems exhibiting resonance. SIAM Review17, 82 (1975)
Meinhardt, H., Gierer, A.: Applications of a Theory of Biological Pattern Formation Based on Lateral Inhibition. J. Cell Science15, 321–376 (1974)
Meinhardt, H.: The Spatial Control of Cell Differentiation by Autocatalysis and Lateral Inhibition. In: Synergetics, a Workshop. (H. Haken, ed.) Berlin, Heidelberg, New York: Springer, 1977
Sattinger, D. H.: Cooperative Effects in Fluid Problems. In: Synergetics, a Workshop. Berlin, Heidelberg, New York: Springer, 1977
Segel, L. A., Levin, S. A.: Application of nonlinear stability theory to the study of the effects of diffusion on predator-prey interactions. AIP Conference Proceedings27, 123–152 (1976)
Turing, A. M.: The Chemical Basis of Morphogenesis. Phil. Transactions of the Royal SocietyB237, 37–72 (1952)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Haken, H., Olbrich, H. Analytical treatment of pattern formation in the Gierer-Meinhardt model of morphogenesis. J. Math. Biology 6, 317–331 (1978). https://doi.org/10.1007/BF02462997
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02462997