Abstract
In 1983 Oster et al. proposed a model for morphogenesis consisting of a system of partial differential equations in which the dispersion relation for the problem linearised about the zero solution has a singularity. That is, the initial growth rate σ of a small perturbation of wave number k from the zero solution tends to positive or negative infinity as k tends to some critical value k c from above or below respectively. We consider here as a caricature of the model a single partial differential equation with a similar dispersion relation in a bounded one-dimensional domain. The wave number, or equivalently the domain size, may be thought of as a bifurcation parameter. For the Neumann problem a phenomenon arises in which, as the domain size l increases past a critical value l l ,the linear stability of the n-th mode jumps from one solution to a remote solution. That is, for l<l n the trivial solution is unstable and a certain non-trivial solution is stable to perturbations of mode n, whereas for l>l n the opposite is true. For the Dirichlet or the Robin problem a linear stability change in the trivial solution occurs, but no corresponding change in any other solution has been found. The corresponding initial boundary value problems are then considered. An asymptotic analysis is performed in the weakly nonlinear limit in the particular case in which only one mode is unstable and gives an asymptotic solution for two classes of nonlinearity, one symmetric and the other asymmetric about u=0. A development of the method of harmonic balance is then used to obtain approximate solutions in the strongly nonlinear case and when more than one mode may be unstable.
Similar content being viewed by others
References
Britton, N. F.: Reaction-diffusion equations and their applications to biology. London: Academic Press 1985
Hagedorn, P.: Nonlinear oscillations. Oxford: Clarendon Press 1981
Hayashi, C.: Nonlinear oscillations in physical systems. New York: McGraw-Hill 1964
Jordan, D. W., Smith, P.: Nonlinear ordinary differential equations. Oxford: Clarendon Press 1979
Murray J. D., Oster, G. F.: Cell traction models for generating pattern and form in morphogenesis. J. Math. Biol. 19, 265–279 (1984a)
Murray, J. D., Oster, G. F.: Generation of biological pattern and form. IMA J. Maths. Appl. Med. Biol. 1, 51–76 (1984b)
Oster G. F., Murray, J. D., Harris, A. K.: Mechanical aspects of mesenchymal morphogenesis. J. Embryol. Exp. Morph. 78, 83–125 (1983)
Oster, G. F., Murray, J. D., Maini, P. K.: A model for chondrogenic condensations in the developing limb: the role of extracellular matrix and cell tractions. J. Embryol. Exp. Morph. 89, 93–112 (1985)
Smoller, J. A.: Shock waves and reaction-diffusion equations. Berlin Heidelberg New York: Springer 1983
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Britton, N.F. A singular dispersion relation arising in a caricature of a model for morphogenesis. J. Math. Biology 26, 387–403 (1988). https://doi.org/10.1007/BF00276369
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00276369