Ngwa, G. A. and Maini, P. K. (1995) Spatio-temporal patterns in a mechanical model for mesenchymal morphogenesis. Journal of Mathematical Biology, 33 (5). pp. 489-520.
We present an in-depth study of spatio-temporal patterns in a simplified version of a mechanical model for pattern formation in mesenchymal morphogenesis. We briefly motivate the derivation of the model and show how to choose realistic boundary conditions to make the system well-posed. We firstly consider one-dimensional patterns and carry out a nonlinear perturbation analysis for the case where the uniform steady state is linearly unstable to a single mode. In two-dimensions, we show that if the displacement field in the model is represented as a sum of orthogonal parts, then the model can be decomposed into two sub-models, only one of which is capable of generating pattern. We thus focus on this particular sub-model. We present a nonlinear analysis of spatio-temporal patterns exhibited by the sub-model on a square domain and discuss mode interaction. Our analysis shows that when a two-dimensional mode number admits two or more degenerate mode pairs, the solution of the full nonlinear system of partial differential equations is a mixed mode solution in which all the degenerate mode pairs are represented in a frequency locked oscillation.
|Uncontrolled Keywords:||Spatio-temporal - Degenerate modes - Periodic patterns - Hopf bifurcation|
|Subjects:||A - C > Biology and other natural sciences|
|Research Groups:||Centre for Mathematical Biology|
|Deposited By:||Philip Maini|
|Deposited On:||10 Dec 2006|
|Last Modified:||20 Jul 2009 14:21|
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