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Stripe selection: An intrinsic property of some pattern‐forming models with nonlinear dynamics
Author(s) -
Lyons Michael J.,
Harrison Lionel G.
Publication year - 1992
Publication title -
developmental dynamics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.634
H-Index - 141
eISSN - 1097-0177
pISSN - 1058-8388
DOI - 10.1002/aja.1001950306
Subject(s) - nonlinear system , pattern formation , property (philosophy) , quadratic equation , domain (mathematical analysis) , dynamics (music) , morphogen , range (aeronautics) , selection (genetic algorithm) , biological system , statistical physics , biology , bistability , mathematical analysis , physics , mathematics , geometry , computer science , artificial intelligence , philosophy , biochemistry , genetics , materials science , epistemology , quantum mechanics , gene , acoustics , composite material
In two‐dimensional pattern formation, the genesis of striped rather than spotted patterns may involve preexisting spatial asymmetries, such as unidirectional gradients or asymmetric shape of the pattern‐forming domain. In the absence of such asymmetries, some kinds of nonlinear dynamics still lead to striped rather than spotted patterns. We have studied the latter effect both by extensive computer experiments on a range of nonlinear models and by mathematical analysis. We conclude that, when the dynamic equations are written in terms of departure from the unpatterned state, the presence of nonlinearities which are odd functions of these departures (e.g., cubic terms) together with absence of even nonlinearities (e.g., quadratic terms) ensures stripe formation. In computer experiments, we have studied the dynamics of two‐morphogen reaction‐diffusion models. The mathematical analysis presented in the Appendix shows that the same property exists in more generalized models for pattern formation in the primary visual cortex. © 1993 Wiley‐Liss, Inc.