Stripe selection: an intrinsic property of some pattern-forming models with nonlinear dynamics

In two-dimensional pattern formation, the genesis of stripped 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.