English

Patterns in Time and Space from a Single Morphogen via Nonlinear Layering

Pattern Formation and Solitons 2026-05-07 v1

Abstract

Spatial and temporal pattern formation in reaction-diffusion systems is typically studied with two or more equations, as scalar reaction-diffusion equations confined to convex domains do not admit stable inhomogeneous states in time or space on long timescales. Here, we show that a single morphogen diffusing across layered two-dimensional media, with nonlinear coupling between layers, is able to generate stable patterns in time and space. This NN-layer model is analysed via a thin-domain limit, which reduces to an NN-component reaction-diffusion system on a homogeneous one-dimensional domain. This reduced model can be analysed via linear stability techniques, showing that non-diffusive, or reactive, coupling between regions is necessary for pattern-forming instabilities, at least in the reduced model. This reduced system can exhibit Turing, Hopf, and Turing-wave instabilities, with emergent structures that are numerically shown to persist even away from the thin-domain regime of the full 2D single-morphogen system. These results suggest that heterogeneous stratification and nonlinear coupling can broaden the class of systems which exhibit complex spatiotemporal behaviours, which may be relevant in scenarios where only a single morphogen is known to act.

Keywords

Cite

@article{arxiv.2605.05063,
  title  = {Patterns in Time and Space from a Single Morphogen via Nonlinear Layering},
  author = {N. Mahashri and Andrew L. Krause and M. Chandru and Thomas E. Woolley},
  journal= {arXiv preprint arXiv:2605.05063},
  year   = {2026}
}

Comments

19 pages, 9 figures

R2 v1 2026-07-01T12:53:04.479Z