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arxiv: 2605.05063 · v1 · submitted 2026-05-06 · 🌊 nlin.PS

Recognition: unknown

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

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Pith reviewed 2026-05-08 15:08 UTC · model grok-4.3

classification 🌊 nlin.PS
keywords couplingmodelmorphogennonlinearreaction-diffusionreducedsinglespace
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The pith

Nonlinear coupling across layers allows a single morphogen to produce Turing, Hopf, and Turing-wave instabilities in a reduced one-dimensional model that persists in the full system.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Morphogens are signaling chemicals that help cells organize into patterns during growth. Standard math models need at least two interacting chemicals to create stable spots or stripes. This work instead stacks thin layers of tissue and lets one chemical move between them with nonlinear rules. A mathematical simplification turns the layered setup into a simpler chain of equations on a line. Analysis of those equations reveals conditions where patterns emerge and stay stable, including waves that move. Computer checks confirm the patterns survive when the layers are not perfectly thin.

Core claim

a single morphogen diffusing across layered two-dimensional media, with nonlinear coupling between layers, is able to generate stable patterns in time and space.

Load-bearing premise

The thin-domain limit reduction to an N-component one-dimensional reaction-diffusion system faithfully represents the instabilities and persistent patterns of the original two-dimensional layered model.

read the original 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 $N$-layer model is analysed via a thin-domain limit, which reduces to an $N$-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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the thin-domain asymptotic reduction and on the assumption that reactive (non-diffusive) coupling is required for instability; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The thin-domain limit reduces the 2D layered system to an N-component 1D reaction-diffusion system whose linear stability determines the pattern-forming instabilities.
    Invoked to obtain the reduced model that is then analyzed.

pith-pipeline@v0.9.0 · 5509 in / 1090 out tokens · 31319 ms · 2026-05-08T15:08:51.907153+00:00 · methodology

discussion (0)

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