Paper 18: Local Taylor-Jet Estimation of a Scale-Dependent Turing Instability Boundary in the Gierer–Meinhardt System

Classical Turing analysis of the Gierer–Meinhardt activator-inhibitor model determines whether a homogeneous equilibrium is linearly unstable to diffusion-driven perturbations at a single, chosen parameter value; it does not, on its own, say up to which point along a continuously varying set of transport parameters that instability persists. We investigate how the corresponding linear-instability condition changes as the transport parameters vary along a hierarchy coordinate spanning a biological hierarchy (cell, tissue, organ), and call the boundary at which instability is lost the scale-extinction problem. We formulate it as a local hierarchy-jet problem, solve it exactly for a controlled analytic benchmark, and quantify how accurately the first boundary crossing can be extrapolated from a finite calibration window. Using the standard two-species instability criterion evaluated with s-dependent coefficients, we define a scale-dependent Turing margin M(s) – organizing its local Taylor jet in the CNRS-H notation, which represents smooth scale-dependent functions as Taylor-coefficient (”digit”) sequences with structural properties (digit-shift differentiation and Cauchy convolution multiplication, derived self-containedly in Appendix A) that propagate the jet through the instability condition; M(s) is the natural two-component specialisation of the general diffusion-driven-instability condition of Villar-Sepulveda and Champneys (2023) – and show, for the analytic diffusion profiles used in the illustrative example below, that successive orders of its CNRS-H jet, centred at one reference scale s0, recover the scale sexit at which Turing instability ceases with rapidly decreasing error, from a finite local jet and without knowledge of the diffusion coefficients away from s0 — reaching sub-percent accuracy at fifth order with exact pointwise derivatives. That behaviour does not survive measurement error. Estimating the jet instead from noisy samples over a narrow calibration window, we find the ordering inverts beyond an optimal retained order that falls as precision degrades — from fifth order for noise-free fitted data to third order at σ= 0.1% and first order at σ= 5% log-space measurement noise (approximately relative noise for small σ) — so that retaining more coefficients becomes actively harmful once they are dominated by noise. For this benchmark and estimator, at an extrapolation ratio of about two, reliable recovery requires approximately 0.1% log-space measurement precision in the diffusion coefficients, which we state as a requirement the method imposes rather than one current practice meets. For one illustrative choice of scale-dependent diffusion profiles and a fully self-consistent Gierer–Meinhardt parameter set — an illustrative pair of unequal effective-transport attenuation profiles — Turing instability is active at and below the reference cellular scale and becomes extinct above a scale-profile-specific threshold (sexit ≈0.538 nats, ℓ≈17 µm for this profile; the numerical value is a property of the chosen illustration, not a predicted biological constant).