Geometric early warning indicator from stochastic separatrix structure in a random two-state ecosystem model

TL;DR

This paper introduces a geometric early warning indicator based on stochastic separatrix structure in a bistable ecosystem model, offering a robust alternative to traditional signals in noisy Arctic environments.

Contribution

It develops a novel geometric indicator derived from the committor function that scales predictably with noise, improving early warning capabilities in high-variability ecosystems.

Findings

The geometric indicator scales linearly with noise strength.

It correlates logarithmically with mean first passage time.

The indicator remains reliable when traditional methods fail due to rapid transitions.

Abstract

Under-ice blooms in the Arctic can develop rapidly under conditions where conventional early warning signals based on critical slowing down fail due to strong noise or limited observational records. We analyze noise-induced transitions in a temperature phytoplankton stochastic differential equation model exhibiting bistability between background and bloom states. The committor function defines a stochastic separatrix as its 1/2-isocommittor, and the normal width of the associated transition layer yields a geometric indicator via arc-length averaging. Under systematic variation of noise intensity, this indicator scales linearly with noise strength, while the logarithm of the mean first passage time follows the Freidlin-Wentzell asymptotic law. Eliminating the noise parameter produces an affine scaling between the logarithmic transition time and the inverse square of the geometric indicator. The relation is robust under variations in discretization, neighborhood definition, and diffusion structure, and holds in the weak noise regime where the transition-layer width scales linearly with noise strength. Unlike variance or lag-one autocorrelation, the geometric indicator remains well defined when rapid transitions preclude reliable time-series estimation. These results provide a geometrically interpretable precursor of bloom onset that may support model-based ecological monitoring in high-variability Arctic systems.