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

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.

Figures (9)

• Figure 1: Expectation based stochastic bifurcation diagram. Black curves represent the deterministic equilibrium branches: solid lines denote stable equilibria ($E_1$ and $E_3$), while the dashed line denotes the unstable saddle ($E_2$). Coloured solid curves depict the stationary expectation $\bar{u}(b_1,\sigma)$ for noise intensities $\sigma\in\{0.005, 0.010, 0.020\}$. Shaded bands indicate the corresponding $10\%$--$90\%$ quantile ranges of the stationary distribution.
• Figure 2: Approximate stationary marginal probability density functions (PDFs). Left column (a, c, e): Temperature marginals $p_T(T)$, where the inset in (a) shows a 20-fold vertical magnification of the region $T \in [0.4, 0.55]$ to resolve the weak $E_3$ peak. Right column (b, d, f): Biomass marginals $p_u(u)$ on a logarithmic vertical axis. Rows from top to bottom correspond to control parameter values $b_1=2.0$, $b_1=2.1$, and $b_1=2.2$. Curves represent noise intensities $\sigma=0.005$ (green dotted), $0.010$ (blue dash-dotted), and $0.020$ (orange dashed).
• Figure 3: Schematic representation of the stochastic separatrix and its associated transition layer. The stochastic separatrix $\Gamma = \{q = 1/2\}$ (black curve) separates the background basin ($q \approx 0$) from the bloom basin ($q \approx 1$). The transition layer (shaded region) corresponds to $q \in [0.4,0.6]$ with $\alpha = 0.1$. The local width $w_\alpha(x)$ is defined as the normal distance between the $q=0.4$ and $q=0.6$ boundaries. The gradient $\nabla q(x)$ points toward the bloom state. The geometric early warning indicator $\mathrm{EWS}_{\mathrm{geom}}$ is the arc-length average of this local width along $\Gamma$.
• Figure 4: Deterministic and stochastic separatrices in the $(T,u)$ phase plane for (a) $b_1=2.0$, (b) $b_1=2.1$, and (c) $b_1=2.2$. The deterministic separatrix is shown as a solid black curve, $T$- and $u$-nullclines are indicated by gray lines. Coloured curves denote the stochastic separatrices $\Gamma(b_1,\sigma)$ for $\sigma=0.005$ (green dotted), $\sigma=0.010$ (blue dashed--dotted), and $\sigma=0.020$ (orange dash). The transition layer ($q\in[0.4,0.6]$) is shown as shaded regions using the same colour scheme. Symbols $E_1$ (solid circle), $E_3$ (open circle), and $E_2$ (open square) represent stable equilibria and the saddle point, respectively.
• Figure 5: Geometric measures under varying noise intensity. (a) Positional shift measures $MDB$ (solid lines) and $MDS$ (dotted lines) as functions of $b_1$. (b) $EWS_{\mathrm{geom}}$ versus $b_1$, showing the BIC-selected optimal breakpoints $\hat{b}_1$ (vertical dashed lines) and the corresponding warning intervals (shaded bands, $\Delta \mathrm{BIC} \le 2$). Curves correspond to $\sigma=0.005$ (green), $\sigma=0.010$ (blue), and $\sigma=0.020$ (orange). Inset: BIC profile for $\sigma = 0.010$.