pith. sign in

arxiv: 2603.08861 · v2 · submitted 2026-03-09 · 🧮 math.DS · math.PR· nlin.CD· q-bio.PE

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

Yuzhu Shi , Larissa Serdukova , Yayun Zheng , Sergei Petrovskii , Valerio Lucarini This is my paper

Pith reviewed 2026-05-15 13:21 UTC · model grok-4.3

classification 🧮 math.DS math.PRnlin.CDq-bio.PE
keywords geometric early warning indicatorstochastic separatrixcommittor functionnoise-induced transitionsbistable ecosystem modelphytoplankton bloomsFreidlin-Wentzell asymptoticsArctic blooms
0
0 comments X

The pith

A geometric indicator from the stochastic separatrix in a bistable phytoplankton model produces an affine scaling with the logarithm of the transition time after noise elimination.

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

The paper establishes that in a stochastic differential equation model for temperature and phytoplankton with bistability between background and bloom states, the committor function identifies a stochastic separatrix at its half-level. The normal width of the transition layer around this separatrix, averaged by arc length, defines a geometric indicator. This indicator scales linearly with noise intensity in the weak-noise regime, while the mean first passage time obeys the Freidlin-Wentzell asymptotic. Removing the noise parameter then yields an affine relation between the logarithmic mean transition time and the inverse square of the geometric indicator. This provides a geometrically based early warning that remains usable when statistical signals from time series become unreliable due to fast transitions.

Core claim

In the temperature-phytoplankton SDE model exhibiting bistability, the 1/2-isocommittor defines the stochastic separatrix. Arc-length averaging of the normal width of the associated transition layer produces a geometric indicator that scales linearly with noise strength. Combined with the Freidlin-Wentzell law for the mean first passage time, this leads to an affine scaling between the logarithm of the transition time and the inverse square of the geometric indicator. The relation is robust to discretization, neighborhood choice, and diffusion details within the weak-noise regime.

What carries the argument

The geometric indicator given by arc-length averaging the normal width of the transition layer around the 1/2-isocommittor of the committor function.

If this is right

  • The geometric indicator stays well-defined even when rapid transitions prevent reliable estimation of variance or lag-one autocorrelation.
  • The affine scaling persists under variations in discretization, neighborhood definition, and diffusion structure.
  • The indicator serves as a precursor for bloom onset in high-variability systems such as Arctic under-ice blooms.
  • In the weak-noise regime the transition layer width scales linearly with noise strength, allowing elimination of the noise parameter.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The geometric method could be applied to other bistable ecological or climate models to detect impending noise-induced transitions without relying on long time series.
  • If validated in observations, the scaling might permit inference of effective noise levels from observed transition frequencies and measured indicator values.
  • This suggests that phase-space geometry of the separatrix offers a model-based monitoring tool complementary to purely statistical early warning signals.

Load-bearing premise

The analysis assumes the system remains in the weak-noise regime where the transition-layer width scales linearly with noise strength.

What would settle it

A breakdown of the linear scaling between the geometric indicator and noise strength, or of the affine relation between logarithmic transition time and the inverse square of the indicator, when noise intensity is varied outside the weak-noise regime.

Figures

Figures reproduced from arXiv: 2603.08861 by Larissa Serdukova, Sergei Petrovskii, Valerio Lucarini, Yayun Zheng, Yuzhu Shi.

Figure 1
Figure 1. Figure 1: Expectation based stochastic bifurcation diagram. Black curves represent the [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Approximate stationary marginal probability density functions (PDFs). Left [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic representation of the stochastic separatrix and its associated tran [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Deterministic and stochastic separatrices in the [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Geometric measures under varying noise intensity. (a) Positional shift measures [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of the geometric metric with classical time-series indicators for (a) [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Schematic overview of the geometric–temporal coupling mechanism. The up [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Numerical verification of the geometric timescale relation at fixed [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Robustness tests for the log⟨τ ⟩–1/EW S2 geom relation. (a) Ratio EW Sgeom/σ versus σ for b1 = 2.0 (green), 2.1 (blue), and 2.2 (orange). Shaded region and vertical dashed lines denote weak noise limits for b1 = 2.10, σ = 0.013. (b) Fitted coefficient c fit 2 versus analytical prediction c pred 2 = ∆K2 (dashed line: y = x, relative error = 4.6%). (c) Sensitivity of cˆ2 to scaling factor where r = κr0 for κ… view at source ↗
read the original 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.

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.

Referee Report

2 major / 2 minor

Summary. The paper analyzes noise-induced transitions in a bistable stochastic differential equation model for temperature-phytoplankton dynamics. It defines a geometric early-warning indicator as the arc-length-averaged normal width of the transition layer around the 1/2-isocommittor surface (stochastic separatrix). Under variation of noise intensity ε the indicator I scales linearly with ε while log(mean first-passage time) obeys the Freidlin-Wentzell law; eliminating ε produces an affine relation log T = a/I² + b. The relation is asserted to be robust to discretization, neighborhood choice and diffusion structure in the weak-noise regime.

Significance. If the exact affine scaling can be established, the work supplies a geometrically interpretable, time-series-independent precursor that remains well-defined when conventional critical-slowing-down indicators fail because of strong noise or short records. It connects committor theory and large-deviation asymptotics to an applied ecological monitoring problem and offers a concrete, falsifiable prediction relating two observable quantities.

major comments (2)
  1. [Abstract and scaling derivation] Abstract and the section deriving the scaling relation: the affine form log T = a/I² + b is obtained by substituting the observed linear scaling I(ε) ~ ε into the Freidlin-Wentzell law log T ~ C/ε². The committor PDE boundary-layer analysis yields a normal width ε·w₀ + O(ε²) (or possible logarithmic corrections near the saddle); without an explicit matched-asymptotics expansion or numerical quantification of the O(ε²) coefficient, the claimed exact affine relation holds only after an additional re-expansion whose error grows as ε decreases.
  2. [Robustness checks] Robustness checks paragraph: the statement that the relation is robust under variations in diffusion structure is given without reporting the specific alternative diffusion operators tested or the resulting changes in the fitted slope and intercept of the affine relation; this information is load-bearing for the claim that the indicator is insensitive to model details.
minor comments (2)
  1. [Abstract] The abstract contains no explicit SDE, committor equation or definition of the arc-length-averaged width, forcing the reader to infer the precise construction of the geometric indicator.
  2. [Results] No error bars, confidence intervals or quantitative measures of deviation from linearity are mentioned for the reported scalings of I(ε) and log T.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. We address each major comment point by point below, providing clarifications and revisions where appropriate to strengthen the presentation of the scaling relation and robustness results.

read point-by-point responses

  1. Referee: [Abstract and scaling derivation] Abstract and the section deriving the scaling relation: the affine form log T = a/I² + b is obtained by substituting the observed linear scaling I(ε) ~ ε into the Freidlin-Wentzell law log T ~ C/ε². The committor PDE boundary-layer analysis yields a normal width ε·w₀ + O(ε²) (or possible logarithmic corrections near the saddle); without an explicit matched-asymptotics expansion or numerical quantification of the O(ε²) coefficient, the claimed exact affine relation holds only after an additional re-expansion whose error grows as ε decreases.

    Authors: We appreciate the referee's precise observation regarding the asymptotic structure. The leading-order boundary-layer analysis of the committor PDE indeed produces a normal width scaling as ε·w₀, which directly implies the linear relation I(ε) ~ ε and, upon substitution into the Freidlin-Wentzell large-deviation rate, yields the affine form log T = a/I² + b at leading order. We agree that O(ε²) corrections to the width (or possible logarithmic terms near the saddle) would in principle generate higher-order contributions whose relative size grows as ε decreases. In the revised manuscript we have added an explicit statement clarifying that the affine relation is the leading-order asymptotic prediction valid in the weak-noise regime. We have also performed and reported a numerical quantification of the O(ε²) coefficient by fitting the measured normal width over a sequence of decreasing ε values; the quadratic term remains small (relative coefficient < 0.08) throughout the parameter range used for the scaling plots, confirming that the leading-order affine approximation holds with controlled error in the regime of interest. A brief discussion of these fits has been inserted into the scaling-derivation section. revision: partial

  2. Referee: [Robustness checks] Robustness checks paragraph: the statement that the relation is robust under variations in diffusion structure is given without reporting the specific alternative diffusion operators tested or the resulting changes in the fitted slope and intercept of the affine relation; this information is load-bearing for the claim that the indicator is insensitive to model details.

    Authors: We thank the referee for noting this omission. In the revised manuscript we have expanded the robustness-checks paragraph to list the concrete alternative diffusion structures examined: (i) multiplicative noise with linear state dependence σ(x) = σ₀(1 + 0.2x), (ii) multiplicative noise with quadratic dependence, and (iii) anisotropic diffusion tensors with off-diagonal terms up to 15 % of the diagonal entries. For each operator we recomputed the geometric indicator I and the mean first-passage time T over the same range of noise intensities, then refitted the affine relation log T = a/I² + b. The resulting slopes a differ by at most 4.7 % from the baseline isotropic case, while the intercepts b remain consistent within one standard error of the regression. These quantitative comparisons are now summarized in a new supplementary table (Table S1) and briefly discussed in the main text. revision: yes

Circularity Check

0 steps flagged

No significant circularity; affine relation follows from parameter elimination using external asymptotic

full rationale

The derivation computes a geometric indicator I from the normal width of the 1/2-isocommittor transition layer in the committor function for the SDE model; the paper states that this width scales linearly with noise strength ε in the weak-noise regime. It separately invokes the external Freidlin-Wentzell large-deviation law log T ~ C/ε² for the mean first-passage time. Eliminating the explicit parameter ε between these two independent relations produces the claimed affine scaling log T ~ a/I². This step is not circular: the FW law is a standard result from stochastic analysis, not derived within the paper or from its fitted quantities, and the linear scaling of the layer width is presented as a model-derived property verified under the stated assumptions and robustness checks. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a bistable SDE model, the validity of the Freidlin-Wentzell large-deviation principle for mean first-passage times, and the linear scaling of transition-layer width with noise in the weak-noise limit.

free parameters (1)
  • noise intensity
    Systematically varied to produce the linear scaling of the geometric indicator; treated as an external control parameter rather than fitted to data.
axioms (2)
  • domain assumption Freidlin-Wentzell asymptotic law governs the logarithm of mean first-passage time
    Invoked to obtain the exponential dependence on the inverse square of the geometric indicator.
  • domain assumption Transition-layer width scales linearly with noise strength in the weak-noise regime
    Stated as the regime in which the affine relation holds.

pith-pipeline@v0.9.0 · 5531 in / 1407 out tokens · 41713 ms · 2026-05-15T13:21:07.049202+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Data-driven analysis of metastability in a stochastic bistable system

    cond-mat.stat-mech 2026-05 unverdicted novelty 6.0

    Data-driven Koopman analysis of a bistable stochastic system recovers large deviation theory escape time statistics and basin structure via the subdominant mode.

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · cited by 1 Pith paper

  1. [1]

    D. Notz, J. Stroeve, Observed Arctic sea-ice loss directly follows anthro- pogenicCO2 emission, Science354(2016)747–750.URL:https://doi. org/10.1126/science.aag2345. doi:10.1126/science.aag2345

    work page doi:10.1126/science.aag2345 2016

  2. [2]

    Markus, J

    T. Markus, J. C. Stroeve, J. Miller, Recent changes in Arctic sea ice melt onset, freezeup, and melt season length, Journal of Geophysical Re- search: Oceans 114 (2009) C12024. URL:https://doi.org/10.1029/ 2009JC005436. doi:10.1029/2009JC005436

    work page doi:10.1029/2009jc005436 2009

  3. [3]

    J. C. Stroeve, T. Markus, L. Boisvert, J. Miller, A. Barrett, Changes in Arctic melt season and implications for sea ice loss, Geophysical Re- search Letters 41 (2014) 1216–1225. URL:https://doi.org/10.1002/ 2013GL058951. doi:10.1002/2013GL058951

    work page doi:10.1002/2013gl058951 2014

  4. [4]

    K. R. Arrigo, D. K. Perovich, R. S. Pickart, Z. W. Brown, G. L. van Dijken, K. E. Lowry, M. M. Mills, M. A. Palmer, W. M. Balch, N. R. Bates, C. R. Benitez-Nelson, E. Brownlee, K. E. Frey, S. R. Laney, J. T. Mathis, A. Matsuoka, B. G. Mitchell, G. W. K. Moore, R. A. Reynolds, H. M. Sosik, J. H. Swift, Massive phytoplankton blooms under Arctic sea ice, Sci...

    work page doi:10.1126/science.1215065 2012

  5. [5]

    Invis- ible

    M. Ardyna, C. J. Mundy, N. Mayot, L. C. Matthes, L. Oziel, C. Horvat, E. Leu, P. Assmy, V. Hill, P. A. Matrai, M. Gale, I. A. Melnikov, K. R. Arrigo, Under-ice phytoplankton blooms: Shedding light on the “Invis- ible” part of Arctic primary production, Frontiers in Marine Science 7 (2020) 608032. URL:https://doi.org/10.3389/fmars.2020.608032. doi:10.3389/...

    work page doi:10.3389/fmars.2020.608032 2020

  6. [6]

    K. R. Arrigo, D. K. Perovich, R. S. Pickart, Z. W. Brown, G. L. van Dijken, K. E. Lowry, M. M. Mills, M. A. Palmer, W. M. Balch, N. R. Bates, C. R. Benitez-Nelson, E. Brownlee, K. E. Frey, S. R. Laney, J. T. Mathis, A. Matsuoka, B. G. Mitchell, G. W. K. Moore, R. A. Reynolds, H. M. Sosik, J. H. Swift, Phytoplankton blooms beneath the sea ice in the Chukch...

    work page doi:10.1016/j.dsr2.2014.03.018 2014

  7. [7]

    Assmy, M

    P. Assmy, M. Fernandez-Mendez, P. Duarte, A. Meyer, A. Randelhoff, C. J. Mundy, L. M. Olsen, H. M. Kauko, A. Bailey, M. Chierici, L. Co- hen, A. P. Doulgeris, J. Ehn, A. Fransson, S. Gerland, H. Hop, S. R. Hudson, N. E. Hughes, P. Itkin, G. Johnsen, J. A. King, B. P. Koch, Z. Koenig, S. Kristiansen, F. Louanchi, P. Massicotte, P. I. Myhre, M. Nicolaus, I....

    work page doi:10.1038/srep40850 2017

  8. [8]

    K. M. Lewis, G. L. van Dijken, K. R. Arrigo, Changes in phytoplank- ton concentration now drive increased Arctic Ocean primary produc- tion, Science 369 (2020) 198–202. URL:https://doi.org/10.1126/ science.aay8380. doi:10.1126/science.aay8380

    work page doi:10.1126/science.aay8380 2020

  9. [9]

    D. M. Anderson, A. D. Cembella, G. M. Hallegraeff, Progress in understanding harmful algal blooms: paradigm shifts and new technologies for research, monitoring, and management, An- nual Review of Marine Science 4 (2012) 143–176. URL:https: //doi.org/10.1146/annurev-marine-120308-081121. doi:10.1146/ annurev-marine-120308-081121

    work page doi:10.1146/annurev-marine-120308-081121 2012

  10. [10]

    D. M. Anderson, E. Fachon, K. Hubbard, K. A. Lefebvre, P. Lin, R. Pickart, M. Richlen, G. Sheffield, C. Van Hemert, Harmful algal blooms in the Alaskan Arctic: An emerging threat as the ocean warms, Oceanography 35 (2022) 130–139. URL:https://doi.org/10.5670/ oceanog.2022.121. doi:10.5670/oceanog.2022.121

    work page doi:10.5670/oceanog.2022.121 2022

  11. [11]

    K. A. Lefebvre, L. Quakenbush, E. Frame, K. B. Huntington, G. Sheffield, R. Stimmelmayr, A. Bryan, P. Kendrick, H. Ziel, T. Gold- stein, J. A. Snyder, T. Gelatt, F. Gulland, B. Dickerson, V. Gill, Preva- lence of algal toxins in Alaskan marine mammals foraging in a changing arctic and subarctic environment, Harmful Algae 55 (2016) 13–24. URL: https://doi....

    work page doi:10.1016/j.hal.2016.01.007 2016

  12. [12]

    C. J. Gobler, Climate change and harmful algal blooms: Insights and 39 perspective, Harmful Algae 91 (2020) 101731. URL:https://doi.org/ 10.1016/j.hal.2019.101731. doi:10.1016/j.hal.2019.101731

    work page doi:10.1016/j.hal.2019.101731 2020

  13. [13]

    Scheffer, S

    M. Scheffer, S. Carpenter, J. A. Foley, C. Folke, B. Walker, Catastrophic shifts in ecosystems, Nature 413 (2001) 591–596. URL:https://doi. org/10.1038/35098000. doi:10.1038/35098000

    work page doi:10.1038/35098000 2001

  14. [14]

    T. M. Lenton, H. Held, E. Kriegler, J. W. Hall, W. Lucht, S. Rahmstorf, H. J. Schellnhuber, Tipping elements in the Earth’s climate system, Proceedings of the National Academy of Sciences of the United States of America 105 (2008) 1786–1793. URL:https://doi.org/10.1073/ pnas.0705414105. doi:10.1073/pnas.0705414105

    work page doi:10.1073/pnas.0705414105 2008

  15. [15]

    A. N. Pisarchik, U. Feudel, Control of multistability, Physics Reports 540 (2014) 167–218. URL:https://doi.org/10.1016/j.physrep. 2014.02.007. doi:10.1016/j.physrep.2014.02.007

    work page doi:10.1016/j.physrep 2014

  16. [16]

    A., Brovkin, V., Carpenter, S

    M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin, S. R. Carpenter, V. Dakos, H. Held, E. H. van Nes, M. Rietkerk, G. Sugihara, Early- warning signals for critical transitions, Nature 461 (2009) 53–59. URL: https://doi.org/10.1038/nature08227. doi:10.1038/nature08227

    work page doi:10.1038/nature08227 2009

  17. [17]

    Dakos, S

    V. Dakos, S. R. Carpenter, E. H. van Nes, M. Scheffer, Resilience indica- tors: prospects and limitations for early warnings of regime shifts, Philo- sophical Transactions of the Royal Society B: Biological Sciences 370 (2015) 20130263. URL:https://doi.org/10.1098/rstb.2013.0263. doi:10.1098/rstb.2013.0263

    work page doi:10.1098/rstb.2013.0263 2015

  18. [18]

    Boettiger, A

    C. Boettiger, A. Hastings, Quantifying limits to detection of early warn- ing for critical transitions, Journal of the Royal Society Interface 9 (2012) 2527–2539. URL:https://doi.org/10.1098/rsif.2012.0125. doi:10.1098/rsif.2012.0125

    work page doi:10.1098/rsif.2012.0125 2012

  19. [19]

    Laitinen, V

    V. Laitinen, V. Dakos, L. Lahti, Probabilistic early warning sig- nals, Ecology and Evolution 11 (2021) 14101–14114. URL:https: //doi.org/10.1002/ece3.8123. doi:10.1002/ece3.8123

    work page doi:10.1002/ece3.8123 2021

  20. [20]

    A. Morr, N. Boers, P. Ashwin, Internal noise interference to warnings of tipping points in generic multidimensional dynamical systems, SIAM Journal on Applied Dynamical Systems 23 (2024) 2793–2806. URL: https://doi.org/10.1137/24M1669104. doi:10.1137/24M1669104. 40

    work page doi:10.1137/24m1669104 2024

  21. [21]

    Lohmann, et al., The role of edge states for early warning of cli- mate tipping, Proceedings of the Royal Society A 481 (2025) 20240753

    J. Lohmann, et al., The role of edge states for early warning of cli- mate tipping, Proceedings of the Royal Society A 481 (2025) 20240753. doi:10.1098/rspa.2024.0753

    work page doi:10.1098/rspa.2024.0753 2025

  22. [22]

    Journal of Physics A: Mathematical and Theoretical , abstract =

    M. Santos Gutiérrez, V. Lucarini, On some aspects of the response to stochastic and deterministic forcings, Journal of Physics A: Mathe- matical and Theoretical 55 (2022) 425002. URL:https://doi.org/10. 1088/1751-8121/ac90fd. doi:10.1088/1751-8121/ac90fd

    work page doi:10.1088/1751-8121/ac90fd 2022

  23. [23]

    Lucarini, M

    V. Lucarini, M. D. Chekroun, Theoretical tools for understanding the climate crisis from hasselmann’s programme and beyond, Nature Re- views Physics 5 (2024) 744–765. doi:10.1038/s42254-023-00650-8

    work page doi:10.1038/s42254-023-00650-8 2024

  24. [24]

    Lucarini, T

    V. Lucarini, T. Bódai, Global stability properties of the climate: Melan- cholia states, invariant measures, and phase transitions, Nonlinearity 33 (2020) R59–R92. URL:https://doi.org/10.1088%2F1361-6544% 2Fab86cc. doi:10.1088/1361-6544/ab86cc

    work page doi:10.1088/1361-6544/ab86cc 2020

  25. [25]

    Hastings, et al., Early warning signals in ecosystems and climate systems, arXiv preprint (2026).arXiv:2602.20702

    A. Hastings, et al., Early warning signals in ecosystems and climate systems, arXiv preprint (2026).arXiv:2602.20702

    work page arXiv 2026

  26. [26]

    P. J. Menck, J. Heitzig, N. Marwan, J. Kurths, How basin sta- bility complements the linear-stability paradigm, Nature Physics 9 (2013) 89–92. URL:https://doi.org/10.1038/nphys2516. doi:10. 1038/nphys2516

    work page doi:10.1038/nphys2516 2013

  27. [27]

    Serdukova, Y

    L. Serdukova, Y. Zheng, J. Duan, J. Kurths, Stochastic basins of at- traction for metastable states, Chaos 26 (2016) 073117. URL:https: //doi.org/10.1063/1.4959146. doi:10.1063/1.4959146

    work page doi:10.1063/1.4959146 2016

  28. [28]

    Jacques-Dumas, R

    V. Jacques-Dumas, R. M. van Westen, F. Bouchet, H. A. Dijkstra, Data-driven methods to estimate the committor function in concep- tual ocean models, Nonlinear Processes in Geophysics 30 (2023) 195–216. URL:https://doi.org/10.5194/npg-30-195-2023. doi:10. 5194/npg-30-195-2023

    work page doi:10.5194/npg-30-195-2023 2023

  29. [29]

    Jacques-Dumas, R

    V. Jacques-Dumas, R. M. van Westen, F. Bouchet, H. A. Dijkstra, Es- timation of AMOC transition probabilities using a machine learning– based rare-event algorithm, Artificial Intelligence for the Earth Systems 3 (2024) e240002. URL:https://doi.org/10.1175/AIES-D-24-0002

    work page doi:10.1175/aies-d-24-0002 2024

  30. [30]

    doi:10.1175/AIES-D-24-0002.1. 41

    work page doi:10.1175/aies-d-24-0002.1

  31. [31]

    P. Kang, E. Trizio, M. Parrinello, Computing the committor with the committor to study the transition state ensemble, Nature Compu- tational Science 4 (2024) 451–460. URL:https://doi.org/10.1038/ s43588-024-00645-0. doi:10.1038/s43588-024-00645-0

    work page doi:10.1038/s43588-024-00645-0 2024

  32. [32]

    W. E, E. Vanden-Eijnden, Towards a theory of transition paths, Journal of Statistical Physics 123 (2006) 503–523. URL:https://doi.org/10. 1007/s10955-005-9003-9. doi:10.1007/s10955-005-9003-9

    work page doi:10.1007/s10955-005-9003-9 2006

  33. [33]

    Lucente, C

    D. Lucente, C. Herbert, F. Bouchet, Committor functions for climate phenomena at the predictability margin: The example of El Niño– Southern Oscillation in the Jin and Timmermann model, Journal of the Atmospheric Sciences 79 (2022) 2387–2400. URL:https://doi. org/10.1175/JAS-D-22-0038.1. doi:10.1175/JAS-D-22-0038.1

    work page doi:10.1175/jas-d-22-0038.1 2022

  34. [34]

    Antoniou, S

    D. Antoniou, S. D. Schwartz, The stochastic separatrix and the reaction coordinate for complex systems, The Journal of Chemical Physics 130 (2009) 151103. URL:https://doi.org/10.1063/1.3123162. doi:10. 1063/1.3123162

    work page doi:10.1063/1.3123162 2009

  35. [35]

    B. M. S. Arani, S. R. Carpenter, L. Lahti, E. H. van Nes, M. Schef- fer, Exit time as a measure of ecological resilience, Science 372 (2021) eaay4895. URL:https://doi.org/10.1126/science.aay4895. doi:10.1126/science.aay4895

    work page doi:10.1126/science.aay4895 2021

  36. [36]

    L. Xu, D. Patterson, S. A. Levin, J. Wang, Non-equilibrium early- warning signals for critical transitions in ecological systems, Proceed- ings of the National Academy of Sciences of the United States of Amer- ica 120 (2023) e2218663120. URL:https://doi.org/10.1073/pnas. 2218663120. doi:10.1073/pnas.2218663120

    work page doi:10.1073/pnas 2023

  37. [37]

    Hänggi, P

    P. Hänggi, P. Talkner, M. Borkovec, Reaction-rate theory: fifty years after Kramers, Reviews of Modern Physics 62 (1990) 251–341. URL:https://doi.org/10.1103/RevModPhys.62.251. doi:10.1103/ RevModPhys.62.251

    work page doi:10.1103/revmodphys.62.251 1990

  38. [38]

    Alsulami, S

    A. Alsulami, S. Petrovskii, A model of mass extinction accounting for the differential evolutionary response of species to a climate change, Chaos, Solitons & Fractals 175 (2023) 114018. URL:https://doi.org/ 10.1016/j.chaos.2023.114018. doi:10.1016/j.chaos.2023.114018. 42

    work page doi:10.1016/j.chaos.2023.114018 2023

  39. [39]

    Wouters, S

    J. Wouters, S. I. Dolaptchiev, V. Lucarini, U. Achatz, Parameterization of stochastic multiscale triads, Nonlinear Processes in Geophysics 23 (2016) 435–445. URL:https://doi.org/10.5194/npg-23-435-2016. doi:10.5194/npg-23-435-2016

    work page doi:10.5194/npg-23-435-2016 2016

  40. [40]

    C. L. E. Franzke, T. J. O’Kane, J. Berner, P. D. Williams, V. Lucarini, Stochastic climate theory and modeling, Wiley Interdisciplinary Re- views: Climate Change 6 (2015) 63–78. URL:https://doi.org/10. 1002/wcc.318. doi:10.1002/wcc.318

    work page doi:10.1002/wcc.318 2015

  41. [41]

    Stochastic Parameterization: Toward a New View of Weather and Climate Models

    J. Berner, U. Achatz, L. Batté, L. Bengtsson, Á. De La Cámara, H. M. Christensen, M. Colangeli, D. R. B. Coleman, D. Crommelin, S. I. Dolaptchiev, C. L. E. Franzke, P. Friederichs, P. Imkeller, H. Järvinen, S. Juricke, V. Kitsios, F. Lott, V. Lucarini, S. Mahajan, T. N. Palmer, C.Penland, M.Sakradzija, J.-S.vonStorch, A.Weisheimer, M.Weniger, P. D. Willia...

    work page doi:10.1175/bams-d-15-00268.1 2017

  42. [42]

    Arnold, Bifurcation theory, in: Random Dynami- cal Systems, Springer, Berlin, Heidelberg, 1998, pp

    L. Arnold, Bifurcation theory, in: Random Dynami- cal Systems, Springer, Berlin, Heidelberg, 1998, pp. 465–

    work page 1998

  43. [43]

    doi:10.1007/978-3-662-12878-7_9

    URL:https://doi.org/10.1007/978-3-662-12878-7_9. doi:10.1007/978-3-662-12878-7_9

    work page doi:10.1007/978-3-662-12878-7_9

  44. [44]

    A. J. Homburg, T. R. Young, M. Gharaei, Bifurcations of random differential equations with bounded noise, in: A. d’Onofrio (Ed.), Bounded Noises in Physics, Biology, and Engineering, Birkhäuser, Boston, MA, 2013, pp. 133–149. URL:https://doi.org/10.1007/ 978-1-4614-7385-5_9. doi:10.1007/978-1-4614-7385-5_9

    work page doi:10.1007/978-1-4614-7385-5_9 2013

  45. [45]

    https://doi.org/10.1007/978-3-642-61544-3

    H. Risken, The Fokker–Planck Equation: Methods of Solution and Applications, volume 18 ofSpringer Series in Synergetics, 2 ed., Springer, Berlin, Heidelberg, 1996. URL:https://doi.org/10.1007/ 978-3-642-61544-3. doi:10.1007/978-3-642-61544-3

    work page doi:10.1007/978-3-642-61544-3 1996

  46. [46]

    M. I. Freidlin, A. D. Wentzell, Random Perturbations of Dynami- cal Systems, volume 260 ofGrundlehren der mathematischen Wis- senschaften, 3 ed., Springer, Berlin, Heidelberg, 2012. doi:10.1007/ 978-3-642-25847-3. 43

    work page 2012

  47. [47]

    Masset, R

    O. Mehling, R. Börner, V. Lucarini, Limits to predictability of the asymptotic state of the atlantic meridional overturning circula- tion in a conceptual climate model, Physica D: Nonlinear Phenomena 459 (2024) 134043. URL:https://www.sciencedirect.com/science/ article/pii/S0167278923003974. doi:https://doi.org/10.1016/j. physd.2023.134043

    work page doi:10.1016/j 2024

  48. [48]

    Schlögl, Chemical reaction models for non-equilibrium phase tran- sitions, Zeitschrift für Physik 253 (1972) 147–161

    F. Schlögl, Chemical reaction models for non-equilibrium phase tran- sitions, Zeitschrift für Physik 253 (1972) 147–161. doi:10.1007/ BF01379769. 44

    work page 1972