Abstract
Ecological systems often exhibit complex nonlinear dynamics like oscillations, chaos, and regime shifts. Universal dynamic equations have shown promise in modeling complex dynamics by combining known functional forms with neural networks that represent unknown relationships. However, these methods do not yet accommodate the forms of uncertainty common to ecological datasets. To address this limitation, we developed state-space universal dynamic equations by combining universal difference and differential equations with a state-space modeling framework, accounting for uncertainty. We tested this framework on three simulated and two empirical case studies and found that this method can recover nonlinear biological interactions that produce complex behaviors including chaos and regime shifts. Their forecasting performance is context-dependent, with the best performance on chaotic and oscillating time series. This innovative approach leveraging both ecological theory and data-driven machine learning offers a promising new way to make accurate and useful predictions of ecosystem change.
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Data availability
The data used in this paper were obtained from two publicly accessible data repositories: The Jornada Experimental Range LTER (https://doi.org/10.6073/pasta/63bfa45df4858db674bf37b52ee5ff44) and the RAM Legacy Stock Assessment Database https://doi.org/10.17616/R34D2X.
Code availability
Software to implement the State-space universal dynamic equation method presented in this papers has been published by the authors on the Julia programming language registry (https://github.com/JuliaRegistries/General) and on GitHub (https://github.com/Jack-H-Buckner/UniversalDiffEq.jl).
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Acknowledgements
This research was supported by the National Science Foundation awards #2233982 and #2233983 to JRW and LCM on Model Enabled Machine Learning for Predicting Ecosystem Regime Shifts. This paper is a product of the model-enabled machine learning for ecology working group, which includes the authors of the paper as well as Cheyenne Jarman, Kunal Rathore, and Emerson Arehart, all of whom provided valuable contributions to the intellectual environment that led to this paper. We would also like to thank Chris Rackauckas for help working with Julia Scientific Machine learning tools and the Hawaiʻi Institute of Marine Biology for hosting a workshop where the ideas for this project were developed.
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J.H.B.: Conceptualization, formal analysis, methodology, software, writing-original draft. Z.D.M.: Conceptualization, methodology, software, writing-review and editing. J.A.E.: Conceptualization, funding acquisition, software, writing-review and editing. N.F.: methodology, software, writing-review and editing. A.G.: Conceptualization, writing-review and editing. L.C.M.: Funding acquisition, project administration, writing-review and editing. JRW: Conceptualization, funding acquisition, project administration, writing-review and editing.
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Communications Earth and Environment thanks the anonymous reviewers for their contribution to the peer review of this work. Primary Handling Editors: Lifen Jiang and Mengjie Wang. [A peer review file is available].
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Buckner, J.H., Meunier, Z.D., Arroyo-Esquivel, J. et al. Recovering complex ecological dynamics from time series using state-space universal dynamic equations.
Commun Earth Environ (2026). https://doi.org/10.1038/s43247-025-03130-2
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DOI: https://doi.org/10.1038/s43247-025-03130-2
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