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Chaos is not rare in natural ecosystems

Abstract

Chaotic dynamics are thought to be rare in natural populations but this may be due to methodological and data limitations, rather than the inherent stability of ecosystems. Following extensive simulation testing, we applied multiple chaos detection methods to a global database of 172 population time series and found evidence for chaos in >30%. In contrast, fitting traditional one-dimensional models identified <10% as chaotic. Chaos was most prevalent among plankton and insects and least among birds and mammals. Lyapunov exponents declined with generation time and scaled as the −1/6 power of body mass among chaotic populations. These results demonstrate that chaos is not rare in natural populations, indicating that there may be intrinsic limits to ecological forecasting and cautioning against the use of steady-state approaches to conservation and management.

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Fig. 1: Chaotic dynamics in relation to variability, predictability, nonlinearity and non-stationarity.
Fig. 2: Chaotic dynamics by taxonomic group and model dimensionality.
Fig. 3: Chaotic dynamics in relation to generation time.
Fig. 4: Positive LEs in relation to body mass.

Data availability

The GPDD data are available on KNB with identifier https://doi.org/10.5063/F1BZ63Z8. Zooplankton data were obtained for Oneida Lake from KNB (identifier kgordon.17.67), for Lake Zurich from Wasserversorgung Zürich and for Lake Geneva from the Observatory on LAkes (OLA-IS, AnaEE-France, INRA of Thonon-les-Bains, CIPEL; https://doi.org/10.4081/jlimnol.2020.1944). The simulated datasets and generating code are available in the code repository. The specific GPDD time series used and associated metadata (including compiled generation time and mass data) are available in the code repository.

Code availability

All analysis code is available at https://doi.org/10.5281/zenodo.6499470.

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Acknowledgements

We thank S. Salinas, S. Newkirk, A. Hein, N. Lustenhouwer, A. M. Kilpatrick and M. O’Farrell for comments that improved the manuscript and C. Symons for assisting with access to the lake data. This work was supported by the NOAA Office of Science and Technology (T.L.R. and S.B.M.), SeaGrant no. NA19OAR4170353 (B.J.J.) and the Lenfest Oceans Program (S.B.M.).

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T.L.R., B.J.J. and S.B.M. all contributed to study design, simulations, data analysis and writing. T.L.R. made the figures.

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Correspondence to Tanya L. Rogers or Stephan B. Munch.

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Nature Ecology & Evolution thanks Stephen Ellner, Jef Huisman, Joshua Garland and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Classification error rates for each chaos detection method, marginalized by time series length and noise level.

Results for the test dataset and validation dataset #1 are shown. DLE = direct Lyapunov exponent, JLE = Jacobian-based Lyapunov exponent, RQA = recurrence quantification analysis, PE = permutation entropy, HVG = horizontal visibility graphs, CDT = chaos decision tree.

Extended Data Fig. 2 Chaotic dynamics in relation to predictability and monotonic trend.

(a) Proportion of time series classified as chaotic using the Jacobian method and (b) values of the Lyapunov exponent (LE) in relation to leave-one-out prediction R2 for abundance. (c) Proportion of time series classified as chaotic using the Jacobian method and (d) values of the LE in relation to monotonic trend, as measured by the squared Spearman rank correlation coefficient. In (A) and (C), the line is a logistic regression, associated band is the 95% confidence interval, and points are vertically jittered to reduce overlap. Point colour indicates taxonomic group.

Extended Data Fig. 3 1-d models fit to the empirical GPDD dataset.

Table includes the average R2 and Lyapunov exponent (LE) across all time series and the proportion of time series classified as chaotic. The HLM II model extends the model of17 to allow for adult survival analogous to what19 did with the Ricker model85.

Extended Data Fig. 4 Probability of chaotic dynamics by location.

(a) Number of time series per location for the 57 different locations in the GPDD dataset. (b) Probability that a location is chaotic, given the observed proportion of chaotic series using the Jacobian method and total error rates for the Jacobian method in the simulated datasets. These results assume that a location represents a single well-mixed ecosystem where species interact of similar timescales, which is not necessarily true. These results should also be interpreted with caution as the error rates depend the particular suite of simulations used, and it is impossible to know whether this suite is a good reflection of ecological reality. Colour indicates taxonomic group(s) from each location.

Extended Data Fig. 5 Distribution of Lyapunov exponents (LEs) for 3 locations with more than 8 time series.

Colour indicates taxonomic group for each time series.

Extended Data Fig. 6 Random sample of a chaotic and non-chaotic time series from each taxonomic group from the GPDD dataset.

Top to bottom: birds, bony fishes, insects, mammals, phytoplankton, zooplankton. Left panels were classified as chaotic using the Jacobian method, right panels as not chaotic. The number in parentheses is the database ID (MainID). Beyond illustrating the data, these plots corroborate the well-known fact that chaotic and non-chaotic series cannot be reliably differentiated by visual inspection3.

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Supplementary Notes 1–5, Figs. 1–9 and Tables 1–13.

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Rogers, T.L., Johnson, B.J. & Munch, S.B. Chaos is not rare in natural ecosystems. Nat Ecol Evol 6, 1105–1111 (2022). https://doi.org/10.1038/s41559-022-01787-y

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