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 â1/6 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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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.
References
May, R. M. Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos. Science 186, 645â647 (1974).
Beddington, J. R., Free, C. A. & Lawton, J. H. Dynamic complexity in predatorâprey models framed in difference equations. Nature 255, 58â60 (1975).
Hastings, A., Hom, C. L., Ellner, S., Turchin, P. & Godfray, H. C. J. Chaos in ecology: is Mother Nature a strange attractor? Annu. Rev. Ecol. Syst. 24, 1â33 (1993).
Cressie, N. & Wikle, C. K. Statistics for Spatio-Temporal Data (John Wiley & Sons, 2011).
The State of World Fisheries and Aquaculture 2020 (FAO, 2020).
Hastings, A. & Powell, T. Chaos in a three-species food chain. Ecology 72, 896â903 (1991).
Huisman, J. & Weissing, F. J. Biodiversity of plankton by species oscillations and chaos. Nature 402, 407â410 (1999).
Doebeli, M. & Ispolatov, I. Chaos and unpredictability in evolution. Evolution 68, 1365â1373 (2014).
Pearce, M. T., Agarwala, A. & Fisher, D. S. Stabilization of extensive fine-scale diversity by ecologically driven spatiotemporal chaos. Proc. Natl Acad. Sci. USA 117, 14572â14583 (2020).
Costantino, R. F., Desharnais, R. A., Cushing, J. M. & Dennis, B. Chaotic dynamics in an insect population. Science 275, 389â391 (1997).
Becks, L., Hilker, F. M., Malchow, H., Jürgens, K. & Arndt, H. Experimental demonstration of chaos in a microbial food web. Nature 435, 1226â1229 (2005).
Benincá, E. et al. Chaos in a long-term experiment with a plankton community. Nature 451, 822â825 (2008).
Tilman, D. & Wedin, D. Oscillations and chaos in the dynamics of a perennial grass. Nature 353, 653â655 (1991).
Turchin, P. & Ellner, S. P. Living on the edge of chaos: population dynamics of fennoscandian voles. Ecology 81, 3099â3116 (2000).
Ferrari, M. J. et al. The dynamics of measles in sub-Saharan Africa. Nature 451, 679â684 (2008).
Benincà , E., Ballantine, B., Ellner, S. P. & Huisman, J. Species fluctuations sustained by a cyclic succession at the edge of chaos. Proc. Natl Acad. Sci. USA 112, 6389â6394 (2015).
Hassell, M. P., Lawton, J. H. & May, R. M. Patterns of dynamical behaviour in single-species populations. J. Anim. Ecol. 45, 471â486 (1976).
Sibly, R. M., Barker, D., Hone, J. & Pagel, M. On the stability of populations of mammals, birds, fish and insects. Ecol. Lett. 10, 970â976 (2007).
Shelton, A. O. & Mangel, M. Fluctuations of fish populations and the magnifying effects of fishing. Proc. Natl Acad. Sci USA. 108, 7075â7080 (2011).
Salvidio, S. Stability and annual return rates in amphibian populations. Amphib. Reptil. 32, 119â124 (2011).
Snell, T. W. & Serra, M. Dynamics of natural rotifer populations. Hydrobiologia 368, 29â35 (1998).
Gross, T., Ebenhöh, W. & Feudel, U. Long food chains are in general chaotic. Oikos 109, 135â144 (2005).
Ispolatov, I., Madhok, V., Allende, S. & Doebeli, M. Chaos in high-dimensional dissipative dynamical systems. Sci. Rep. 5, 12506 (2015).
Clark, T. J. & Luis, A. D. Nonlinear population dynamics are ubiquitous in animals. Nat. Ecol. Evol. 4, 75â81 (2020).
Sivakumar, B., Berndtsson, R., Olsson, J. & Jinno, K. Evidence of chaos in the rainfall-runoff process. Hydrol. Sci. J. 46, 131â145 (2001).
Hanski, I., Turchin, P., Korpimäki, E. & Henttonen, H. Population oscillations of boreal rodents: regulation by mustelid predators leads to chaos. Nature 364, 232â235 (1993).
Turchin, P. & Taylor, A. D. Complex dynamics in ecological time series. Ecology 73, 289â305 (1992).
Munch, S. B., Brias, A., Sugihara, G. & Rogers, T. L. Frequently asked questions about nonlinear dynamics and empirical dynamic modelling. ICES J. Mar. Sci. 77, 1463â1479 (2020).
Sugihara, G. & May, R. M. Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series. Nature 344, 734â741 (1990).
Ellner, S. P. & Turchin, P. Chaos in a noisy world: new methods and evidence from time-series analysis. Am. Nat. 145, 343â375 (1995).
Nychka, D., Ellner, S., Gallant, A. R. & McCaffrey, D. Finding chaos in noisy systems. J. R. Stat. Soc. B 54, 399â426 (1992).
Webber, C. L. & Zbilut, J. P. Dynamical assessment of physiological systems and states using recurrence plot strategies. J. Appl. Physiol. 76, 965â973 (1994).
Bandt, C. & Pompe, B. Permutation entropy: a natural complexity measure for time series. Phys. Rev. Lett. 88, 174102 (2002).
Luque, B., Lacasa, L., Ballesteros, F. & Luque, J. Horizontal visibility graphs: exact results for random time series. Phys. Rev. E 80, 46103 (2009).
Toker, D., Sommer, F. T. & DâEsposito, M. A simple method for detecting chaos in nature. Commun. Biol. 3, 11 (2020).
Pikovsky, A. & Politi, A. Lyapunov Exponents: A Tool to Explore Complex Dynamics (Cambridge Univ. Press, 2016).
Rosenstein, M. T., Collins, J. J. & De Luca, C. J. A practical method for calculating largest Lyapunov exponents from small data sets. Physica D 65, 117â134 (1993).
Dämmig, M. & Mitschke, F. Estimation of Lyapunov exponents from time series: the stochastic case. Phys. Lett. A 178, 385â394 (1993).
Prendergast, J., Bazeley-White, E., Smith, O., Lawton, J. & Inchausti, P. The Global Population Dynamics Database (KNB, 2010); https://doi.org/10.5063/F1BZ63Z8
Thibaut, L. M. & Connolly, S. R. Hierarchical modeling strengthens evidence for density dependence in observational time series of population dynamics. Ecology 101, e02893 (2020).
Knape, J. & de Valpine, P. Are patterns of density dependence in the Global Population Dynamics Database driven by uncertainty about population abundance? Ecol. Lett. 15, 17â23 (2012).
Takens, F. in Dynamical Systems and Turbulence (eds Rand, D. A. & Young, L. S.) 366â381 (Springer, 1981).
Sugihara, G. Nonlinear forecasting for the classification of natural time series. Philos. Trans. R. Soc. A 348, 477â495 (1994).
Loh, J. et al. The Living Planet Index: using species population time series to track trends in biodiversity. Philos. Trans. R. Soc. B 360, 289â295 (2005).
Kendall, B. E. Cycles chaos, and noise in predatorâprey dynamics. Chaos Solitons Fractals 12, 321â332 (2001).
Anderson, C. N. K. et al. Why fishing magnifies fluctuations in fish abundance. Nature 452, 835â839 (2008).
Anderson, D. M. & Gillooly, J. F. Allometric scaling of Lyapunov exponents in chaotic populations. Popul. Ecol. 62, 364â369 (2020).
Graham, D. W. et al. Experimental demonstration of chaotic instability in biological nitrification. ISME J. 1, 385â393 (2007).
Turchin, P. Nonlinear time-series modeling of vole population fluctuations. Res. Popul. Ecol. 38, 121â132 (1996).
Becks, L. & Arndt, H. Different types of synchrony in chaotic and cyclic communities. Nat. Commun. 4, 1359 (2013).
Becks, L. & Arndt, H. Transitions from stable equilibria to chaos, and back, in an experimental food web. Ecology 89, 3222â3226 (2008).
Rezende, E. L., Albert, E. M., Fortuna, M. A. & Bascompte, J. Compartments in a marine food web associated with phylogeny, body mass, and habitat structure. Ecol. Lett. 12, 779â788 (2009).
Krause, A. E., Frank, K. A., Mason, D. M., Ulanowicz, R. E. & Taylor, W. W. Compartments revealed in food-web structure. Nature 426, 282â285 (2003).
The IUCN Red List of Threatened Species Version 2020-2 (IUCN, 2020); https://www.iucnredlist.org
Freckleton, R. P. & Watkinson, A. R. Are weed population dynamics chaotic? J. Appl. Ecol. 39, 699â707 (2002).
May, R. M. Simple mathematical models with very complicated dynamics. Nature 261, 459â467 (1976).
Mora, C., Tittensor, D. P., Adl, S., Simpson, A. G. B. & Worm, B. How many species are there on Earth and in the ocean? PLoS Biol. 9, e1001127 (2011).
Munch, S. B., Giron-Nava, A. & Sugihara, G. Nonlinear dynamics and noise in fisheries recruitment: a global meta-analysis. Fish Fish. 19, 964â973 (2018).
Boettiger, C., Harte, T., Chamberlain, S. & Ram, K. rgpdd: R Interface to the Global Population Dynamics Database. https://docs.ropensci.org/rgpdd, https://github.com/ropensci/rgpdd (2019).
Brook, B. W., Traill, L. W. & Bradshaw, C. J. A. Minimum viable population sizes and global extinction risk are unrelated. Ecol. Lett. 9, 375â382 (2006).
Baars, J. W. M. Autecological investigations of marine diatoms, 2. Generation times of 50 species. Hydrobiol. Bull. 15, 137â151 (1981).
Lavigne, A. S., Sunesen, I. & Sar, E. A. Morphological, taxonomic and nomenclatural analysis of species of Odontella, Trieres and Zygoceros (Triceratiaceae, Bacillariophyta) from Anegada Bay (Province of Buenos Aires, Argentina). Diatom Res. 30, 307â331 (2015).
Anderson, D. M. & Gillooly, J. F. Physiological constraints on long-term population cycles: a broad-scale view. Evol. Ecol. Res. 18, 693â707 (2017).
Janes, M. J. Oviposition studies on the chinch bug, Blissus leucopterus (Say). Ann. Entomol. Soc. Am. 28, 109â120 (1935).
Cook, L. M. Food-plant specialization in the moth Panaxia dominula L. Evolution 15, 478â485 (1961).
Casey, T. M. Flight energetics of sphinx moths: power input during hovering flight. J. Exp. Biol. 64, 529â543 (1976).
Kobayashi, A., Tanaka, Y. & Shimada, M. Genetic variation of sex allocation in the parasitoid wasp Heterospilus prosopidis. Evolution 57, 2659â2664 (2003).
Hozumi, N. & Miyatake, T. Body-size dependent difference in death-feigning behavior of adult Callosobruchus chinensis. J. Insect Behav. 18, 557â566 (2005).
Huntley, M. E. & Lopez, M. D. G. Temperature-dependent production of marine copepods: a global synthesis. Am. Nat. 140, 201â242 (1992).
Cohen, R. E. & Lough, R. G. Lengthâweight relationships for several copepods dominant in the Georges BankâGulf of Maine area. J. Northwest Atl. Fish. Sci. 2, 47â52 (1981).
World Register of Marine Species (WoRMS, accessed 1 November 2020); https://doi.org/10.14284/170
Nakamura, Y. Growth and grazing of a large heterotrophic dinoflagellate, Noctiluca scintillans, in laboratory cultures. J. Plankton Res. 20, 1711â1720 (1998).
Boulding, E. G. & Platt, T. Variation in photosynthetic rates among individual cells of a marine dinoflagellate. Mar. Ecol. Prog. Ser. 29, 199â203 (1986).
Rimet, F. et al. The Observatory on LAkes (OLA) database: sixty years of environmental data accessible to the public. J. Limnol. https://doi.org/10.4081/jlimnol.2020.1944 (2020).
Rudstam, L. Zooplankton Survey of Oneida Lake, New York, 1964 to Present (KNB, 2020); https://knb.ecoinformatics.org/view/kgordon.17.99https://knb.ecoinformatics.org/knb/metacat/kgordon.17.67/default
Dumont, H. J., Van de Velde, I. & Dumont, S. The dry weight estimate of biomass in a selection of Cladocera, Copepoda and Rotifera from the plankton, periphyton and benthos of continental waters. Oecologia 19, 75â97 (1975).
Geller, W. & Müller, H. Seasonal variability in the relationship between body length and individual dry weight as related to food abundance and clutch size in two coexisting Daphnia species. J. Plankton Res. 7, 1â18 (1985).
Branstrator, D. K. Contrasting life histories of the predatory cladocerans Leptodora kindtii and Bythotrephes longimanus. J. Plankton Res. 27, 569â585 (2005).
Rosen, R. A. Lengthâdry weight relationships of some freshwater zooplankton. J. Freshw. Ecol. 1, 225â229 (1981).
Peters, R. H. & Downing, J. A. Empirical analysis of zooplankton filtering and feeding rates. Limnol. Oceanogr. 29, 763â784 (1984).
Eckmann, J. P., Kamphorst, S. O. & Ruelle, D. Recurrence plots of dynamical systems. Europhys. Lett. 4, 973â977 (1987).
Luque, B., Lacasa, L., Ballesteros, F. J. & Robledo, A. Analytical properties of horizontal visibility graphs in the Feigenbaum scenario. Chaos 22, 013109 (2012).
McCaffrey, D. F., Ellner, S., Gallant, A. R. & Nychka, D. W. Estimating the Lyapunov exponent of a chaotic system with nonparametric regression. J. Am. Stat. Assoc. 87, 682â695 (1992).
Brown, J. H., Gillooly, J. F., Allen, A. P., Savage, V. M. & West, G. B. Toward a metabolic theory of ecology. Ecology 85, 1771â1789 (2004).
Ricker, W. E. Stock and recruitment. J. Fish. Board Can. 11, 559â623 (1954).
Acknowledgements
We thank S. Salinas, S. Newkirk, A. Hein, N. Lustenhouwer, A. M. Kilpatrick and M. OâFarrell 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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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. DLEâ=âdirect Lyapunov exponent, JLEâ=âJacobian-based Lyapunov exponent, RQAâ=ârecurrence quantification analysis, PEâ=âpermutation entropy, HVGâ=âhorizontal visibility graphs, CDTâ=âchaos 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. 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 1â5, Figs. 1â9 and Tables 1â13.
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Rogers, T.L., Johnson, B.J. & Munch, S.B. Chaos is not rare in natural ecosystems. Nat Ecol Evol 6, 1105â1111 (2022). https://doi.org/10.1038/s41559-022-01787-y
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DOI: https://doi.org/10.1038/s41559-022-01787-y
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