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Nonlinear population dynamics are ubiquitous in animals


Nonlinear dynamics, where a change in the input is not proportional to a change in the output, are often found throughout nature, for example in biochemical kinetics. Because of the complex suite of interacting abiotic and biotic variables present in ecosystems, animal population dynamics are often thought to be driven in a nonlinear, state-dependent fashion. However, so far these have only been identified in model organisms and some natural systems. Here we show that nonlinear population dynamics are ubiquitous in nature. We use nonlinear forecasting to analyse 747 datasets of 228 species to find that insect population trends were highly nonlinear (74%), followed by mammals (58%), bony fish (49%) and birds (35%). This indicates that linear, equilibrium-based model assumptions may fail at predicting population dynamics across a wide range of animal taxa. We show that faster-reproducing animals are more likely to have nonlinear and high-dimensional dynamics, supporting past ecological theory. Lastly, only a third of time series were predictable beyond two years; therefore, the ability to predict animal population trends using these methods may be limited. Our results suggest that the complex dynamics necessary to cause regime shifts and other transitions may be inherent in a wide variety of animals.

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Fig. 1: Animal time series with linear or nonlinear population dynamics.
Fig. 2: Dimensionality in animal time series by taxonomic class/superclass.
Fig. 3: Forecast skill of animal time series by taxonomic class/superclass.

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Funding for this project was provided by the National Science Foundation (NSF) Graduate Research Fellowship under grant no. 366280 (T.J.C.), the Idaho Department of Fish and Game (T.J.C.) and NSF grant no. 1836793 (A.D.L.). We thank H. Wilhelm Martin for his superb assistance with the analysis and M. Hebblewhite and J. Maron for comments on earlier drafts.

Author information




T.J.C. collected and analysed the data and wrote the manuscript. A.D.L. supervised the project and edited the manuscript.

Corresponding author

Correspondence to T. J. Clark.

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Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Principal components analysis of life-history traits.

PC1 explains 72.2% of variation in our data (axes 1, 2, and 3), representing body length (mm), minimum age at first reproduction (months), and lifespan (months), respectively. PC2 explains 24.9% of variation in our data (axis 4), representing fertility (# of young per year). Colored ellipses represent 95% probability that data for each taxonomic classification fall within the ellipse.

Extended Data Fig. 2 Final model results of nonlinearity.

E is the embedding dimensionality, ρ is the forecast skill, and CV is the coefficient of variation of a time-series. An asterisk indicates coefficients that were significant at P ≤ 0.05.

Extended Data Fig. 3 Animal time-series with linear, nonlinear, or non-predictable population dynamics.

a, Animal time-series arranged by taxonomic class. b, Animal time-series were arranged by taxonomic order where sample size ≥ 10. Bolded numbers show sample size.

Extended Data Fig. 4 Final model results of dimensionality (E).

PC1 is the first principal component of life history traits, representing a combination of body length (mm), minimum age of first reproduction (months), and longevity (months) of animals (positive coefficient estimates = faster life histories; Extended Data Fig. 1). N is the time-series length. An asterisk indicates coefficients that were significant at P ≤ 0.05.

Extended Data Fig. 5 Final model results of forecast skill (ρ).

PC1 is the first principal component of life history traits, representing a combination of body length (mm), minimum age of first reproduction (months), and longevity (months) of animals. Linear, nonlinear, and not predictable represent the categorization of population dynamics. CV is the coefficient of variation of a time-series. An asterisk indicates coefficients that were significant at P ≤ 0.05.

Extended Data Fig. 6 Repeating zeroes in datasets do not change likelihood of nonlinearity.

Proportion of linearity/nonlinearity in animal time-series, arranged by taxonomic class. Due to some time-series having long sequences of zeroes, we filtered out time-series with strings of zeroes. a, Time-series with no filtering. b, Strings of zeroes > 20 filtered. c, Strings of zeroes > 5 filtered. d, Strings of zeroes > 1 filtered.

Extended Data Fig. 7 Summary statistics for animal time-series by taxonomic class.

Median time-series length represents median number of time-series data for the final dataset. Predictable datasets were categorized if the Pearson correlation coefficient of out-of-sample prediction was significant at P ≤ 0.05.

Extended Data Fig. 8 Examples of predictable time-series.

a, Standardized abundance of woodcock (Scolopax minor) over time. b, Standardized abundance of grey red-backed voles (Myodes rufocanus) over time. c, Standardized abundance of dover soles (Solea solea) over time. d, Standardized abundance of woolly beech aphids (Phyllaphis fagi) over time. Grey lines represent the observed abundance, blue lines represent predicted abundance.

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Clark, T.J., Luis, A.D. Nonlinear population dynamics are ubiquitous in animals. Nat Ecol Evol 4, 75–81 (2020).

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