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# Thresholds for ecological responses to global change do not emerge from empirical data

## Abstract

To understand ecosystem responses to anthropogenic global change, a prevailing framework is the definition of threshold levels of pressure, above which response magnitudes and their variances increase disproportionately. However, we lack systematic quantitative evidence as to whether empirical data allow definition of such thresholds. Here, we summarize 36 meta-analyses measuring more than 4,600 global change impacts on natural communities. We find that threshold transgressions were rarely detectable, either within or across meta-analyses. Instead, ecological responses were characterized mostly by progressively increasing magnitude and variance when pressure increased. Sensitivity analyses with modelled data revealed that minor variances in the response are sufficient to preclude the detection of thresholds from data, even if they are present. The simulations reinforced our contention that global change biology needs to abandon the general expectation that system properties allow defining thresholds as a way to manage nature under global change. Rather, highly variable responses, even under weak pressures, suggest that ‘safe-operating spaces’ are unlikely to be quantifiable.

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## Data availability

All data are available at https://zenodo.org/record/3828869#.XsI4ZmgzaUk.

## Code availability

All code are available at https://zenodo.org/record/3828869#.XsI4ZmgzaUk.

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## Acknowledgements

The data reported in this paper are presented and derived from 36 different meta-analyses; they are archived and available from each of these as indicated in the Supplementary Text. The concept of this paper emerged during scientific discussions with T. Blenckner at Stockholm University, at the UK NERC/BESS Tansley Working Group on ecological stability and the TippingPond EU Biodiversa project. The actual work was funded by the Lower Saxony Ministry of Science and Culture through the MARBAS project to H.H. and the HIFMB, a collaboration between the Alfred-Wegener-Institute, Helmholtz-Center for Polar and Marine Research and the Carl von Ossietzky University Oldenburg, initially funded by the Ministry for Science and Culture of Lower Saxony and the Volkswagen Foundation through the ‘Niedersächsisches Vorab’ grant programme (grant no. ZN3285). The work was finalized with support by Deutsche Forschungsgemeinschaft grant no. HI848/26–1. L. Toaspern helped with gathering data from invasion meta-analyses. P. Ruckdeschel helped with the statistical approach. M. Vilà provided additional information on their published meta-analyses. We acknowledge the comments by U. Feudel, G. Gerlach and the members of the Plankton Ecology Lab at the Carl von Ossietzky University Oldenburg on the manuscript which helped with our argumentation.

## Author information

Authors

### Contributions

H.H. designed the analysis and discussed the framework with I.D. J.M.M., J.A.F. and J.M. developed the statistical approach with input from H.H. and D.H. H.H. assembled the effect size information. J.A.F. and J.M. performed the statistical analysis. M.K., A.M.L. and W.S.H. provided input on palaeoecological and experimental constraints, respectively. H.H. wrote the manuscript together with W.S.H. and J.M.M. as well as input from all co-authors.

### Corresponding author

Correspondence to Helmut Hillebrand.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Peer reviewer reports are available.

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

## Extended data

### Extended Data Fig. 1 Test power as in Fig. 2, but for the “qgam” approach.

Fractions of positive test results (equals test power when test should be positive) for simulated test cases. We analysed the test power for 9 scenarios of responses to pressure in meta-analyses, the derivation of each scenario is described in the supplementary online material, Extended Data Fig. 7. Scenarios a–d do not comprise a threshold, where scenario a is the null model without an effect of the pressure on the response. Scenarios e–i do comprise a threshold, for the latter two combined with intermediate responses. For the three statistical test used in our analyses, the expected outcome is colour-coded, with green representing that the test should be significant. We then tested the proportion of 1000 simulated datasets for which the tests were significant with a probability p = 0.05 (black) and p = 0.01 (blue). We did for increasing noise variance (= inverse signal-to-noise ratio). The three tests together allow perfect detection of thresholds at the absence of noise (scenarios e–h), only if threshold-type and gradual responses are mixed (scenario i), the analysis of multimodality (HD) fails, giving the same output as a gradual increase in mean and variance of the response (scenario d). With increasing noise variance, however, the detection probability for thresholds via HD and QR rapidly decreases.We used default settings for the “qgam” approach due to high runtimes and computational effort, thus settings are not optimized as for the test power calculations based on kernel density estimation. Note: HD is equal to kernel method, because it is not based on different quantile estimations.

### Extended Data Fig. 2 Changes in the response magnitude along increasing pressure strength.

Further meta-analyses testing for changes in the response magnitude along increasing pressure strength. Red and blue shaded regions indicate the (5%-95%) interquantile ranges for the bivariate data (including the pressure gradient) and the univariate LRR data (ignoring the pressure gradient = homogeneous marginal probability), respectively. Solid red and dashed blue thick lines trace the related median (50% quantile). Overlain are the data points and at the bottom the yellow shaded area indicates the distribution px(gx) resultant from a weighted kernel density estimation. Colour codes for habitat (dark blue: freshwater, aquamarine: marine, green: terrestrial), circle size reflect statistical weight.

### Extended Data Fig. 3 Test cases at different noise levels.

In order to assess the power of our statistical tests, we simulated artificial meta-analyses combining prototypical response~stressor relationships with (normally distributed) random fluctuations reflecting natural variability, and compared related statistical test results with expectations. Stressor range (along horizontal range) and deterministic effect sizes (along vertical axis) are normalized to [−0.5,0.5] x [0.5,0.5]. Stressor values are normally distributed with mean zero. The relative intensity of random fluctuations is quantified by inverse signal-to-noise ratio (isnr). A grey background indicates absence of thresholds, yellow background threshold presence. a, (neutral -simple-): Here pressure strength has no impact on the response, which falls into a single response. Thus, we assume that across all “studies” in this “meta-analysis”, there is one main response type and no threshold. b, (neutral -bimodal-): Here pressure strength has no impact on the response, which falls into either of two alternative attractors: a weak and a strong response. Thus, we assume that across all “studies” in this “meta-analysis”, there are two main response types and no threshold. c, (plain trend, proportionate response): A gradual response with no change in variability revealing a trend but no threshold. d, (gradual, no threshold): A nonlinear but smooth increase with smoothly increasing variability. Here we assume that the responses increase with some normally distributed error with the pressure without transgressing any threshold. e, (saddle-node bifurcation): A widely discussed model situation in the context of ‘tipping points’ and ‘catastrophic regime shifts’. f, (strict threshold): Here we assume that across all studies in a meta-analysis, the response switches from weak to strong (as defined in case a) at exactly the same threshold for each study. This assumption is very unrealistic (see below) but makes the case when there are two main response types and a global threshold holding for any single study in the meta-analysis. g, (variable threshold): Here we assume that all studies in a meta-analysis potentially transgress a threshold, but the position of the threshold differs. Thus, the probability that the response switches from weak to strong increases with increasing pressure. Response similar to Case a. h, (variable threshold with intermediates): Here we assumed that not all studies in a meta-analysis potentially transgresses a threshold, but some of the studies show gradual responses. As in Case f, the position of the threshold differs between studies and the probability that the response switches from weak to strong increases with increasing pressure. As for cases a,b,e and f, we assume there are two main response types. This scenario can be distinguished from case d by the abrupt change in variance along the pressure gradient. i, (variable threshold and variable effect sizes below and above threshold): Here we assumed that the position of the threshold differs between studies (as in Case f) and any experiment in the study had a 50% chance that the threshold was crossed, independent of the pressure magnitude. By contrast to cases a, b and e–h, we relax the assumption that there are two main response types, but transgressing the thresholds leads to an increase in effect size, which depended on the position on the pressure gradient. Thus, if a study with a large pressure magnitude transgressed the threshold, the increase in response magnitude was larger than if a study with an overall small pressure did so.

### Extended Data Fig. 4 Permutation example.

An example dataset (a) together with a surrogate dataset based on permuted X values (b); as in Fig. 2 of the main text, colour codes habitat (blue: marine, green: terrestrial), circle size reflects statistical weight, and the yellow shaded area indicates the distribution pX(gx) resultant from a weighted kernel density estimation.

### Extended Data Fig. 5 Two-dimensional probability densities.

Densities are calculated over a grid (gx,gy) for the original dataset (a) and the surrogate dataset (b).

### Extended Data Fig. 6 Conditional probability distribution example.

The conditional probability distribution $$p_{LRR \mid X}\left( {gy \mid gx} \right)$$ for each grid point gx together with the marginal distribution pLRR(gy) (thick black line). a, original dataset, b, surrogate dataset.

### Extended Data Fig. 7 Cumulative distribution example.

The cumulative distribution functions $$F_{LRR \mid X}\left( {gy \mid gx} \right)$$ and FLRR(gy) (thick black line) for the probability profiles shown in Supplementary Fig. 3. a, original, b, surrogate.

### Extended Data Fig. 8 Comparison of kernel density estimation and “qgam”.

Images of the reconstructed statistical structures for an original dataset (MA1.1) and one of its surrogate datasets. a, Quantiles estimated by optimized kernel density estimation; b, Quantiles estimated by “qgam”.

## Supplementary information

### Supplementary Information

Supplementary text and Table 1.

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Hillebrand, H., Donohue, I., Harpole, W.S. et al. Thresholds for ecological responses to global change do not emerge from empirical data. Nat Ecol Evol 4, 1502–1509 (2020). https://doi.org/10.1038/s41559-020-1256-9

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• DOI: https://doi.org/10.1038/s41559-020-1256-9

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