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Slower recovery in space before collapse of connected populations

Abstract

Slower recovery from perturbations near a tipping point and its indirect signatures in fluctuation patterns have been suggested to foreshadow catastrophes in a wide variety of systems1,2. Recent studies of populations in the field and in the laboratory have used time-series data to confirm some of the theoretically predicted early warning indicators, such as an increase in recovery time or in the size and timescale of fluctuations3,4,5,6. However, the predictive power of temporal warning signals is limited by the demand for long-term observations. Large-scale spatial data are more accessible, but the performance of warning signals in spatially extended systems7,8,9,10 needs to be examined empirically3,11,12,13. Here we use spatially extended yeast populations, an experimental system with a fold bifurcation (tipping point)6, to evaluate early warning signals based on spatio-temporal fluctuations and to identify a novel spatial warning indicator. We found that two leading indicators based on fluctuations increased before collapse of connected populations; however, the magnitudes of the increases were smaller than those observed in isolated populations, possibly because local variation is reduced by dispersal. Furthermore, we propose a generic indicator based on deterministic spatial patterns, which we call ‘recovery length’. As the spatial counterpart of recovery time14, recovery length is the distance necessary for connected populations to recover from spatial perturbations. In our experiments, recovery length increased substantially before population collapse, suggesting that the spatial scale of recovery can provide a superior warning signal before tipping points in spatially extended systems.

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Figure 1: Yeast populations with a tipping point: an experimental system to study the collapse of connected populations.
Figure 2: Early warning signals based on fluctuations show suppressed increase in connected populations.
Figure 3: Early warning signals can be classified into four categories by the nature of perturbations and measurements.
Figure 4: Recovery length provides a direct measure of critical slowing down in space.

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Acknowledgements

We would like to thank D. Vorselen, T. Krieger, D. Seekell, M. Pace and members of the Gore laboratory (A. Sanchez, M. Datta, E. Yurtsev, T. Artemova, K. Axelrod and A. Chen) for comments on the manuscript. T. Krieger performed initial simulations for the connected populations. Y. Zhang and O. Ornek collected preliminary data for the experiment to measure recovery length. This work was supported by a Whitaker Health Sciences Fund Fellowship (to L.D.), a Pappalardo Fellowship (to K.S.K.), an NIH R00 Pathways to Independence Award (NIH R00 GM085279-02), an NIH New Innovator Award (NIH DP2), an NSF CAREER Award, a Sloan Research Fellowship, the Pew Scholars Program and the Allen Investigator Program.

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Contributions

L.D., K.S.K. and J.G. designed the study. L.D. performed the experiments and analysis. K.S.K. and J.G. assisted with the analysis. L.D., K.S.K. and J.G. wrote the manuscript.

Corresponding authors

Correspondence to Lei Dai or Jeff Gore.

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The authors declare no competing financial interests.

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This file contains Supplementary Figures 1-11, Supplementary Tables 1-2, Supplementary Notes 1-4 and Supplementary References. (PDF 1034 kb)

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Dai, L., Korolev, K. & Gore, J. Slower recovery in space before collapse of connected populations. Nature 496, 355–358 (2013). https://doi.org/10.1038/nature12071

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