Slower recovery in space before collapse of connected populations

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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.

At a glance


  1. Yeast populations with a tipping point: an experimental system to study the collapse of connected populations.
    Figure 1: Yeast populations with a tipping point: an experimental system to study the collapse of connected populations.

    a, Isolated yeast populations collapse after crossing a tipping point. The distribution of population density around equilibrium is shown in spread points; the red square denotes the mean. Insets are traces of replicate populations at dilution factor 1,000 (stable) and 1,700 (collapsed). b, Yeast populations are spatially connected by controlled daily dispersal. Each circle corresponds to a habitat patch where a local population grows. A fraction of the local population is transferred to each of its two nearest neighbours, and the rest to itself.

  2. Early warning signals based on fluctuations show suppressed increase in connected populations.
    Figure 2: Early warning signals based on fluctuations show suppressed increase in connected populations.

    a, Coefficient of variation (CV). b, Temporal correlation (lag-1 autocorrelation). The coefficient of variation and temporal correlation of both isolated populations (red squares) and connected populations (blue circles) increased before the tipping point. The signals were suppressed in the connected populations, possibly owing to the averaging effect of dispersal. Error bars are standard errors given by bootstrap for isolated populations and standard errors of the mean (n = 4) for connected populations.

  3. Early warning signals can be classified into four categories by the nature of perturbations and measurements.
    Figure 3: Early warning signals can be classified into four categories by the nature of perturbations and measurements.

    a, Recovery time; b, Recovery length; c, Statistical indicators based on temporal fluctuations; d, Statistical indicators based on spatial fluctuations. The unexplored category of early warning signals is the spatial counterpart of recovery time: ‘recovery length’. The recovery length characterizes the spatial scale over which population density recovers from a pulse perturbation in space, such as at a boundary with a region of lower quality (b). The recovery length increases towards the tipping point (Supplementary Note 2) and provides a novel indicator of critical slowing down in spatial data.

  4. Recovery length provides a direct measure of critical slowing down in space.
    Figure 4: Recovery length provides a direct measure of critical slowing down in space.

    a, Connected populations in relatively good regions of dilution factor 750 (left) and 1,200 (right) recover gradually in space to the equilibrium density. Blue circles denote the steady-state profile of population density after averaging over replicates (shown in grey). b, Recovery profiles at dilution factor 500, 750, 1,000, 1,200, 1,350 and 1,400 show an increasing spatial scale of recovery. The profile is normalized by the population density of the patch furthest from the bad region. Lines are shape-preserving interpolations; the position of half-recovery is marked by a red square. c, Two different measures of recovery length increase substantially with dilution factor. Error bars are standard errors given by bootstrap.


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  1. Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

    • Lei Dai,
    • Kirill S. Korolev &
    • Jeff Gore


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.

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    This file contains Supplementary Figures 1-11, Supplementary Tables 1-2, Supplementary Notes 1-4 and Supplementary References.

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