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Quantifying resilience to recurrent ecosystem disturbances using flow–kick dynamics


Shifting ecosystem disturbance patterns due to climate change (for example, storms, droughts and wildfires) or direct human interference (for example, harvests and nutrient loading) highlight the importance of quantifying and strengthening the resilience of desired ecological regimes. Although existing metrics capture resilience to isolated shocks, gradual parameter changes, and continual noise, quantifying resilience to repeated, discrete disturbance events requires different analytical tools. Here, we introduce a mathematical flow–kick framework that uses dynamical systems tools to quantify resilience to disturbances explicitly in terms of their magnitude and frequency. We identify a boundary between disturbance regimes that cause either escape from, or stabilization within, a basin of attraction. We use the boundary to define resilience metrics tailored to repeated, discrete perturbations. The flow–kick model suggests that the distance-to-threshold resilience metric overestimates resilience in the context of repeated perturbations. It also reveals counterintuitive triggers for regime shifts. These include increasing the periods between disturbance events in proportion to increases to disturbance magnitude, and—in systems with multiple dynamic variables—increasing time periods between disturbances of constant magnitude.

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Fig. 1: A flow–kick model of repeated harvests from a fishery.
Fig. 2: Flow–kick resilience boundary in disturbance space.
Fig. 3: Shorter distance to threshold in a flow–kick system.
Fig. 4: Connections between the flow–kick resilience boundary and existing resilience metrics illustrated in a model of lake water quality.
Fig. 5: Flow–kick dynamics in a two-dimensional model of ocean circulation.

Data availability

The MATLAB scripts and functions used in the study are available from the open-source GitHub repository at


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This work was supported by an NSF Graduate Research Fellowship (grant number 00039202 to K.M.), the Mathematics and Climate Change Research Network (NFS grant DMS-0940243) and the Computational Sustainability Network (NSF grant CCS-1521672). Thanks go to A. Shaw, L. Sullivan, J. Cowles, K. Kimmel, E. Strombom, F. Isbell and A. Rossberg for feedback on various stages of the manuscript. We are also indebted to the Ecology Theory Group at the University of Minnesota, and A. Helfgott, S. Lord, D. McGehee and C. Chong for helpful conversations.

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Authors and Affiliations



M.L.Z. and S.I. conceived of the flow–kick model of disturbance. All authors contributed to development of the model and analysis of the resulting dynamics. K.M. wrote the manuscript, and S.I., A.H.-L., I.K., V.L., E.B. and M.L.Z. contributed to the revisions.

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Correspondence to Katherine Meyer.

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Supplementary Tables 1–2, Supplementary Methods, Supplementary Figures 1–3, Supplementary Discussion, Supplementary References

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Meyer, K., Hoyer-Leitzel, A., Iams, S. et al. Quantifying resilience to recurrent ecosystem disturbances using flow–kick dynamics. Nat Sustain 1, 671–678 (2018).

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