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Non-parametric estimation of the structural stability of non-equilibrium community dynamics


Environmental factors are important drivers of community dynamics. Yet, despite extensive research, it is still extremely challenging to predict the effect of environmental changes on the dynamics of ecological communities. Equilibrium- and model-based approaches have provided a theoretical framework with which to investigate this problem systematically. However, the applicability of this framework to empirical data has been limited because equilibrium dynamics of populations within communities are seldom observed in nature and exact equations for community dynamics are rarely known. To overcome these limitations, here we develop a data-driven non-parametric framework to estimate the tolerance of non-equilibrium community dynamics to environmental perturbations (that is, their structural stability). Following our approach, we show that in non-equilibrium systems, structural stability can vary significantly across time. As a case study, we investigate the structural stability of a rocky intertidal community with dynamics at the edge of chaos. The structural stability of the community as a whole exhibited a clear seasonal pattern, despite the persistent chaotic dynamics of individual populations. Importantly, we show that this seasonal pattern of structural stability is causally driven by sea temperature. Overall, our approach provides novel opportunities for estimating the tolerance of ecological communities to environmental changes within a non-parametric framework.

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Fig. 1: Parameter perturbations occurring at different points in time on a chaotic dynamical system can have different effects on its dynamics.
Fig. 2: Perturbations acting on the parameters of a nonlinear system with large and small VCRs induce different effects on the trajectories.
Fig. 3: Validation on synthetic data.
Fig. 4: Structural stability pattern of a rocky intertidal community at the edge of chaos.
Fig. 5: CCM test.

Data availability

The data used in the manuscript can be downloaded from Benincà et al.25 (

Code availability

The code accompanying the manuscript is available on GitHub at The repository contains the code used to generate Figs. 35 and an illustrative example of how to compute the VCR from a multivariate time series.


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We thank C. Song for insightful discussions. Funding was provided by MIT Research Committee funds and the Mitsui Chair (S.S.).

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S.C. and S.S. designed the study. S.C. performed the study. S.S. supervised the study. S.C. and S.S. wrote the paper.

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Correspondence to Simone Cenci.

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The authors declare no competing interests.

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Supplementary Sections 1–3, Supplementary references, Supplementary Figs. 1–10

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Cenci, S., Saavedra, S. Non-parametric estimation of the structural stability of non-equilibrium community dynamics. Nat Ecol Evol 3, 912–918 (2019).

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