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

Nature Ecology & Evolution volume 3, pages 912–918 (2019)Cite this article

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Abstract

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 (http://www.pnas.org/content/112/20/6389/tab-figures-data).

Code availability

The code accompanying the manuscript is available on GitHub at https://github.com/MITEcology/Nature.Eco.Evo-Cenci-Saavedra-2019.git. The repository contains the code used to generate Figs. 3–5 and an illustrative example of how to compute the VCR from a multivariate time series.

References

  1. Levins, R. Evolution in Changing Environments: Some Theoretical Explorations (Monographs in Population Biology) (Princeton Univ. Press, 1968).

  2. Thomas, C. D. et al. Extinction risk from climate change. Nature 427, 145–148 (2004).

    Article  CAS  PubMed  Google Scholar 

  3. Chase, E. & Harwood, V. J. Comparison of the effects of environmental parameters on growth rates of Vibrio vulnificus biotypes I, II, and III by culture and quantitative PCR analysis. Appl. Environ. Microbiol. 77, 4200–4207 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  4. Montoya, J. & Raffaelli, D. Climate change, biotic interactions and ecosystem services. Phil. Trans. R. Soc. B 365, 2013–2018 (2010).

    Article  PubMed  PubMed Central  Google Scholar 

  5. Thackeray, S. J. et al. Phenological sensitivity to climate across taxa and trophic levels. Nature 535, 241–245 (2016).

    CAS  PubMed  Google Scholar 

  6. Sugihara, G. et al. Detecting causality in complex ecosystems. Science 338, 496–500 (2012).

    Article  CAS  PubMed  Google Scholar 

  7. Dai, L., Vorselen, D., Korolev, K. S. & Gore, J. Generic indicators for loss of resilience before a tipping point leading to population collapse. Science 336, 1175–1177 (2012).

    Article  CAS  PubMed  Google Scholar 

  8. Andronov, A. & Pontrjagin, L. Systémes grossiers. Dokl. Akad. Nauk SSSR 14, 247–251 (1937).

    Google Scholar 

  9. Smale, S. Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967).

    Article  Google Scholar 

  10. Thom, R. Structural Stability and Morphogenesis (Addison-Wesley, 1994).

  11. Allen, P. M. Evolution, population dynamics, and stability. Proc. Natl Acad. Sci. USA 73, 665–668 (1976).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  12. Rohr, R. P., Saavedra, S. & Bascompte, J. On the structural stability of mutualistic systems. Science 345, 1253497 (2014).

    Article  PubMed  Google Scholar 

  13. Arnoldi, J.-F. & Haegeman, B. Unifying dynamical and structural stability of equilibria. Proc. R. Soc. Lond. A 472, 20150874 (2016).

    Article  Google Scholar 

  14. Cenci, S. & Saavedra, S. Structural stability of nonlinear population dynamics. Phys. Rev. E 97, 012401 (2018).

    Article  CAS  PubMed  Google Scholar 

  15. Song, C., Rohr, R. P. & Saavedra, S. A guideline to study the feasibility domain of multi-trophic and changing ecological communities. J. Theor. Biol. 450, 30–36 (2018).

    Article  PubMed  Google Scholar 

  16. Wood, S. N. & Thomas, M. B. Super–sensitivity to structure in biological models. Proc. R. Soc. Lond. B 266, 565–570 (1999).

    Article  Google Scholar 

  17. Perretti, C. T., Munch, S. B. & Sugihara, G. Model free forecasting outperforms the correct mechanistic model for simulated and experimental data. Proc. Natl Acad. Sci. USA 110, 5253–5257 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  18. Ye, H. et al. Equation-free mechanistic ecosystem forecasting using empirical dynamic modeling. Proc. Natl Acad. Sci. USA 112, E1569–E1576 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  19. Benincà, E. et al. Chaos in a long-term experiment with a plankton community. Nature 451, 822–825 (2008).

    Article  PubMed  Google Scholar 

  20. Bjørnstad, O. N. & Grenfell, B. T. Noisy clockwork: time series analysis of population fluctuations in animals. Science 293, 638–643 (2001).

    Article  PubMed  Google Scholar 

  21. Ushio, M. et al. Fluctuating interaction network and time-varying stability of a natural fish community. Nature 554, 360–363 (2018).

    Article  CAS  PubMed  Google Scholar 

  22. Costantino, R. F., Desharnais, R. A., Cushing, J. M. & Dennis, B. Chaotic dynamics in an insect population. Science 275, 389–391 (1997).

    Article  CAS  PubMed  Google Scholar 

  23. Turchin, P. & P. Ellner, S. Living on the edge of chaos: population dynamics of fennoscandian voles. Ecology 81, 3099–3116 (2000).

    Article  Google Scholar 

  24. Becks, L., Hilker, F. M., Malchow, H., Jürgens, K. & Arndt, H. Experimental demonstration of chaos in a microbial food web. Nature 435, 1226–1229 (2005).

    Article  CAS  PubMed  Google Scholar 

  25. Benincà, E., Ballantine, B., Ellner, S. P. & Huisman, J. Species fluctuations sustained by a cyclic succession at the edge of chaos. Proc. Natl Acad. Sci. USA 112, 6389–6394 (2015).

    Article  PubMed  PubMed Central  Google Scholar 

  26. Deyle, E, R., May, R. M., Munch, S. B. & Sugihara, G. Tracking and forecasting ecosystem interactions in real time. Proc. R. Soc. Lond. B 283, 20152258 (2016).

    Article  Google Scholar 

  27. Sugihara, G. Nonlinear forecasting for the classification of natural time series. Phil. Trans. R. Soc. Lond. A 348, 477–495 (1994).

    Article  Google Scholar 

  28. Cenci, S., Sugihara, G. & Saavedra, S. Regularized S-map for inference and forecasting with noisy ecological time series. Methods Ecol. Evol. https://doi.org/10.1111/2041-210X.13150 (2019).

    Article  Google Scholar 

  29. Cenci, S. & Saavedra, S. Uncertainty quantification of the effects of biotic interactions on community dynamics from nonlinear time-series data. J. R. Soc. Interface 15, 20180695 (2018).

    Article  PubMed  PubMed Central  Google Scholar 

  30. Helmuth, B. S. T. & Hofmann, G. E. Microhabitats, thermal heterogeneity, and patterns of physiological stress in the rocky intertidal zone. Biol. Bull. 201, 374–384 (2001).

    Article  CAS  PubMed  Google Scholar 

  31. Harley, C. D. G. & Helmuth, B. S. T. Local and regional scale effects of wave exposure, thermal stress, and absolute versus effective shore level on patterns of intertidal zonation. Limnol. Oceanogr. 48, 1498–1508 (2003).

    Article  Google Scholar 

  32. Hastie, T., Tibshirani, R. & Friedman, J. The Elements of Statistical Learning: Data Mining, Inference, and Prediction (Springer, 2001).

  33. Kantz, H. & Schreiber, T. Nonlinear Time Series Analysis (Cambridge Univ. Press, 2004).

  34. Theiler, J., Eubank, S., Longtin, A., Galdrikian, B. & Farmer, J. D. Testing for nonlinearity in time series: the method of surrogate data. Physica D 58, 77–94 (1992).

    Article  Google Scholar 

  35. Mees, A. Nonlinear Dynamics and Statistics (Birkhäuser, 2011).

  36. Mayo-Wilson, C. Structural chaos. Phil. Sci. 82, 1236–1247 (2015).

    Article  Google Scholar 

  37. Strogatz, S. Nonlinear Dynamics and Chaos. With Applications to Physics, Biology, Chemistry, and Engineering (Studies in Nonlinearity) (CRC Press, 2014).

  38. Saavedra, S. et al. A structural approach for understanding multispecies coexistence. Ecol. Monogr. 87, 470–486 (2017).

    Article  Google Scholar 

  39. Nese, J. M. Quantifying local predictability in phase space. Physica D 35, 237–250 (1989).

    Article  Google Scholar 

  40. Song, C. & Saavedra, S. Structural stability as a consistent predictor of phenological events. Proc. R. Soc. B 285, 20180767 (2018).

    Article  PubMed  PubMed Central  Google Scholar 

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Acknowledgements

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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  1. Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

    Simone Cenci & Serguei Saavedra

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  1. Simone Cenci
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Contributions

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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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). https://doi.org/10.1038/s41559-019-0879-1

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