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Reinterpreting the relationship between number of species and number of links connects community structure and stability

Nature Ecology & Evolution volume 5, pages 1102–1109 (2021)Cite this article

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Abstract

For 50 years, ecologists have examined how the number of interactions (links) scales with the number of species in ecological networks. Here, we show that the way the number of links varies when species are sequentially removed from a community is fully defined by a single parameter identifiable from empirical data. We mathematically demonstrate that this parameter is network-specific and connects local stability and robustness, establishing a formal connection between community structure and two prime stability concepts. Importantly, this connection highlights a local stability–robustness trade-off, which is stronger in mutualistic than in trophic networks. Analysis of 435 empirical networks confirmed these results. We finally show how our network-specific approach relates to the classical across-network approach found in literature. Taken together, our results elucidate one of the intricate relationships between network structure and stability in community networks.

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Fig. 1: The network-specific link–species relationship in two empirical networks.
Fig. 2: Prediction of the number of species lost on the basis of the number of species removed.
Fig. 3: Robustness (averaged Rob0.5 over 10,000 simulations) of 435 empirical networks and its prediction on the basis of the parameter b shaping the link–species relationship.
Fig. 4: Distribution of the parameter b for of the 435 empirical networks used in this study.
Fig. 5: Prediction of local stability on the basis of robustness for trophic (red) and mutualistic (blue) networks.

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Data availability

The empirical network matrices that support the findings of this study are available from seven datasets published on Dryad56,57,58,59,60,61,62 and from three online databases: The GlobalWeb (http://globalwebdb.com) hosted by the University of Canberra; Interaction Web Database (http://www.ecologia.ib.usp.br/iwdb) hosted by the National Center for Ecological Analysis and Synthesis (University of California); This work has used the Web of Life dataset (www.web-of-life.es), a service created by R. Ortega, M. Angel Fortuna and J. Bascompte and provided by the Bascompte Lab at the Spanish Research Council. The empirical networks used are listed in Supplementary Data. This file also contains the metrics computed on the basis of the in-silico experiments and the equations: these data allow the reproduction of Figs. 2c, 3, 4 and 5.

Code availability

Code needed to reproduce the results presented in the article is available at Zenodo with the identifier https://doi.org/10.5281/zenodo.4671579.

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Acknowledgements

Computational resources have been provided by the Consortium des Équipements de Calcul Intensif (CÉCI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under grant no. 2.5020.11 and by the Walloon Region. F.D.L. received support from the Fund for Scientific Research, FNRS (PDRT.0048.16). G.B. acknowledges funding from the Swedish Research Council (Vetenskapsrådet), grant no. VR 2017-05245.

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Authors and Affiliations

  1. Research Unit in Environmental and Evolutionary Biology, Institute of Life, Earth and the Environment, Namur Institute of Complex Systems, University of Namur, Namur, Belgium

    Camille Carpentier, Jürg Werner Spaak & Frederik De Laender

  2. Department of Physics, Chemistry and Biology, Division of Theoretical Biology, Linköping University, Linköping, Sweden

    György Barabás

  3. MTA-ELTE Theoretical Biology and Evolutionary Ecology Research Group, Budapest, Hungary

    György Barabás

  4. Department of Ecology and Evolutionary Biology, Cornell University, Ithaca, NY, USA

    Jürg Werner Spaak

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  1. Camille Carpentier
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Contributions

C.C. conceived the presented ideas, developed the theory and performed the analytic calculations and the in-silico experiments. G.B., J.W.S. and F.D.L. contributed to the analytical methods. F.D.L. and C.C. wrote the first version of the manuscript. All authors discussed the results and contributed to the final manuscript.

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Correspondence to Camille Carpentier.

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The authors declare no competing interests.

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Peer review information Nature Ecology & Evolution thanks Ulrich Brose, Ian Donohue, Christian Guill and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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Extended data

Extended Data Fig. 1 Analytical approximation of b and prediction of the L~S relationship of empirical networks.

a, The shape of the L~S relationship of 435 empirical networks based on equation (2) (y-axis) or estimated through log-log regression (x-axis). b, R2 obtained for the predictions of the number of links along 10,000 decompositions of each empirical network based on its b-value (equation (1)) compared to the R2 of the log-log regression method. In both panels, the dotted line corresponds to the identity relationship.

Extended Data Fig. 2 Distribution of the determination coefficients of equation (1) in scenario 1 (z =0, dark grey) and scenario 2 (z > 0, light grey).

R2 obtained for the predictions of the number of links along 10,000 decompositions of each empirical network based on its b-value and equation (1). The R2 for the first scenario correspond to the \(R_b^2\) presented in Extended Data Fig. 1.

Extended Data Fig. 3 Prediction of Robx at various threshold x.

a, Example for the Lake Crescent trophic network65: the observed values are averaged on 10,000 decompositions with z = 0.5 (fraction of independent species); error bars represent one standard deviation. The predicted values are obtained through equation (3). b, R2 between observed and predicted Robx with \(x \in \left[ {0.01,1.00} \right]\) (step of 0.025). Observed values are averaged on 10,000 decomposition per networks; the predicted Robx are computed based on equation (3). To compute R2 average value and variation at each robustness threshold x, we randomly sampled our networks database 1,000 times: each time, 200 networks were randomly selected and the R2 between the 200 predicted and the 200 simulated Robx was computed. Doing so allows calculation of the \(\overline {R^2}\) and estimations of the variation around this value when the networks considered change: the upper limit of the shaded area corresponds to the 75th percentile of the 1,000 samples, the lower limit to 25th percentile. We performed this analysis by simulating network decompositions following two scenarios: z = 0 (black) and z = fraction of species having only outgoing links (grey). All \(\overline {R^2}\) are higher than 0.5 whenever \(x\in\left[0.23,\ 0.65\right]\). For higher robustness thresholds (x > 0.65), the R2 drops due to the approximation made to estimate \({\widehat {\Delta S}}\) (Supplementary Equation 2): we tend to overestimate robustness when xeff is high.

Extended Data Fig. 4 Results of statistical tests on b, S and β.

a, p-values of Welch t-tests allowing the comparison of the mean value of S (white) and b (grey) between η mutualistic (M) and η trophic (T) networks (η being the number of networks randomly drawn from the 249 mutualistic and the 186 trophic networks of the dataset). We performed 100 tests per sampling size η. While \(\bar{\mathbf{S}}\) does not significantly differ between the two interaction types, \(\overline {{\mathbf{b}}_{\mathrm{M}}} < \overline {{\mathbf{b}}_{\mathrm{T}}}\) when η > 80. b, p-values of Levene tests allowing the comparison of the variance of S and b between the two interaction types. While var(S) does not significantly differ, var(bM) < var(bT) for all tests. c, p-values of two-sample Kolmogorov–Smirnov tests allowing the comparison of the distribution of b and S between the two interaction types. The full distribution of b can be compared between the two interaction types if their distribution of S is similar enough. This is fulfilled when η ≤ 140: while the interaction types have the same distribution of S, their distribution of b differs when \(\eta \in [60,140]\). d, p-values of Welch t-tests allowing the comparison of the slope of the b ~ S relationship in η mutualistic and η trophic networks obtained through linear regression. The tests were performed on 100 slopes for each sample size η: slopeM > slopeT when η > 20. e, p-values of Welch t-tests allowing the comparison of the slope β of the L ~ S relationship in η mutualistic and η trophic networks obtained through log-log regressions. The tests were performed on the 100 slopes computed for each sample size η: βM < βT when η > 100. On all panels, the dotted horizontal line indicates the p-value = 0.001 threshold. The centre line of the box-plot elements corresponds to the median, the box limits to the upper and lower quartiles, the whiskers to 1.5x interquartile range and the diamonds to the outliers.

Extended Data Fig. 5 Mean value of b, variance of b, slope of the b ~ S relationship and the across-networks slope β in samples of mutualistic (blue) and trophic (red) networks.

The vertical lines correspond to the observed values for the full dataset (249 mutualistic and 186 trophic networks) while the histograms illustrate the distribution of these values in 100 subsets of 2×100 networks randomly sampled from the complete dataset and this, for each interaction type. a, The mean value of b is significantly lower in mutualistic networks than in trophic ones (Welch t-test, see Extended Data Figure 4a). b, The variance of b is significantly lower in mutualistic networks than in trophic ones (Levene test, see Extended Data Figure 4b). c, The rate at which b decreases when S increases (slope of the b ~ S relationship) is significantly higher (less negative) in mutualistic networks than in trophic ones (Welch t-test, see Extended Data Figure 4d). d, The slope β of the across-network log-log regression does not significantly differ between the mutualistic and the trophic networks (Welch t-test, see Extended Data Figure 4e).

Extended Data Fig. 6 Distribution of the number of species S and the number of links L for the 435 empirical networks used in this study.

The black line indicates the distribution of the number of species (a) and the number of links (b) when all networks are considered while the blue histogram corresponds to the distribution of the variables in mutualistic networks and the red histogram to the one in trophic networks.

Supplementary information

Supplementary Information

Supplementary Equations (1)–(4) and references.

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Supplementary Data

List of the empirical networks used.

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Carpentier, C., Barabás, G., Spaak, J.W. et al. Reinterpreting the relationship between number of species and number of links connects community structure and stability. Nat Ecol Evol 5, 1102–1109 (2021). https://doi.org/10.1038/s41559-021-01468-2

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