Stability criteria for complex ecosystems

Journal name:
Nature
Volume:
483,
Pages:
205–208
Date published:
DOI:
doi:10.1038/nature10832
Received
Accepted
Published online

Forty years ago, May proved1, 2 that sufficiently large or complex ecological networks have a probability of persisting that is close to zero, contrary to previous expectations3, 4, 5. May analysed large networks in which species interact at random1, 2, 6. However, in natural systems pairs of species have well-defined interactions (for example predator–prey, mutualistic or competitive). Here we extend May’s results to these relationships and find remarkable differences between predator–prey interactions, which are stabilizing, and mutualistic and competitive interactions, which are destabilizing. We provide analytic stability criteria for all cases. We use the criteria to prove that, counterintuitively, the probability of stability for predator–prey networks decreases when a realistic food web structure is imposed7, 8 or if there is a large preponderance of weak interactions9, 10. Similarly, stability is negatively affected by nestedness11, 12, 13, 14 in bipartite mutualistic networks. These results are found by separating the contribution of network structure and interaction strengths to stability. Stable predator–prey networks can be arbitrarily large and complex, provided that predator–prey pairs are tightly coupled. The stability criteria are widely applicable, because they hold for any system of differential equations.

At a glance

Figures

  1. Distributions of the eigenvalues and corresponding stability profiles.
    Figure 1: Distributions of the eigenvalues and corresponding stability profiles.

    a, For X~N(0,σ2), S = 250, C = 0.25 and σ = 1, we plot the eigenvalues of 10 matrices (colours) with −d = −1 on the diagonal and off-diagonal elements, following the random, predator–prey or mixture prescriptions. The black ellipses are derived analytically in the text. b, Numerical simulations for the corresponding stability profiles. For the random case, starting from S = 250, C = 0.5, σ = 0.1 and d = 1, we systematically varied C (crosses) or σ (plus signs) to obtain spanning [0.5,,1.0,,1.5] of the critical value for stability (indicated in red, 1 in the case of random matrices). The profiles were obtained by computing the probability of stability out of 1,000 matrices. The predator–prey case is as the random but with σ = 0.5 and critical value π/(π2). The mixture case is as the random but with critical value π/(π+2). In all cases, the phase transition between stability and instability is accurately predicted by our derivation.

  2. Stability criteria for different types of interaction.
    Figure 2: Stability criteria for different types of interaction.

    We fixed θ = d/σ = 4, and for a given connectance C we solved for the largest integer S that satisfies the stability criterion for each type of interactions. Combinations of S and C below each curve lead to stable matrices with a probability close to 1. The interaction types form a strict hierarchy from mutualism (most unlikely to be stable) to predator–prey (most likely to be stable).

  3. Distribution of the eigenvalues for the three types of mutualism.
    Figure 3: Distribution of the eigenvalues for the three types of mutualism.

    a, Unstructured mutualism. b, Bipartite mutualism. c, Nested and bipartite mutualism. In all cases, S = 250, σ = 0.1, C = 0.2 and d = 1. Note that the bipartite case does produce extreme negative real eigenvalues (green arrow) coupled with positive ones, but the row sum (and thus the rightmost eigenvalue, red arrow) is equal to that of the unstructured mutualistic case. The nested matrices, in which generalist species yield (on average) larger row and column sums, have larger rightmost eigenvalues. Thus, highly nested matrices are less likely than the other two cases to be stable.

References

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Affiliations

  1. Department of Ecology and Evolution, University of Chicago, 1101 East 57th Street, Chicago, Illinois 60637, USA

    • Stefano Allesina &
    • Si Tang
  2. Computation Institute, University of Chicago, 5735 South Ellis Avenue, Chicago, Illinois 60637, USA

    • Stefano Allesina

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All authors contributed equally.

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The authors declare no competing financial interests.

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  1. Supplementary Information (1.1M)

    This file contains Supplementary Text and Data, Supplementary Table 1, Supplementary Figures 1-7 with legends and additional references.

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