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, 111 (50), 17923-8

Trophic Coherence Determines Food-Web Stability

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Trophic Coherence Determines Food-Web Stability

Samuel Johnson et al. Proc Natl Acad Sci U S A.

Abstract

Why are large, complex ecosystems stable? Both theory and simulations of current models predict the onset of instability with growing size and complexity, so for decades it has been conjectured that ecosystems must have some unidentified structural property exempting them from this outcome. We show that trophic coherence--a hitherto ignored feature of food webs that current structural models fail to reproduce--is a better statistical predictor of linear stability than size or complexity. Furthermore, we prove that a maximally coherent network with constant interaction strengths will always be linearly stable. We also propose a simple model that, by correctly capturing the trophic coherence of food webs, accurately reproduces their stability and other basic structural features. Most remarkably, our model shows that stability can increase with size and complexity. This suggests a key to May's paradox, and a range of opportunities and concerns for biodiversity conservation.

Keywords: May's paradox; complex networks; diversity–stability debate; dynamical stability; food webs.

Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.
Scatter plots of stability (as measured by R, the real part of the leading eigenvalue of the interaction matrix) against several network properties in a dataset of 46 food webs; Pearson’s correlation coefficient is shown in each case. (A) Stability against S, where S is the number of species (r2=0.064). (B) Stability against K, where K is the mean degree (r2=0.461). (C) Stability against incoherence parameter q (r2=0.596). (D) Stability after all self-links (representing cannibalism) have been removed (Rnc) against incoherence parameter q (r2=0.804).
Fig. 2.
Fig. 2.
(A) Three networks with differing trophic coherence, the height of each node representing its trophic level. The networks on the left and right were generated with the PPM, with T=0.01 and T=10 yielding a maximally coherent structure (q=0) and a highly incoherent one (q=0.5), respectively. The network in the middle is the food web of a stream in Troy, Maine, which has q=0.18 (43). All three have the same numbers of species, basal species, and links. (B) Incoherence parameter, q, against T for PPM networks with the parameters of the Troy food web (green); and against c for generalized niche model networks with the same parameters (blue). The dashed line indicates the empirical value of q. (C) Stability (as given by R, the real part of the leading eigenvalue of the interaction matrix) for the networks of B. Also shown is the stability of networks generated with the same models and parameters, but after removing self-links (empty circles). In B and C, the dashed line represents the empirical value of R, and bars on the symbols are for 1 SD.
Fig. 3.
Fig. 3.
(A) Mean absolute deviations (MAD) from empirical values of the incoherence parameter, q, for each food-web model—cascade (CM), generalized niche (GNM), niche (NM), nested hierarchy (NHM), minimum potential niche (MPNM), and PPM—compared with a dataset of 46 food webs. (B) MAD from empirical values of stability, R, for the same models and food webs as in A. (C) MAD from empirical values of stability, R, after removing self-links, for the same models and food webs as in A and B. (D) Scaling of stability, R, with size, S, in networks generated with each of the models of previous panels except for the PPM. Mean degree is K=S. The dashed line indicates the slope predicted for random matrices by May (1), and the dotted curve is from Allesina and Tang (27). (E) Scaling of stability, R, with size, S, in PPM networks generated with different values of T. In descending order, T=10, 0.5, 0.3, 0.2 and 0.01. B=0.25S. (Inset) Slope, γ, of the stability-size line against T for α=0.55, 0.5, and 0.4, where the mean degree is K=Sα. In D and E, bars on the symbols are for 1 SD.

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