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Estimating comparable distances to tipping points across mutualistic systems by scaled recovery rates

Nature Ecology & Evolution volume 6, pages 1524–1536 (2022)Cite this article

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Abstract

Mutualistic systems can experience abrupt and irreversible regime shifts caused by local or global stressors. Despite decades of efforts to understand ecosystem dynamics and determine whether a tipping point could occur, there are no current approaches to estimate distances (in state/parameter space) to tipping points and compare the distances across various mutualistic systems. Here we develop a general dimension-reduction approach that simultaneously compresses the natural control and state parameters of high-dimensional complex systems and introduces a scaling factor for recovery rates. Our theoretical framework places various systems with entirely different dynamical parameters, network structure and state perturbations on the same scale. More importantly, it compares distances to tipping points across different systems on the basis of data on abundance and topology. By applying the method to 54 real-world mutualistic networks, our analytical results unveil the network characteristics and system parameters that control a system’s resilience. We contribute to the ongoing efforts in developing a general framework for mapping and predicting distance to tipping points of ecological and potentially other systems.

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Fig. 1: Early-warning signals indicate distance to tipping points of mutualistic networks.
Fig. 2: The resilience function of mutualistic networks to structural perturbations.
Fig. 3: Detecting the resilience of mutualistic networks.
Fig. 4: Detecting signals from real ecosystems.
Fig. 5: Early-warning signals predict distances to tipping points of ecosystems.

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Data availability

All network and abundance data can be accessed through https://doi.org/10.5281/zenodo.6784072.

Code availability

All code used in this study is available on Zenodo at https://doi.org/10.5281/zenodo.6784072.

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Acknowledgements

W.Z. acknowledges support from the National Natural Science Foundation of China (grant nos. 61702200, 61473183, U1509211 and 61627810) and National Key R&D Program of China grant no. 2017YFE0128500. J.G. acknowledges the support of the USA National Science Foundation under Grant No. 2047488, and the Rensselaer-IBM AI Research Collaboration.. Q.W. was partially supported by the US National Science Foundation (grant no. 1761950 and 2125326). We sincerely thank J. Bascompte for early discussion and detailed suggestions that helped improve our paper.

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Authors and Affiliations

  1. Department of Automation, Shanghai Jiao Tong University, Shanghai, China

    Huixin Zhang & Weidong Zhang

  2. Department of Civil and Environmental Engineering, Northeastern University, Boston, MA, USA

    Qi Wang

  3. Department of Physics, Bar-Ilan University, Ramat-Gan, Israel

    Shlomo Havlin

  4. Department of Computer Science, Rensselaer Polytechnic Institute, Troy, NY, USA

    Jianxi Gao

  5. Network Science and Technology Center, Rensselaer Polytechnic Institute, Troy, NY, USA

    Jianxi Gao

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  1. Huixin Zhang
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Contributions

All authors designed and conducted the research. H.Z. and J.G. performed the analytical and numerical calculations. H.Z., Q.W. and J.G. carried out analysis and interpreted the data. J.G. led the writing of the manuscript.

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Correspondence to Jianxi Gao.

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Extended data

Extended Data Fig. 1 54 real ecological networks with abundance values from 8 locations.

We obtained 54 pollination networks with real proportional abundances from references listed above (‘Ref.’), including 51 weighted networks (Table S3) and 3 unweighted ones (Table S4), spread in 8 locations world-wide. We sorted the effective abundance ratio decently as A - H. ‘ID’ is the network’s code shown in ‘web of life’66, ‘LAT’ refers to the latitude of one location, and ‘LONG’ refers to its longitude.

Extended Data Fig. 2 The characteristics of the real networks analysed in the paper.

‘ID’ is the identity of one network recorded in the website66. ‘∣P∣’, ‘∣A∣’ denotes the number of plant and animal species respectively. ‘∣links∣’ is the number of links in a network. ‘C’ is the connectivity of a network. ‘HP’ and ‘HA’ are the heterogeneity of the plant and animal sub-networks respectively. ‘γ’ is the effective mutual strength. ‘\(\left\langle {{{\rm{err}}}}\right\rangle\)’ is the prediction error of our method for each network. For data sources and references, see Supplementary Prediction error.

Extended Data Fig. 3 The robustness of the proposed resilience function for non-transition case.

The proposed resilience function (the gray curve, theoretical prediction obtained from equation (1) with βs = 100 and β = 10) predicts well the resilience of 28 weighted networks (see supplementary Table S2) without transition, under different parameters: α = 2, α = 1, α = − 1, h = 0.8, h = 0.5 and h = 0.2. Moreover, it is robust to competition typologies by assigning three different linking probabilities: 20% (light blue), 50% (blue), and 100% (red).

Extended Data Fig. 4 Prediction fails with large inter-competition strengths and high heterogeneity.

a-b, Test how inter-competition strength influence the prediction on a synthetic system with a plants’ inter-competition network A (a 100*100 ER network), an animals’ inter-competition matrix C (a 150*150 ER network), and a mutualistic network (a 100*150 ER network), with homogeneous inter-competition βij and mean degree 5. a, The simulation results of the effective abundance for all species (‘S’, obtained by solving the high-dimensional equations equation (2)) fit well with the theoretical prediction (‘T’, obtained by numerically solving 1D function \(\frac{dx}{dt}=x(\alpha -{\beta }_{w}x+\frac{\gamma x}{1+h\gamma x})\) directly) when the inter-specific competition strength βij is small (βij < 0.2 for certain mutual strength γ0 = [3, 5, 15]). However, the prediction fails with large inter-competition βij. b, For one γ0, we calculate the prediction error (difference between ‘S’ and ‘T’) for βij = [0.1, 0.2, 0.5, 1] respectively. In each box, there are 15 points (γ0 = [1: 15]). The central mark indicates the median, the bottom and top edges of the box indicate the 25th and 75th percentiles respectively, while the whiskers extend to the most extreme data points not considered outliers (the outliers are marked with ‘+’). c, We show how the heterogeneity of a mutualistic network influences the prediction accuracy applied to 39 real mutualistic networks in Fig. 2q. Firstly, we project each system into two networks31 and then calculate the heterogeneity of the two networks that is, HP and HA, respectively. d, We compare the calculation time between the full system (equation (1)) and the decoupled system (first solve equation (3) and then put xeff and xeco into equation (4)), for 10 random graphs. Data are presented as mean values +/- SEM. CPU: Single core in Intel(R) Xeon(R) CPU E5-2683 v4 @ 2.10GHz. Parameter setting: h = 0.2, \({\alpha }^{(P)}=-0.{3}_{{N}^{{{{\rm{I}}}}}\times 1}\), \({\alpha }^{(A)}=-0.{3}_{{N}^{{{{\rm{II}}}}}\times 1}\), \({\beta }_{S}^{(P)}={{{\rm{diag}}}}({3}_{{N}^{{{{\rm{I}}}}}\times 1})\), \({\beta }_{S}^{(A)}={{{\rm{diag}}}}({3}_{{N}^{{{{\rm{II}}}}}\times 1})\), \({\gamma }_{ij}={\varepsilon }_{ij}{\gamma }_{0}/{({s}_{i})}^{\delta }\), δ = 0.5.

Extended Data Fig. 5 The prediction of resilience in supply chain networks.

The resilience is tested against: (1) perturbation in links (a-d, for mutualistic network), and (2) perturbation in nodes (e-h). Link perturbation has two variations: increasing/decreasing (that is competition/mutualism) the average weight to a fraction fw of their original value; and extinction of a fraction fL of links. Node perturbation also has two variations: extinction of a fraction fsN of suppliers; extinction of a fraction fmN of manufacturers. i, All data (red points) in a-h uniformly collapse onto the resilience function (blue surface), indicating that regardless of the network structures and the forms of perturbation, the state of the system is captured by β and γ (see dynamics for supply chain networks in Supplementary Note 7).

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Supplementary Figs. 1–26, Discussion and Tables 1–6.

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Zhang, H., Wang, Q., Zhang, W. et al. Estimating comparable distances to tipping points across mutualistic systems by scaled recovery rates. Nat Ecol Evol 6, 1524–1536 (2022). https://doi.org/10.1038/s41559-022-01850-8

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