Philippa Kaur

Field experiment
Study design
In 2017, eight unmanaged meadows were selected in the Prealps of Switzerland. This region has low levels of light emission with a radiance lower than 0.25 × 10-9 W sr-1 cm-2 (data from https://www.lightpollutionmap.info). Meadows had an average linear distance to the nearest site of 1.45 ± 0.34 km. The sites were located in the middle of the meadows on as homogenous vegetation as possible, so that there was no influence by elements like bushes or forest edges. The most abundant and widespread plant species on the meadows was Cirsium oleraceum (Asteraceae), followed by other plant species being abundant but not present on all sampling sites: Angelica sylvestris (Apiaceae), Eupatorium cannabinum (Asteraceae), Erigeron annuus s.l. (Asteraceae) and Filipendula ulmaria (Rosaceae). On four out of the eight meadows we experimentally installed a LED street lamp (Schréder GmbH, type: AMPERA MIDI 48 LED, colour temperature: neutral white (4,000 K), nominal LED flux: 6,800 lm) on 6 m high poles. Street lamps were installed on one side of the meadows, which resulted in an experimental set-up, where during nighttime a part of the meadow was illuminated by a cone of light. The part of the meadow further from the experimentally set-up street lamp was not illuminated and its darkness corresponded to the darkness measured on the control meadows that had no artificial light source in the vicinity. In other words, the four meadows were divided by artificial light into two parts, one directly illuminated by the lamp and the other being dark but adjacent to the illuminated part. Subsequently, we refer to the two parts as two sites, even though they were part of the same meadow, i.e., the illuminated part is further referred to as illuminated site, the dark part adjacent to the illuminated part as adjacent site (see Fig. 1). It is important to notice, that the street lamp was experimentally established, i.e., there was no systematic bias in terms of other landscape structures (such as roads, forest edges or hedges) where the illuminated part of the meadow was, adjacent, respectively. Thus, landscape structures that were different between the illuminated and dark part of a meadow potentially influenced the results in a non-systematic way and increased variance, but did not create a systematic bias. The remaining four meadows were left completely dark (further referred to as dark control sites), but they were equipped with a fake street lamp to provide comparable conditions. Light intensity on illuminated sites followed a negative exponential curve as function of the distance from the lamp dropping from 75.73 ± 1.54 lx just under the lamp ( More

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Sampling
Killer whale biopsy samples of skin and blubber from 38 individuals were collected year-round from August 2017 to July 2018 in northern Norway. All whales were sampled according to relevant guidelines and regulations, and conducted under the permit FOTS-ID 10176 issued by Mattilsynet (the Norwegian Food Safety Authority, report nr. 2016/179856). Details of seasonal sampling locations, stable isotope dietary descriptors and classification of sampled individuals are described in a previous study14. In the current study, total Hg was analysed in skin from all individuals (n = 38), whereas organohalogen contaminants (OHC) was analysed in blubber of 31 individuals due to insufficient blubber for the remaining 7 individuals.
OHC analysis
OHC analysis was conducted at the Laboratory of Environmental Toxicology at the Norwegian University of Life Sciences, Oslo, Norway. We analysed a total of 83 OHCs: 49 organochlorines (OCs), including 34 PCBs and 15 organochlorine pesticides (OCPs), 18 brominated flame retardants (BFRs), including newer and unregulated compounds, and 16 hydroxylated metabolites (OH-metabolites) of PCBs and polybrominated diphenylethers (PBDEs). A full list of analysed compounds can be found in Supplementary Table S1.
We analysed OCs and BFRs using a multicomponent method, first described in 197842, and since modified for a range of compounds and biological matrices43,44,45,46. The analysis of the OH-metabolites was conducted according to previously published methods47,48. An outline of the method is described in the Supplementary Information. Reported concentrations were blank corrected based on the average concentration detected within blank samples. The limit of detection (LOD) was defined as three times the average noise in chromatograms, and ranged from 0.40 to 11.10 ng/g w.w. for OCs, 0.012 to 0.362 ng/g w.w. for BFRs and 0.013 to 0.040 ng/g w.w. for OH-metabolites (see Supplementary Table S2). Internal reference materials for OCs and BFRs (contaminated seal blubber, MTref01) and OH-metabolites (contaminated seal blood, MTref03) were also extracted in conjunction with sample material to assess method performance. Internal standard recoveries are listed in Supplementary Table S2.
Hg analysis
We analysed total Hg by atomic absorption spectrometry at the University of Oslo, using a Direct Mercury Analyser (DMA-80, Milestone Srl, Soirsole, Italy). Killer whale skin samples were freeze dried in a Leybold-Heraeus GT2 freeze dryer with a Leybold Vakuum GmbH vacuum pump (Leybold, Cologne, Germany) and then homogenised to a fine powder using an agate pestle and mortar. Approximately 0.002 g of killer whale skin were analysed in parallel with sample blanks and certified reference material (DORM-4, fish protein; DOLT-5, dogfish liver, National Research Council, Ottawa, Canada). If enough material, samples were analysed in duplicates to ensure precision of measurements and the arithmetic mean value used. Average recoveries of the certified reference materials were within 10% of the reported values. The detection limit of the instrument was 0.05 ng mercury.
Data treatment
We included OHC compounds found in levels above the instrument’s LOD in a minimum of 65% of the individual whale samples for statistical analysis (see Supplementary Table S1, Supporting Information for pollutants excluded). For individual concentrations below the LOD, we imputed left-censored data by replacing missing values with a random number between 0 and the LOD assuming a beta distribution (α = 5, β = 1) to retain the pattern of the dataset. In total, 95 values below the LOD were replaced, representing 6.52% of the OHC dataset. All total Hg samples were above the LOD.
We defined the ΣPCBs as the sum of all 28 PCB congeners detected in more than 65% of the whale samples (PCB-28, -66, -74, -87, -99, -101, 105, -110, -114, -118, -128, -137, -138, -141, -149, -151, -153, -156-, 157, 170, -180, -183, -187, -189, 194, -196, -206, -209). The definition for ΣPCBs varies within killer whale literature, with some studies analysing only a few core PCB congeners35, some all 209 of the possible congeners36, and others not providing a definition (e.g. for thresholds for possible health effects7). There will therefore inevitably be some errors in comparisons. However, since the ΣPCBs in killer whales is dominated by a few commonly reported congeners, typically PCB-153 and -13816,37, it is unlikely that the inclusion of other minor constituents will have a major influence on the total load. PCBs were further grouped according to the number of chlorine substitutions per molecule, i.e. homologue group to compare the pattern of PCBs. ΣDDTs was defined as the sum of p,p′-DDT, p,p′-DDD and p,p′-DDE, the ΣPBDEs as the sum of BDE-28, -47, -99, -100, -153 and -154 and the sum of chlordanes (ΣCHLs) as the sum of oxychlordane, trans-chlordane, cis-chlordane, trans-nonachlor and cis-nonachlor.
Statistical analyses
Statistical analyses were performed using R v. 3.4.149. The significance level was to set to α = 0.05, except in cases where the value was adjusted due to multiple testing, and was two-tailed. In addition to visual inspection, normality was tested using the Shapiro–Wilk’s test50 and homogeneity of variance by Levene’s test51 using the R package car52.
Whale dietary groups
The dietary groups used in this study are based on a previous study, which used stable isotope values inputted into a Gaussian mixture model to assign sampled individuals to two fish-eating groups: Herring-eaters and Lumpfish-eaters and one mammal-eating group Seal-eaters14. The three dietary groups were characterised by disparate, non-overlapping isotopic niches that were consistent with predatory field observations. The seal-eating group was defined by higher δ15N values than the two fish-eating groups.
We found that the herring and lumpfish-eating killer whales did not differ in either their OHC levels (Tukey’s HSD: p = 0.49) or total Hg levels (pairwise Welch’s t-test: p = 0.67). In this study, we thus combined the dietary groups Herring-eaters and Lumpfish-eaters into the group Fish-eaters, to enable easier comparison to the seal-eating killer whales.
We then used Welch’s t-test to compare the ΣPCB levels in the seal-eating and fish-eating dietary groups (using a log10 transformation), and to compare the total Hg levels in the skin between the two dietary groups.
OHC dataset
We used multivariate analysis to compare and visualise the differences in all the OHCs between the dietary groups, age and sex classes using the vegan package in R53. Principle Component Analysis (PCA) was used to visualise the main structure of the data: reducing the dimensions to two new, uncorrelated, latent variables termed principle components 1 and 2 (PC1 and PC2). We log-10 transformed contaminant levels to ensure normality and homogeneity of variance, and the presence of any influential outliers were checked by the Cook’s distance test. Redundancy Analysis (RDA) was used to extract and summarise the variation in the OHC levels constrained, and thereby explained, by a set of explanatory variables54. Significant associations between response variables and the explanatory variables were identified by an RDA based forward model selection, followed by a Monte Carlo forward permutation test (1,000 unrestricted permutations). The samples’ scores along PC1 were subject to one-way Analysis of Variance (ANOVA) followed by Tukey’s honestly significant difference post hoc test (Tukey’s HSD) to analyse differences between the three dietary groups. PC1 scores were also used to evaluate correlation to total Hg levels in the skin using a Spearman’s rank correlation test. Absolute concentrations were subject to PCA with lipid % as a covariate, after checking its significance using RDA, as lipid normalising data in inferential statistics can often lead to misleading conclusions55.
We lipid-normalised OHC values when comparing levels to threshold values for toxicity or other killer whale populations, and used the geometric mean as the average for each dietary group to reflect the log normal distribution of the data. In accordance with convention, efforts were made to only compare adult males with other worldwide populations, as reproductive female whales are known to transfer a substantial portion of their OHC burden to their calves35,36,38. In any case of comparison, similar metrics were compared (i.e. arithmetic mean, geometric mean, median) and all variables kept similar (i.e. sex, age, biopsy/stranded animals). We make the assumption in this study that the killer whales sampled in 2002 in Norway were fish-eaters for the following reasons: firstly, the whales were sampled on herring overwintering grounds, feeding on herring, and photographs were taken of five of the eight adults sampled and were identified as herring-eaters from previous field observations16. Secondly, the PCB pattern in the sampled whales showed 76% of ΣPCBs higher chlorinated congeners (hexaCBs or higher), which is more similar to the fish-eaters from our study (80% higher chlorinated congeners) than the seal-eaters (87% higher chlorinated congeners). Thirdly, the upper 95% confidence range of all pollutants reported in the 2002 killer whales falls below both the geometric and arithmetic mean values for seal-eaters from this study.
Total Hg dataset
The normal distribution of the data within each dietary group meant we used the arithmetic mean as an average. The three dietary groups (Herring-eaters, Lumpfish-eaters and Seal-eaters) were compared using a pairwise Welch’s t-test with a Benjamini–Hochberg False Discovery Rate correction to adjust for multiple testing. Because we found no difference between the Herring-eaters and Lumpfish-eaters (p = 0.67), we combined these two groups to a group called “Fish-eaters” for easier comparison with the seal-eaters. The total Hg levels in the skin of the two groups, Fish-eaters and Seal-eaters were compared using Welch’s t-test.
There is a strong positive correlation between Hg levels in the skin and liver in toothed whales, and this can be used to compare Hg levels measured in skin with hepatic toxicity threshold values56,57,58. To extrapolate to liver from skin in our samples, we chose an equation based on a model using concentrations in the liver (Hgliver μg/g w.w) and skin (Hgskin μg/g w.w) of bottlenose dolphins (Tursiops truncatus) (Eq. 1)58. We converted dry weight to wet weight using the water content for each individual whale measured during freeze drying.

$$ln left( {Hg_{liver} } right) = 1.6124 times ln left( {Hg_{skin} } right) + 2.0346$$
(1)

When comparing Hg concentrations to other worldwide populations, both male and female whales were included. This was due to a lack of information of sex in one of the populations for comparisons and because killer whales are unlikely to pass on Hg burdens to calves5,59. More

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Minimal fatal shock
The first step in identifying the MiFaS for a given system is to define a desired state (mathbf {X_0}). We then assume that, prior to perturbations, the system resides on (mathbf {X_0}) and that a shock—applied at (t=0)—kicks the system’s state instantaneously to (mathbf {X}(0)). A shock—now defined as (mathbf {x}(0) = mathbf {X}(0)-mathbf {X_0})—is said to be fatal if (mathbf {X}(0)) is located outside the basin of (mathbf {X_0}) and non-fatal if (mathbf {X}(0)) is located within the basin of (mathbf {X_0}). Accordingly, the MiFaS is a vector which displays the shortest distance between the desired state and its basin boundary and the corresponding direction in state space (Fig. 1a).
Figure 1

Representation of the Minimal Fatal Shock and the related search algorithm. (a) The MiFaS (red arrow) is the smallest perturbation to the desired state (mathbf {X_0}) which puts the system outside the basin of (mathbf {X_0}) and into the basin of an alternative attractor (mathbf {X_a}). (b) The search algorithm starts with a relatively large perturbation magnitude. The related subspace of allowed initial conditions is given by the largest circle and the direction of maximum amplification is displayed by the green arrow. As the magnitude of allowed perturbations is reduced, the direction of maximum amplification converges towards the MiFaS. Color coding marks the objective function (distance to the desired state after a short integration time) with dark colors displaying large values and bright colors small values. This figure was generated using MATLAB version R2020a (https://www.mathworks.com/).

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The second essential step, is defining a norm for the perturbation size. It is important to note that the use of a certain norm is not only a technical but also an interpretative decision. Throughout this work, we use the Euclidean distance to the desired state (mathbf {X_0}) to quantify the magnitude d of a perturbation

$$begin{aligned} d ; = , ||mathbf {x}(0) || , = , ||mathbf {X}(0) – mathbf {X_0}||. end{aligned}$$
(1)

To determine the MiFaS, we develop a search algorithm which is based on the minimal seed approach41 and which can be divided into two stages, the global random initialization (stage I) and the local non-random optimization (stage II).
In stage I, we randomly draw initial conditions from a shrinking subspace in state space to find a fatal shock with a preferably small magnitude d (see “Methods” and Supplementary Fig. S1). Stage II starts with the smallest fatal shock received from stage I (Supplementary Fig. S1). From this point on, we take two seemingly opposing steps. First, we adapt the direction of (mathbf {x}(0)) in order to move (mathbf {X}(0)) away from the basin of (mathbf {X_0}) while keeping d fixed. Second, we move (mathbf {X}(0)) towards the basin by reducing d by a step size (Delta d). By repeating these two steps iteratively, we attain smaller and smaller fatal shocks which finally converge towards a local MiFaS (see Fig. 1b and Supplementary Fig. S1). It is important to note that the outcome of the search—and thus the achieved local MiFaS—is dependent on the initialization in stage I. Accordingly, to attain the global MiFaS, we need to run the search algorithm multiple times and select the minimum of the local MiFaS as the global one.
Figure 2

Minimal Fatal Shock for an exemplary plant–pollinator network. (a) Direction of the MiFaS. The perturbation vector is scaled to a length of 1. The relative contribution of each element of the vector (node in the network) to the overall perturbation is represented by the area and the color saturation of the respective squares and circles. A pink coloring denotes a loss and a green coloring a gain in species abundance at the respective node. Squares portray pollinators and circles plants. Species being lost after the perturbation are marked by the yellow shaded region. Placement of the vertices is based on the Kamada–Kawai algorithm66 obtained from python-igraph version 0.7.1 (https://igraph.org/). (b) Transient behavior following the MiFaS. Dark gray area shows the situation before the perturbation (desired state). Lighter gray area shows how the state variables are altered due to the perturbation. Light gray area depicts the transient behavior after the system has been perturbed. (c) Evolution over a longer time span. Vertical line displays the time interval shown in (b). The figure was generated using MATLAB version R2020a (https://www.mathworks.com/).

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The centerpiece of the outlined algorithm is the adaptation of the direction of (mathbf {x}(0)) during stage II, which aims at maximizing the distance between (mathbf {X}(0)) and the basin boundary of (mathbf {X_0}). However, since this distance is not easily accessible, it is approximated by an objective function which can be maximized within a constraint optimization. For the two applications we present here, the objective function can be thought of as the amplification of the shock over a preselected time T (see “Methods” for specific definition). The mechanism behind this is that trajectories close to the basin boundary stay close to it for long times as they move along the stable manifold of a saddle-type state while trajectories far off the boundary approach an alternative attractor faster and thus lead to earlier and stronger amplifications.
In summary, as a result of the optimization procedure we obtain the magnitude of the smallest distance to the basin boundary which can be utilized as a quantitative measure of global stability and the direction of the perturbation in the high-dimensional phase space.
Plant–pollinator networks
In our first example, we consider a simple model of mutualism which captures the crucial aspects of a system of plants and their corresponding pollinators43,45. The mutualistic system is described as a bipartite network, with one set of nodes representing a number of (N_P) plant species and one set representing a number of (N_A) animal species whose dynamics are given by

$$begin{aligned} frac{mathrm {d} P_i}{mathrm {d} t} ,&= , alpha P_i , – , sum _{k=1}^{N_P} beta _{ik} P_i P_k , + , frac{sum _{j=1}^{N_A} gamma _{ij} A_j P_i}{1 + h sum _{j=1}^{N_A} gamma _{ij} A_j},nonumber \ frac{mathrm {d} A_j}{mathrm {d} t} ,&= , alpha A_j , – , sum _{l=1}^{N_A} {tilde{beta }}_{jl} A_j A_l , + , frac{sum _{i=1}^{N_P} {tilde{gamma }}_{ji} P_i A_j}{1 + h sum _{i=1}^{N_P} {tilde{gamma }}_{ji} P_i}, end{aligned}$$
(2)

where (P_i) denotes the abundance of plant species i ((i=1, ldots , N_P)) and (A_j) the abundance of animal species j ((j=1, ldots , N_A)). In Eq. (2), the parameter (alpha) gives the intrinsic growth rate, (beta _{ik}) (({tilde{beta }}_{jl})) the competitive pressure of plant (animal) species k (l) on plant (animal) species i (j), (gamma _{ij}) (({tilde{gamma }}_{ji})) the benefit plant (animal) species i (j) obtains from animal (plant) species j (i) and h the handling time for pollination. As a general principle, we assume the benefit a species gains from pollination to be obligatory for its own growth, an assumption which is necessary to obtain multistability in this model57. Therefore, we choose the net growth rate (alpha le 0).
In order to keep the parametrization as simple as possible, we set (alpha), (beta _{ii}) (({tilde{beta }}_{jj})) and h to be equal for all species. To reduce the complexity of the overall interaction pattern, we assume all-to-all coupling for the interspecific competition between species within one set, whereby (beta _{ik}=beta _0/(N_{P}-1)) for (i ne k) (({tilde{beta }}_{jl}=beta _0/(N_{A}-1)) for (j ne l)). By contrast, a mutualistic interaction between an animal and a plant species can either be absent, in which case (gamma _{ij}=0) (({tilde{gamma }}_{ji}=0)), or present, in which case (gamma _{ij}=gamma _0/kappa _i) (({tilde{gamma }}_{ji}=gamma _0/{tilde{kappa }}_j)), where (kappa _i) (({tilde{kappa }}_j)) denotes the degree or the number of mutualistic partners of plant (animal) species i (j). This formulation corresponds to a full trade-off between the benefit a species attains from one partner and the number of partners this species has45. An important aspect of the chosen parametrization is that species solely differ on account of their position in the mutualistic network. In the following, we determine the MiFaS for realistic plant–pollinator networks from the Web of Life Database58 representing networks from different geographic locations across various climate zones (see Supplementary Fig. S5 and Supplementary Table S2). With (alpha = -0.3), (beta _{ii}=1.0), (beta _0 = 1.0), (gamma _0 = 4.5) and (h=0.1), we choose the model parameters in a way that ensures that each of the studied systems possesses a state in which all species coexist. This ’desired’ state (mathbf {X_0}) is opposed to multiple ’undesired’ states in which one or more species are gone extinct (the MiFaS is actually fatal).
To interpret the results, it is useful to state some general considerations first. Due to the mutualism, the growth of a species depends on the abundance of its mutualistic partners. As the growth of these partners can also depend on further other partners, these further partners indirectly support the growth of the first species. We could continue building this chain of dependencies but essential is that species being close to each other within the network and especially those sharing partners benefit from each other. On the other hand, due to competition high abundances of one species directly impede the growth of all species within the same group (animals or plants). Hence, the net effect which an increase or decrease of a species’ abundance has on another species depends on the interplay between the two processes. The indirect benefits can either balance or enhance the negative effects due to competition depending on whether species are close (balance) or far apart (enhance).
At first, we compute the minimal fatal shock (MiFaS) for an exemplary network from Morant Point in Jamaica (Fig. 2a). The topology of this system is characterized by an asymmetric division into a small tree-like part and a large core, i.e. a large mostly well connected component. This topological division is mirrored in the direction of the MiFaS which is visualized by the color-coding. A small negatively perturbed part consisting of the tree-like periphery (nodes within the yellow shaded region in Fig. 2a) plus its single non-peripheral neighbor is opposed to the rest of the network which is positively perturbed. This division exemplifies how the mutualistic and competitive interactions between species shape the system’s response to perturbations. In the tree-shaped part of the network, all species are close to each other but far away from most other species. Furthermore, due to the sole connection between the two characteristic structural parts of the network, the share of partners between the two is minimal. As a result, the interdependency of species within the tree-shaped part is extremely high. Accordingly, the loss of abundance of any species in the tree-like structure—as it is the case in the MiFaS (Fig. 2)—significantly affects all other species in this tree-like periphery. On the contrary, the competitive stress due to species within the large component is high as it is not balanced by the indirect benefits. It is actually even enhanced as the increase of abundance of one species boosts the growth of its partners which again enhances the competive stress on the peripheral tree-like structure.
Figure 3

Magnitudes of 59 and direction of six MiFaS in plant–pollinator networks. The 59 networks are ordered, from low to high, and labeled according to their respective magnitude of the MiFaS. In addition, the direction of the MiFaS is shown for six exemplary networks. Perturbation vectors are scaled to a length of 1. The relative contribution of each element of the vector (node in the network) to the overall perturbation is represented by the area and the color saturation of the respective squares and circles. A pink coloring denotes a loss and a green coloring a gain in species abundance at the respective node. Squares portray pollinators and circles plants. Species being lost after the perturbation are marked by the yellow shaded region. Placement of the vertices is based on the Kamada–Kawai algorithm66 obtained from python-igraph version 0.7.1 (https://igraph.org/). The figure was generated using MATLAB version R2020a (https://www.mathworks.com/).

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After the system has been hit by the MiFaS, all ten species within the tree-like periphery are lost in the long run (Fig. 2c and yellow shaded region in Fig. 2a). The remaining species—except for the single neighbor of the periphery—tend to higher abundances as the competitive pressure on them is relaxed. Accordingly, the new asymptotic state (Fig. 2c) again shows that the net impact of the peripheral species on most other species has been negative. Apart from the new asymptotic state, the transient leading there (Fig. 2b,c) is of interest as well. In fact, the transient behavior is typical for an initial state close to the basin boundary which is made up by the stable manifold of a saddle point. The transient at first moves towards the saddle fast (Fig. 2b), stays in its vicinity for some time as the repulsion is weak and finally settles on an attractor which, in this case, is the undesired state of partial extinction (Fig. 2c).
Figure 4

Minimal Fatal Shock in the Great Britain power grid. (a) Direction of the MiFaS. The perturbation vector is scaled to a length of 1. The relative contribution of each element of the vector (node in the network) to the overall perturbation is represented by the area and the color saturation of the respective squares and circles. A blue coloring denotes a deceleration and a pink coloring an acceleration at the respective node. Squares portray consumers and circles generators. Width of transmission line scales with respective initial transmission load. (b) Blow-up of tree-like structure in (a). (c, d) Transient behavior following the MiFaS. (c) Time series of the loads on the transmission lines included in (b). Colors of highlighted loads correspond to colors of transmission lines in (b), remaining loads are depicted in white. (d) Time series of the frequency deviations of all oscillators, color coding corresponds to perturbation magnitude and direction at each node. The figure was generated using MATLAB version R2020a (https://www.mathworks.com/).

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Overall, we examine the MiFaS for a total of 59 plant–pollinator systems, each being based on one of the real-world network topologies. For comparison, we order the networks from sensitive to robust according to the magnitude of their respective MiFaS and depict the direction of the MiFaS for five further exemplary systems (Fig. 3).
Figure 5

Local Minimal Fatal Shocks in the Great Britain power grid. Direction of the local MiFaS. The perturbation vectors are scaled to a length of 1. The relative contribution of each element of the vector (node in the network) to the overall perturbation is represented by the area and the color saturation of the respective squares and circles. A blue coloring denotes a deceleration and a pink coloring an acceleration at the respective node. Squares portray consumers and circles generators. (a–d) Blow-ups of the significantly perturbed area of four local MiFaS which correspond to different outcomes of the optimization process. Highlighted edges represent the trigger transmission line of the particular perturbation. The figure was generated using MATLAB version R2020a (https://www.mathworks.com/).

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Some characteristics found for the MiFaS of the exemplary network (Fig. 2) prove to be generally valid. For each system, the division of the MiFaS into a small negatively perturbed part and a larger but weaker positively perturbed part displays how mutualistic interdependency and competition shape the system’s response to perturbations. In this context, the negatively perturbed part marks the weakest point of the network at whose outer edge the extinction occurs. Speaking in ecological terms, we find these weak points always being associated with specialization and the distribution of negative perturbations depends on the nature of the caused interdependency: in the exemplary system (network 1 in Fig. 3), where the specialization among all species within the tree-like structure is rather mutual, all involved species are significantly perturbed (the same for network 13 and partly for network 4, Fig. 3). However, the more asymmetric the specialization gets—meaning that many specialists are connected to a single generalist—the stronger the negative perturbation focuses on this generalist (networks 4 (rightarrow) 26 (rightarrow) 27 (rightarrow) 49, Fig. 3). This perturbation structure proofs to be efficient as the dependency of the generalist on each single specialist is low but its cumulated dependency on all specialized partners is high. A perturbation at the generalist therefore induces a negative feedback whose strength also depends on the number of connections the generalist has to other-non-specialized species. Accordingly, network 49 is much more robust than network 26 as the decisive generalist is highly connected to the core.
The positive contribution to the overall MiFaS marks the impact of competitive forces which depends on the global interdependency among species. In the case of a single well-connected core and a periphery which only consists of specialists being directly connected to this core, indirect positive effects between species balance competive effects as all species are close and well connected. Accordingly, we do not find any significant contribution of positive perturbations to the overall MiFaS (networks 37, 49, Fig. 3). The contrary is the case if the core is not well build, meaning that only a few connections between important hub nodes exist (networks 4, 26) or if—due to strong reciprocal specialization—a larger peripheral structure exists (networks 1, 13). In such cases, positive perturbations at rather central core-species contribute significantly to the overall MiFaS and thus to the extinction of peripheral species. In summary, a strong global interdependency among all species favors a system’s robustness whereas a strong local interdependency paired with a weak global interdependency depicts the worst case scenario.
Great Britain power grid
As a second example we consider a coarse-grained model of a power grid which exhibits synchronization dynamics. In this framework, a power grid is described as a network of Kuramoto-like13 second order phase oscillators whose dynamics are given by

$$begin{aligned} frac{mathrm {d} phi _i}{mathrm {d} t}&= omega _i nonumber \ frac{mathrm {d} omega _i}{mathrm {d} t}&= P_i – alpha omega _i + sum ^N_{j=1} K_{ji} , sin (phi _j-phi _i), end{aligned}$$
(3)

where (phi _i) and (omega _i) denote the phase and frequency deviation of oscillator i from a grid’s rated frequency (which will hereinafter be referred to as phase and frequency). The parameters (alpha) and (P_i) are the grid’s damping constant and the net power input/output of oscillator i, respectively. The capacities of the transmission lines and therefore also the topology of the grid are contained in the matrix K, with (K_{ji}=K_{ij} >0) if oscillators i and j are connected and (K_{ij}=0) otherwise.
As an example, we consider the Great Britain power grid which consists of 120 nodes and 165 transmission lines59. For reasons of simplification, we assume one half of the oscillators to be generators ((P_i=+P_0)) and one half to be consumers ((P_i=-P_0)) whose distribution within the grid we draw randomly (see Fig. 5). Furthermore, we choose the same maximum capacity for all transmission lines, either (K_{ij}=K_0) or (K_{ij}=0). In a realistic parameter setting of this model, one ’desired’ synchronized state ((phi _i=const) and (omega _i=0) for all i) representing stable operation competes with several ’undesired’ non-synchronized states. With (alpha =0.1), (P_0=1.0) and (K_0=5.0), we choose the model parameters accordingly. In this setting, the MiFaS represents the smallest perturbation to the synchronous state which induces a shift to one of the non-synchronous states interpreted as a power outage.
The combination of frequencies and phases is actually problematic when determining the MiFaS since they differ in units. We therefore only take into account perturbations in the frequencies (omega). In this context, choosing the frequencies (omega) instead of the phases (phi) seems reasonable as disturbances usually occur due to fluctuations in the power generation or consumption60. Such parametric disturbances would first affect the frequencies via (mathrm {d}omega /mathrm {d}t) (Eq. 3). Furthermore, considering only frequencies allows a clearer depiction of the MiFaS, since the corresponding vector contains exactly one entry per node of the power grid.
Examining one random realization of the power grid (Fig. 4a), we find that, like in the exemplary plant–pollinator network, the MiFaS is associated with a tree-like structure including the most peripheral nodes of the network (according to the resistance centrality proposed by61, see Supplementary Fig. S7). In fact, the same structure is highlighted by some of the eigenmodes of the graph Laplacian (see Supplementary Fig. S8). However, apart from the observation that the MiFaS is orthogonal to a neutral perturbation affecting all oscillators in the same way which is equivalent to its first eigenmode, we find no simple connection to the graph Laplacian (see Supplementary Information).
In order to understand the effectiveness of the MiFaS, it is instructive to have a closer look at how the desynchronization occurs after the system has been hit by the MiFaS (Fig. 4c,d). The desynchronization is triggered by an overload on the transmission line which connects the seven northermost oscillators to the rest of the grid (Fig. 4b). Due to the accumulation of consumers within this tree-like structure (5 consumers towards 2 generators), already in the unperturbed state, the load—(K sin (phi _j – phi _i)) for the line connecting nodes j and i—on the ’trigger transmission line’ is comparatively close to its maximum capacity K (see Fig. 4c). Intuitively, a strong deceleration of oscillators inside plus an acceleration of oscillators outside the tree-like structure seems to be an efficient way to induce an overload. Indeed, we find the strongest negative perturbations at the seven oscillators within (Fig. 4b) as well as positive perturbations at several oscillators outside the tree-like structure. However, in the northern part of the grid, the overall MiFaS roughly follows a broad gradient distribution with negative perturbations on both sides of the trigger transmission line and the strongest positive perturbations at rather distant nodes in the northwest of Great Britain. This distribution is efficient as the perturbations in frequencies first have to be transferred into phase deviations to induce an overload. A relatively smooth gradient ensures that the arising phase deviations are balanced slowly and thus a large transmission load can build up.
This transfer can be observed in the first stage of the transient following the MiFaS (Fig. 4c,d). In this stage, the system evolves rather smoothly towards a point where the frequency deviations of all oscillators are close to zero but where, at the same time, the transmission load on the trigger line (red line in Fig. 4) has passed its maximum capacity. The system subsequently enters a stage in which both transmission loads as well as frequencies oscillate erratically until the oscillations suddenly collapse and the system settles on an undesired attractor. It is remarkable that the final overload (green line in Fig. 4) is not located on the line which triggered the desynchronization but on a line deeper in the tree-like structure (Fig. 4c). The final overload is similar to a cutoff of two consumers from the rest of the grid, as the frequencies in the two departed components evolve more or less independently. It is however important to note that this particular undesired state represents only one of several possible outcomes. Indeed, already the slightest variation (smaller than the finite precision of the search algorithm) of the initial perturbation can lead to a different non-synchronous asymptotic state, although the trigger transmission line is always the same. Such high sensitivity is often an indicator for complexly intervowen basins of attraction, characteristic to many highly multistable systems62.
In order to gain more insights into how certain topological features harm a power grid’s stability against shocks, we examine some of the local MiFaS inducing power outages (Fig. 5). These local minima correspond to different outcomes of the applied optimization scheme for the same network topology and parametrization and thus represent further close but less crucial distances between the desired state and its basin boundary. As we are interested in distinct topological weak points of the grid, we take into account only those local minima which differ in the involved trigger transmission line (highlighted edges in Fig. 5).
The local MiFaS, and in particular the examination of the associated trigger transmission lines, reveal two mutually reinforcing sources for the emergence of weak points. Firstly, desynchronization events are triggered on transmission bottlenecks which result from the loose connection between a peripheral subgraph and the rest of the grid. Four out of five of the shown local MiFaS (Fig. 4 and Fig. 5a–c) are actually related to the most pronounced case of such a bottleneck which is a bridge, i.e. a single edge connecting two subnetworks. Secondly, the accumulation of oscillators of the same type within a subgraph induces a local mismatch between power generation and consumption (Fig. 4 and Fig. 5a–d). We find each of the shown local MiFaS to be related to such a local mismatch. Already in the unperturbed state, this mismatch has to be balanced by a high initial load on the connecting transmission line(s) which in turn results in a low threshold for an overload (Fig. 5d). This overload is then triggered by the MiFaS by reinforcing the generation/consumption imbalance between the two subgraphs. Accordingly, all fatal shocks involve strong frequency perturbations with a sign according to the already established power mismatch in the peripheral subgraph and frequency perturbations in the opposite direction in adjacent areas of the grid. However, as in the global MiFaS, the boundary between positive and negative perturbations is not sharp but more (Fig. 5a,c,d) or less (Fig. 5b) follows a kind of gradient.
Of particular interest is the local MiFaS shown in Fig. 5c as its underlying topological motif is quite common in the network: a node with degree 1, also termed ’dead end’32. Apart from the two dead ends within trees (Fig. 5a,b), the portrayed dead end is the one being most sensitive to perturbations despite or seemingly because it is connected to a rather central node of degree 6 (see also Supplementary Fig. S7). For none of the surroundings of the other dead ends, which are all adjacent to lower degree nodes, we find a local MiFaS of similar low magnitude. Accordingly, we conclude that a rather central position of the node from which the peripheral subgraph branches off might actually harm its robustness against particular perturbations. More

Distribution characteristics of REEs in ground water
In this study, ground water samples were collected from 18 ground water monitoring wells around tailings ponds and their chemical characteristics were also having been determined, as showed in Figure S1. Fe, Mn2+, Cl−, SO42−, ammonia nitrogen and total hardness showed the same trend and decreased with distance. The ground water environmental quality standard (III Grade, National Standard Bureau of PR China, GB3838-2002, the water quality above III Grade can be used for living and drinking after treatment, but the water quality below III Grade was bad and cannot be used as drinking water source) was used as the evaluation standard. The ratio of the number of wells with Fe, Mn2+, Cl−, SO42−, ammonia nitrogen and total hardness exceeding the standard in the total number of wells was 33.33%, 61.11%, 66.67%, 77.78%, 100% and 81.25%, respectively.
In order to study the accumulation of REEs in ground water, the concentration of REEs in 18 ground water samples around the tailings pond were measured. The total REEs concentrations in ground water ranged from 0.0820 to 12.3 μg/L, and rare earth in the ground water accumulated in the southeast of the tailings pond (Fig. 2). In addition, the concentrations of REEs in ground water around the tailings pond decreased in the order of Ce  > La  > Nd  > Pr  > Gd  > Sm  > Dy  > Er  > Eu  > Yb  > Tb  > Ho  > Tm  > Lu. Chondrite-normalized REEs patterns for ground waters around the tailings were shown in Fig. 4b and Table 1. The well points have the same normalization pattern with a predominance of LREEs over HREEs.
Figure 2

Distribution of rare earth elements in the ground water surrounding the rare earth tailings pond (μg/L).

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Table 1 Distribution characteristics of REEs in ground water surrounding tailings pond.
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The distribution patterns of REEs in ground water were characterized by obvious fractionation of LREEs and HREEs with the LREEs/HREEs ratios of 2.77 ~ 25.9, and (La/Yb)N of 1.445 ~ 50.67. The degree of LREEs fractionation with (La/Sm)N of 0.5806 ~ 5.216. Most sampling points presented the positive anomaly of Ce and Eu, however, GW1, GW5, GW6, GW9, GW10, GW13 and GW6 were negative anomalies of Ce, while GW1, GW5, GW7 and GW8 were negative anomalies of Eu. Individual anomalies showed differentiation between selected elements (Ce and Eu) and the other REEs (Table 1).
Baotou environmental monitoring station, Inner Mongolia, China detected ground water leakage around the pond, and various degrees of ground water pollution were found with relatively lower metals concentration and higher anionic concentration21,22,23. Therefore, in addition to REEs, for our ground water correlation analysis we chose to also look at Fe, Mn2+, Cl−, SO42−, ammonia nitrogen and some other ions (HCO3−, total hardness). Correlation analysis showed that total hardness (r = 0.541, p  More