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Aquarium experiment and sampling
To examine the effect of changes in species composition on the behaviour of eDNA, we conducted aquarium experiments using two mock fish communities comprising H. neglectus, C. temminckii, O. latipes, R. flumineus, and M. anguillicaudatus. Mock community 1 (MC1) consisted of one individual of each of the five fish species, whereas mock community 2 (MC2) consisted of three H. neglectus individuals and one individual of each of the other four fish species (Fig. 2). We used two aquaria (A and B). Each aquarium was used four times, twice for each mock community, giving two replicates (R1 and R2). This resulted in eight experimental units (2 mock fish communities × 2 aquaria × 2 replicates). Figure 2 shows the experimental setup used in this study.
Figure 2

Experimental setup of the aquarium experiments.

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To set up the aquaria, 20 L of tap water was added into each aquarium (GEX Co. Ltd., Osaka, Japan) and heated with a heater (Spectrum Brands, Wisconsin, US) until the water temperature reached 25 °C. Water in the two aquaria was maintained at 25 °C and constantly circulated with an aeration device. Before adding fish to the aquaria, the water was sampled for the negative control. The first experimental samples (day 0) were taken 1 h after adding the fish and subsequent samples were taken each day until day 4. At each sampling, two 1-L samples of surface water were collected from each aquarium and then 2 L of tap water was added to each aquarium to maintain the volume of water. The weight of individual fish species was measured using an electronic balance immediately after the final sampling. After each experiment, the two aquaria were bleached before being reused.
In Japan, experiments on fish do not require any legal procedures or permission. However, in order to avoid causing pain to the specimens, the experiments in this study were conducted in accordance with the ARRIVE guidelines, Japanese laws and guidelines for mammals, birds, and reptiles as below; Act on Welfare and Management of Animals (Notice of the Ministry of the Environment No. 105 of October 1, 1973), Standards relating to the Care and Keeping and Reducing Pain of Laboratory Animals (Notice of the Ministry of the Environment No. 88 of 2006), Fundamental Guidelines for Proper Conduct of Animal Experiment and Related Activities in Academic Research Institutions under the jurisdiction of the Ministry of Education (Notice of Ministry of Education No. 71, 2006), and Guidelines for Proper Conduct of Animal Experiments (established by the Science Council of Japan on June 1, 2006).
DNA extraction
Each 1-L water sample was filtered immediately through a GF/F glass fibre filter (nominal pore size = 0.7 μm, diameter = 47 mm; GE Healthcare Japan Corporation, Tokyo, Japan). Filter funnels and measuring cups were bleached after filtration to prevent cross-contamination among the water samples. All filters were stored separately at − 20 °C until DNA extraction. Total eDNA was extracted from each filter using a DNeasy Blood and Tissue Kit (QIAGEN, Hilden, Germany) and Salivette tubes (Sarstedt AG & Co. KG, Nümbrecht, Germany). Extraction methods were as previously described18 with modifications. A filter sample was placed in the upper part of the Salivette tube and 220 μL of solution containing Buffer AL (200 μL) and Proteinase K (20 μL) was added. The tube containing the filter was incubated at 56 °C for 30 min, then centrifuged at 5000 × g for 3 min, and the solution was collected in the base of the tube. To increase eDNA yield, 220 μL Tris-EDTA (TE) buffer was added to the filter sample and centrifuged at 5000 × g for 1 min. Then, ethanol (200 μL) was added to the collected solution, and the mixture was transferred to a spin column. Total eDNA was eluted in buffer AE (100 μL), following the manufacturer’s instructions. All eDNA samples were stored at − 20 °C prior to qSeq and dPCR.
Quantitative sequencing
Simultaneous quantification and sequencing of the extracted eDNA were performed by qSeq as previously described15,16. First, SPE was performed. The SPE reaction mixture (20 µL) consisted of 1 × PrimeSTAR Max premix (Takara Bio Inc., Kusatsu, Japan), 300 nM of the primer qSeq-MiFish-U-F (Table 1), and extracted DNA (2 µL). The SPE primer qSeq-MiFish-U-F contains an eight-base length random sequence tag, which creates 65,536 different variations, enabling the quantification of up to approximately 1.0 × 105 copies of DNA15. This amount of variation was sufficient to quantify the abundance of eDNA in this study. SPE was initiated by denaturation at 94 °C for 1 min, followed by cooling to 60 °C at 0.3 °C/s, incubation at 60 °C for 1 min, and final extension at 70 °C for 10 min. Subsequently, the excess primer was completely digested by adding exonuclease I (4 µL, 5 U/µL; Takara Bio Inc.) to the SPE mixture. The digestion was performed at 37 °C for 120 min, followed by inactivation of the exonuclease I at 80 °C for 30 min. The first-round PCR mixture (25 µL) contained PrimeSTAR Max premix (12.5 µL), primers qSeq-MiFish-U-R and F2 (300 nM each; Table 1), and the SPE product (2 µL). Following 40 cycles of amplification at 98 °C for 10 s, 55 °C for 5 s, and 72 °C for 5 s, the amplification product was subjected to agarose gel electrophoresis, and the band of the expected size was removed and purified using Nucleospin Gel and PCR Clean-up column (Takara Bio Inc.). The qSeq-MiFish-U-R primer also contains eight N bases to increase the complexity, which improves the sequencing quality, and thus PhiX was not added in this study. Finally, a 2nd-round PCR was performed to add an index for Illumina sequencing as described elsewhere15. The indexed PCR amplicon was purified using AMPure XP beads (Beckman Coulter, Indianapolis, IN) followed by sequencing using a MiSeq platform with MiSeq Reagent Kit v3 for 600 cycles (Illumina). The sequence data obtained in this study were deposited in the DDBJ database under accession numbers SAMD00219124–SAMD00219214.
Table 1 Oligonucleotide sequences used in this study.
Full size table

Data analysis
First, all sequences were assembled and screened by length and quality of reads using the mothur software package (v1.39.5)22. The processed sequence reads were classified using the MiFish pipeline (http://mitofish.aori.u-tokyo.ac.jp/mifish/), with the parameters as previously described23. Subsequently, the representative sequences of individual operational taxonomic units (OTUs) were extracted using the Usearch program (http://www.drive5.com/usearch/). The random sequence tags (RST) at the end of sequences in the OTUs were counted to quantify the environmental DNA from each fish species as described elsewhere16. For comparison, the relative proportion of eDNA from individual species in each sample was calculated from the composition of the sequences of the fish species obtained by qSeq.
Microfluidic digital PCR
Quantification of eDNA was also performed by microfluidic dPCR using the BioMark Real-time System and 12.765 Digital Array (Fluidigm Corporation, South San Francisco, CA, United States) as previously described13. For each sample, the PCR mixture (6 µL) contained 2 × Probe qPCR mix (3.0 µL; Takara Bio Inc.), 20 × binding dye sample loading reagent (0.6 µL; Fluidigm Corporation), forward and reverse primers (900 nM), TaqMan probe (125 nM), ROX solution (0.015 µL), and sample DNA (1.0 µL). We used three sets of primers and probes to quantify the eDNA of H. neglectus, O. latipes, and M. anguillicaudatus (Table 1). PCR was initiated at 98 °C for 2 min, followed by 50 cycles of 98 °C for 10 s and 60 °C for 1 min. The amplification curves obtained from individual reaction chambers of the microfluidic chip were analysed using Fluidigm Digital PCR analysis software (Fluidigm Corporation) to obtain abundance of DNA molecules.
Statistical analysis
We employed Gaussian Type II regression models with the standardised major axis method to determine the relationship between the log10 eDNA abundances obtained from qSeq and dPCR analyses with the “sma” function of the “smatr” ver. 3.4.8 package in R ver. 3.6.024. Zero values were disregarded for the modelling. We employed the Gaussian Type II model because our preliminary evaluation showed higher R2 values for Type II regression models with a Gaussian distribution than for those with a logarithmic distribution in all cases. We compared the differences in the coefficient values by overlapping the 95% confidence interval (CI) ranges. More

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Observed forest disturbances
We focused on the vulnerability of European forests to three major natural disturbances: forest fires, windthrows and insect outbreaks (bark beetles, defoliators and sucking insects). In order to identify/calibrate/validate vulnerability models (details on model development in the following sections) we used a large number of records of forest disturbances collected over the 2000–2017 period (Supplementary Fig. 1, step1). Fires were retrieved from the European Forest Fire Information System (EFFIS, https://effis.jrc.ec.europa.eu/) and count 15,818 records. Windthrows were acquired from the European Forest Windthrow dataset62 (FORWIND, https://doi.org/10.6084/m9.figshare.9555008) with 89,743 records. Insect outbreaks were retrieved from the National Insect and Disease Survey (IDS, http://foresthealth.fs.usda.gov) database of the United States Department of Agriculture (USDA) which includes 50,777 records. Each disturbance record is represented by a vector feature describing the spatial delineation of the damaged forest patch obtained by visual photointerpretation of aerial and satellite imagery or field surveys.
Even if the study focuses on Europe, for insect diseases we used the IDS-USDA database due to the lack of an analogous monitoring system and related dataset for Europe. Therefore, the models of vulnerability to insect outbreaks were identified/calibrated/validated on US data and then applied in predictive mode to Europe (see following sections for details). To assure the transferability of such models, we developed models for functional groups instead of working on species-specific models. For this purpose, we classified records based on functional groups of the pest (bark beetles, defoliators and sucking insects) and on the PFT of the host tree species. Records were considered if the host plant belonged to the following PFTs: broadleaved deciduous, broadleaved evergreen, needle leaf deciduous and needle leaf evergreen.
Reconstruction of annual biomass time series
In order to evaluate the biomass loss expected given a disturbance event occurs, multi-temporal information of biomass is required. However, there is still no single technology for direct and continuous monitoring of such variable in time. In order to reconstruct the temporal variations in biomass over the 2000–2017 period we integrated a static 100-m above ground biomass map acquired for the year 2010 from multiple Earth Observation systems63 with forest cover changes derived from the Global Forest Change (GFC) maps recorded at 30-m spatial resolution from Landsat imagery21. The GFC maps include three major layers: “2000 Tree Cover”, “Forest Cover Loss” and “Forest Cover Gain”. “2000 Tree Cover” (TC2000) is a global map of tree canopy cover (expressed in percentage) for the year 2000. “Forest Cover Loss” is defined as the complete removal of tree-cover canopy at the Landsat pixel scale (natural or human-driven) and is reported annually. “Forest Cover Gain” reflects a non-forest to forest change and refers to the period 2000–2012 as unique feature without reporting the timing of the gain.
The data integration approach built a on the assumption that changes in biomass are fully conditioned by the changes in tree cover. First, we quantified the percentage of tree cover in 2010 (TC2010) by masking out all pixels where forest loss occurred over the 2000–2010 period from the TC2000 map.
Then, in order to characterize to what extent an increase or decrease in tree cover may affect biomass, we quantified the density of biomass per percentage of tree cover lost (ρloss) and gained (ρgain) as follows:

$$rho _{mathrm{{loss}}} = frac{{B_{2010}}}{{{mathrm{{TC}}}_{2010,{mathrm{{loss}}}}}},$$
(1)

$$rho _{mathrm{{gain}}} = frac{{B_{2010}}}{{{mathrm{{TC}}}_{2010,{mathrm{{gain}}}}}},$$
(2)

where (B_{2010}) is the static biomass map available for the year 2010 (ref. 63). ({mathrm{{TC}}}_{2010,{mathrm{{loss}}}}) is the ({mathrm{{TC}}}_{2010}) masked over the pixels where there has been a forest loss during the 2011–2017 period. This filtering provides a picture of forests that were intact in 2010 but removed since then. Similarly, ({mathrm{{TC}}}_{2010,{mathrm{{gain}}}}) is the ({mathrm{{TC}}}_{2010}) masked over the pixels where there has been a forest gain and identifies the reforested and afforested areas. Since the map of forest gain is a binary map referring to the year 2012, forest gain pixels lack any information on their tree cover as their value in 2000 is zero. We therefore associated to forest gain pixels the maximum of tree cover percentage computed in a moving window with a radius of 2.5 km. This value represents the maximum potential tree cover in the local environmental conditions and refers to the whole 2000–2012 period (({mathrm{{TC}}}_{2012,{mathrm{{gain}}}})). Then, we assumed that forest gain proceeds at a constant rate over time and that the associated tree cover thus grows linearly:

$$frac{{{mathrm{{TC}}}_{2010,{mathrm{{gain}}}}}}{{left( {2010 – 2000} right)}} = frac{{{mathrm{{TC}}}_{2012,{mathrm{{gain}}}}}}{{left( {2012 – 2000} right)}} to {mathrm{{TC}}}_{2010,{mathrm{{gain}}}} = 0.83 cdot {mathrm{{TC}}}_{2012,{mathrm{{gain}}}},$$
(3)

Both ({mathrm{{TC}}}_{2010,{mathrm{{loss}}}}) and ({mathrm{{TC}}}_{2010,{mathrm{{gain}}}}) were resampled to the (B_{2010}) spatial resolution (100 m). Supplementary Figure 13 shows the frequency distribution of (rho _{mathrm{{loss}}}) and (rho _{{mathrm{{gain}}}}) over a test area in Southern Finland. As expected, the density of biomass associated with forest losses is higher than that one associated to forest gain. Indeed, biomass of new forest plantations is generally lower than the biomass of an old one (e.g. a forest that is typically harvested).
The obtained maps of (rho _{{mathrm{{loss}}}}) and (rho _{{mathrm{{gain}}}}) in Eqs. (1) and (2) refer to sparse and isolated pixels where there have been forest gain or loss. To obtain continuous fields, such density values were spatialized by computing their median over a 0.1° grid. Annual maps of biomass were finally obtained at 100 m spatial resolution as follows:

$$B_t = B_{2010} + alpha cdot rho _{{mathrm{{loss}}}} cdot {mathrm{{TC}}}_{t,{mathrm{{loss}}}} – rho _{{mathrm{{gain}}}} cdot {mathrm{{TC}}}_{t,{mathrm{{gain}}}} cdot frac{{left( {2010 – t} right)}}{{10}},$$
(4)

where t is the year (over the 2000–2017 period) and α takes the value of +1 for t  5% were selected (Supplementary Fig. 1, step 3). In the case of windthrows, we noted that maximum wind speeds retrieved from 0.5° spatial resolution of reanalysis data may largely underestimate effective maximum winds. This was particularly evident for tornado events, given their limited spatial extents compared to the grid cell, and the storm event Klaus that occurred in 2009 and for which we noticed an underestimation of the effective wind speed of the 78% (retrieved ~12 ms−1 instead of observed maximum wind speed of 55 ms−1 (ref. 67)). Therefore, such events were excluded from our analysis.
Possible missing data in the environmental variables were corrected by the median value of the variable-specific distributions (Supplementary Fig. 1, step 4). Potential effects of spatial dependence structure in the observational datasets were reduced by resampling ({mathrm{{BL}}}_{{mathrm{{rel}}}}), F, C and L along the gradients of the three principal components (PC) derived from the initial set of predictors. To this aim, we used 20 bins of equal intervals for each PC dimension spanning the full range of values. The resampling procedure was stratified by splitting the records in training and testing sets. For each year between 2000 and 2017, we randomly extracted 60% of the records. The extracted subset (({mathrm{{BL}}}_{{mathrm{{rel}}}}), F, C and L) was then binned in the PC space using the average as aggregation metric weighted by the areal extents of each disturbance record. The remaining 40% of records were similarly processed and used as a separate validation set (Supplementary Fig. 1, steps 5–7). The cover fraction of each PFT was resampled using the same approach and renormalized within each bin. Only bins with at least three records were retained for model development.
The resampled training and testing sets were used to calibrate and validate an “approximate” RF model using the full set of variables (A) as predictors initially identified based on literature review (Supplementary Fig. 1, step 8 and Supplementary Table 1). With the RF algorithm importance scores for each environmental variable can be calculated31. These scores reflect how important each covariate is in determining the fitted values of relative biomass loss. The RF implemented here uses 500 regression trees, whose depth and number of predictors to sample at each node were identified using Bayesian optimization. To reduce potential redundancy effects across predictors and facilitate the interpretability of results, we implemented a feature selection procedure. Based on the “approximate” RF model the importance of each predictor was quantified. We then computed the Spearman correlation between each pair of predictors and when it exceeded 0.8, the predictor with the lower variable importance was excluded (Supplementary Fig. 1, step 9 and Supplementary Table 1). The remaining predictors (I) were then used for a second set of RF runs, in which we iteratively evaluated RF performance on a reduced set of predictors, excluding in each new run the less important variable computed on the new reduced set of features. The set of predictors which maximizes the R2 was finally selected (Q hereafter for short) (Supplementary Fig. 1, step 10 and Supplementary Table 1). The implemented iterative feature selection procedure identifies a reasonable compromise between computing cost and model performance. The general equation describing the vulnerability is as follows:

$${mathrm{{BL}}}_{{mathrm{{rel}}}} = vleft( {{Q}} right),$$
(6)

where v is the vulnerability model implemented in the RF regression algorithm, and describes the relative biomass losses as a function of a selected Q set of environmental variables.
Such automatic feature selection process was complemented with visual interpretation of the PDPs68 based on the RF algorithm. PDP is used to visualize the relationship between explanatory covariates (environmental predictors) and ({mathrm{{BL}}}_{{mathrm{{rel}}}}), independent of other covariates (Supplementary Figs. 2–4). PDP results were analysed in combination with a detailed study of the literature and allowed us to understand and interpret the response functions to natural disturbances (see details in the main text and Fig. 2). Consistency of PDPs at the boundaries of the observational ranges was carefully checked to reduce possible artefacts generated when the models are used to extrapolate outside the range of training conditions.
Vulnerability models were further refined by retrieving v functions separately for each PFT. For PFT-specific vulnerability models, only resampled records in the PC space with a cover fraction >5% were retained and used for the model development (Supplementary Fig. 1, step 11). Model performances were ultimately evaluated on the testing set in terms of coefficient of determination (R2), root mean square error (RMSE), percent bias (PBIAS)69 and RE.
Regarding the insect-related disturbance, we initially implemented specific RF models for different insect groups (bark beetles, sucking insect and defoliators). However, due to the limited sample size of the first two groups, RF was not able to represent their effects on biomass losses reliably. We therefore opted to merge all three groups in a unique insect disturbance class (hereafter referred as insect outbreaks). We recognize that different ecological processes may characterize each insect group and therefore the use of a unique insect class may potentially mask some distinctive features. The resulting vulnerability models can therefore identify only drivers and patterns common to all groups (e.g., susceptibility to temperature anomalies70,71).
Interacting processes
The co-occurrence of multi-dimensional environmental factors resulting from the combination of interacting physical processes (compound events) may amplify or dampen ecosystem responses29. Tree-based models consider all variables together in the model and account for nonlinear feature interactions in the final model31,68. The inherent ability of RF models to detect interacting variables allows avoiding the prescription of specific relations between variables based on “a priori” knowledge—as for instance required in parametric regression frameworks—by letting the model learn automatically these relations from data.
In order to detect feature interactions and assess their strength in the developed RF-based vulnerability models we computed the Friedman’s H-statistic50. Here, we derived the H-statistic to assess second-order interactions by quantifying how much of the variation of the prediction depends on two-way interactions. To speed up the computation, we sampled 50 equally spaced data points over the environmental gradients.
We complemented this analysis by estimating the amplification or dampening effect (({Delta}{{P}})) associated to each feature interaction. To this aim, we quantified the difference in the peak values between the response function which incorporates interacting processes (two-way partial dependences) and those ones decomposed without interactions (one-dimensional partial dependences) and expressed in terms of relative variations.
The H and ({Delta}{{P}}) metrics were computed for each pair of features, and averaged for different combinations of predictor categories (forest, climate, landscape).
Spatial and temporal patterns of vulnerability and its key drivers
The RF models were used to evaluate the vulnerability of forests annually between 1979 and 2018 for each grid cell (0.25°) of the spatial domain covering the geographic Europe (including Turkey and European Russia). To this aim, vulnerability models were used in predictive mode using as input spatial maps of predictors, preliminary resampled to the common resolution, and with results expressed in terms of potential relative biomass loss (({mathrm{{PBL}}}_{{mathrm{{rel}}}})). Estimates of ({mathrm{{PBL}}}_{{mathrm{{rel}}}}) are obtained as the average from all trees in the RF ensemble. The ongoing changes in climate features were also accounted for in our framework. Climate predictors were kept dynamic for backward RF runs, while the remaining forest and landscape features were fixed to their current values averaged over the 2009–2018 period. Doing so, we implicitly assume that the sampling of response variables and predictors is representative for the whole temporal period. However, over longer time periods (from decades to century) additional ecosystem processes may play a role, such as adaptation phenomena driven by species change and shifting biomes, which could also affect vulnerability trends. The lack of multi-temporal monitoring of most of the forest and landscape predictors hampered the integration of their dynamics in the backward RF runs.
Results of PFT-specific vulnerability models were averaged at grid-cell level with weighting based on the cover fractions of PFTs (Supplementary Fig. 1, steps 12–13). This resulted in annual maps of vulnerability to each natural disturbance. Spatial and temporal variations in vulnerability were both expressed in relative and absolute terms. Absolute biomass losses were retrieved by multiplying estimates of potential relative biomass loss by the available biomass. Therefore, vulnerability values in a given grid cell reflect the biomass (relative or absolute) that would be affected if exposed to a disturbance under its specific local and temporal environmental conditions.
Grid-cell uncertainty of predicted vulnerability values were quantified in terms of standard error (SE) derived by dividing standard deviations of the computed responses over the ensemble of the grown trees of the model by the square root of the ensemble size (Supplementary Fig. 7).
We then calculated the “current” vulnerability as the average vulnerability over the 2009–2018 period. To factor out the local dependence of the current vulnerability on each predictor we retrieved the Individual Conditional Expectation72 (ICE) for each grid cell. ICE plots show the relationship between the predicted target variable (({mathrm{{PBL}}}_{{mathrm{{rel}}}})) and one predictor variable for individual cases of the predictor dataset. In our application, an individual case is a specific combination of F, C and L data for a given grid cell. To summarize and map the ICE of each grid cell in a single number, we fitted by linear regression the partial dependence of ({mathrm{{PBL}}}_{{mathrm{{rel}}}}) versus the corresponding predictor variable and mapped the slope of this regression, hereafter referred as “local sensitivity” (Supplementary Figs. 5–7), similarly to the approach presented in ref. 30. The marginal contribution ((Z_{mathrm{{marg}}})) of each environmental category of predictors (F, C and L, hereafter referred as X for short) on the current vulnerability was derived as follows:

$$Z_{{mathrm{{marg}}},X} = 100 times frac{{mathop {sum }nolimits_{i in X} left| {s_i} right|}}{{mathop {sum }nolimits_{j in Q} left| {s_j} right|}},$$
(7)

where s represents the slope of ICE, i runs over all predictors of X, whereas j runs over all available predictors Q. Therefore (Z_{mathrm{{marg}},X}) values range between 0 (no dependence of current vulnerability on X predictors) and 100% (full dependence of current vulnerability on X predictors).
Long-term linear trends in vulnerability ((delta {mathrm{{PBL}}}_{{mathrm{{rel}}}})) were quantified over the 1979–2018 period for each grid cell and their significance evaluated by the two-sided Mann–Kendall test. In order to isolate the key determinants of the emerging trends in vulnerability, a set of factorial simulations was performed. To this aim, we estimated the vulnerability due to the temporal variations in a given k climate predictor (({mathrm{{PBL}}}_{{mathrm{{rel}}}}^k)), by applying the RF models to a data array in which the k climate variable is dynamic while all the remaining features are kept fixed to their “current” value (average value over 2009–2018). The resulting trends in vulnerability associated to the k factor (({mathrm{{PBL}}}_{{mathrm{{rel}}}}^k)) are then calculated by linear regression and subject to the Mann–Kendall test.
Spatial and temporal patterns were visualized at grid-point scale and averaged over geographic macro-regions (Supplementary Fig. 14 and Supplementary Tables 2 and 3). Zonal statistics were obtained by averaging grid-cell results weighted by their forest areal extent. Forests with cover fraction lower than 0.1 were excluded from the analyses. Uncertainty in spatial averages were based on the 95% bootstrap confidence interval computed with 100 bootstrap samples.
In order to derive statistics minimally affected by potential extrapolation errors of the RF models, we replicated the aforementioned analyses by excluding areas outside the observational ranges of climatological temperature and precipitation (Supplementary Fig. 8).
Combining forest vulnerability to multiple natural disturbances
To quantify the total vulnerability to multiple disturbances we defined the OVI, similarly to the multi-hazard index developed in ref. 73. We assumed that the considered disturbances are independent and mutually non-exclusive and the potential biomass loss of single disturbances is spread homogeneously within each grid cell. From the inclusion-exclusion principle of combinatorics the potential biomass loss associated to the OVI can be expressed for a given year as follows:

$${mathrm{{PBL}}}_{{mathrm{{rel}}}}left( {{mathrm{{OVI}}}} right) = mathop {bigcup}nolimits_{p = 1}^D {{mathrm{{PBL}}}_{{mathrm{{rel}}},p}} = mathop {sum }limits_{q = 1}^D left( {left( { – 1} right)^{q – 1} cdot mathop {sum }limits_{{G subset left{ {1, ldots ,D} right}} atop {left| G right| = q} } {mathrm{{PBL}}}_{{mathrm{{rel}}},G}} right),$$
(8)

where p refers to the disturbance-specific ({mathrm{{PBL}}}_{{mathrm{{rel}}}}), D is the number of disturbances considered, the last sum runs over all subsets G of the indices {1, …, D} containing exactly q elements, and

$${mathrm{{PBL}}}_{{mathrm{{rel}}},G}: = mathop {bigcap}nolimits_{p in I} {{mathrm{{PBL}}}_{{mathrm{{rel}}},p}} ,$$
(9)

expresses the intersection of all those ({mathrm{{PBL}}}_{{mathrm{{rel}}},p}) with index in G. Maps of current overall vulnerability and trends were ultimately analysed following the approach adopted for single disturbances.
This approach does not account for the potential reduction in exposed biomass following the occurrence of a given disturbance. Furthermore, possible amplification/dampening effects due to interacting disturbances could also occur3,74. A strong interaction effect has been documented for instance between windthrows and bark beetle disturbances. Uprooted trees are virtually defenseless breeding material supporting the build-up of beetle populations and the consequent increase in vulnerability to insect outbreaks3,59. Insect outbreaks, in turn, may potentially affect the severity of subsequent forest fires by altering the abundance of available fuel60. The magnitude of these effects varies with insect type and outbreak timing. Despite the relevance of these interactions, the lack of reference observational data of compound events hampered the integration of their effects in our modelling framework. Therefore, estimates of OVI can only partially capture the overall vulnerability resulting from multiple disturbances and should be viewed in light of these limitations.
Spatial maps of current overall vulnerability and trends in OVI were then normalized separately based on the min–max method and combined by simple multiplication into a single index, hereafter referred as space-time integrated OVI. High values of space-time integrated OVI depict forest areas that are currently susceptible to multiple disturbances and their vulnerability have experienced a substantial increase over the 1979–2018 period. The space-time integrated OVI is used to identify currently fragile ecosystems that might in the future become even more susceptible to natural disturbances.
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article. More

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