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A. Strain sampling and isolationBradyrhizobium is a commonly occurring genus in soil [21]. Closely related Bradyrhizobium diazoefficiens (previously Bradyrhizobium japonicum) strains were isolated from soil, as previously described [20, 22]. In brief, Bradyrhizobium isolates that formed symbiotic associations with a foundational legume species in the censused region, Acacia acuminata, were isolated from soil sampled along a large region spanning ~300,000 km2 in South West Australia, a globally significant biodiversity hotspot [23]. In total 60 soil samples were collected from twenty sites (3 soil samples per site; Supplementary Fig. S1) and 380 isolates were sequenced (19 isolates per site, 5 or 6 isolates per soil sample, each isolate re-plated from a single colony at least 2 times). Host A. acuminata legume plants were inoculated with field soil in controlled chamber conditions and isolates were cultured on Mannitol Yeast agar plates from root nodules (see [20, 22] for details). A total of 374 strains were included in this study after removing 5 contaminated samples and one sample that was a different Bradyrhizobium species; non- Bradyrhizobium diazoefficiens sample removal was determined from 16S rRNA sequences extracted from draft genome assemblies (Method C) using RNAmmer [24].B. Environmental variation among sampled sitesIn this study, I focus on environmental factors (temperature, rainfall, soil pH and salinity) previously identified to impact either rhizobia growth performance, functional fitness or persistence in soil [25,26,27,28] and where a directionality of rhizobial stress response could be attributed with respect to environmental variation present in the sampled region (i.e. stress occurs at high temperatures, low rainfall, high acidity and high salinity). Each environmental factor was standardised to a mean of 0 and a standard deviation of 1, and pH and rainfall scales were reversed to standardise stress responses directions so that low stress is at low values and high stress is at high values for all factors (Supplementary Fig. S2). Additionally, salinity was transformed using a log transformation (log(x + 0.01) to account for some zeroes) prior to standardisation.C. Isolate sequencing and pangenome annotationIllumina short reads (150 bp paired-end) were obtained and draft genome assemblies were generated using Unicycler from a previous study [29]. Resulting assemblies were of good assembly quality (99.2% of all strains had >95.0% genome completeness score according to BUSCO [30]; Table S1; assembled using reads that contained nominal 0.016 ± 0.00524% non-prokaryotic DNA content across all 374 isolates, according to Kraken classification [31]). Protein coding regions (CDS regions) were identified using Prokka [32] and assembled into a draft pangenome using ROARY [33], which produced a matrix of counts of orthologous gene clusters (i.e. here cluster refers to a set of protein-coding sequences containing all orthologous variants from all the different strains, grouped together and designated as a single putative gene). Gene clusters that occurred in 99% of strains were designated as ‘core genes’ and used to calculate the ‘efficiency of selection’ [34, 35] (measured as dN/dS, Method G.2) and population divergence measured as Fixation Index ‘Fst’, Method H) across each environmental stress factor. The identified gene clusters were then annotated using eggNOG-mapper V2 [36] and the strain by gene cluster matrix was reaggregated using the Seed ortholog ID returned by eggNOG-mapper as the protein identity. Out of the total 2,744,533 CDS regions identified in the full sample of 374 strains, eggNOG-mapper was able to assign 2,612,345 of them to 91,230 unique Seed orthologs. These 91,230 protein coding genes constituted the final protein dataset for subsequent analyses.D. Calculation and statistical analysis of gene richness and pangenome diversity along the stress gradientGene richness was calculated as the total number of unique seed orthologues for each strain (i.e. genome). Any singleton genes that occurred in only a single strain, as well as ‘core’ genes that occurred in every strain (for symmetry, and because these are equally uninformative with respect to variation between strains) were removed, leaving 74,089 genes in this analysis. Gene richness (being count data) was modelled on a negative binomial distribution (glmer.nb function) as a function of each of the four environmental stressors as predictors using the lme4 package in R [37], also accounting for hierarchical structure in the data by including site and soil sample as random effects.To rule out potentially spurious effects of assembly quality (i.e. missed gene annotations due to incomplete draft genomes) on key findings, I confirmed no significant association between gene richness and genome completeness (r = 0.042, p = 0.4224, Fig. S3).Finally, pangenome diversity was calculated as the total number of unique genes that occurred in each soil sample (since multiple strains were isolated from a single soil sample). Pangenome diversity was modelled the same as gene richness, except here soil sample was not included as a random effect.E. Calculation of network and duplication traits for each geneI used the seed orthologue identifier from eggNOG-mapper annotations to query matching genes within StringDB ([38]; https://string-db.org/), which collects information on protein-protein interactions. Out of 91,230 query seed orthologues, 73,126 (~80%) returned a match in STRING. For matching seed orthologue hits, a network was created by connecting any proteins that were annotated as having pairwise interactions in the STRING database using the igraph package in R [39]. Two vertex-based network metrics were calculated for each gene: betweenness centrality, which measures a genes tendency to connect other genes in the gene network, and mean cosine similarity, which is a measure of how much a gene’s links to other genes are similar to other genes.Betweenness centrality was calculated using igraph (functional betweenness). For mean cosine similarity, a pairwise cosine similarity was first calculated between all genes. To do this, the igraph network object was converted into a (naturally sparse yet large) adjacency matrix and the cosSparse function in qlcMatrix in R [40] was used to calculate cosine similarity between all pairs of genes. To obtain an overall cosine similarity trait value for each gene, the average pairwise cosine similarity to all other genes in the network was calculated.Finally, gene duplication level was calculated for each gene as one additional measure of ‘redundancy’, by calculating the average number of gene duplicates found within the same strain. Duplicates were identified as CDS regions with the same Seed orthologue ID.F. Gene distribution modelsTo determine how gene traits predict accessory genome distributions patterns along the stress gradients, I first calculated a model-based metric (hereafter and more specifically a standardised coefficient, ‘z-score’) of the relative tendency of each gene to be found in different soil samples across the four stress gradients (heat, salinity, acidity, and aridity). This was achieved by modelling each gene’s presence or absence in a strain as a function of the four stress gradients, with site and soil sample as a random effect, using a binomial model in lme4 (the structure of the model being the same as the gene richness model, only the response is different). To reduce computational overhead, these models were only run for the set of genes that were used in the gene richness analysis (e.g. after removing singletons and core genes), and which had matching network traits calculated (e.g. they occurred in the STRING database; n = 64,867 genes). Distribution models were run in tandem for each gene using the manyany function in the R package mvabund [41]. Standardised coefficients, or z-scores (coefficient/standard error) indicating the change in the probability of occurrence for each gene across each of the stress gradients were extracted. More negative coefficients correspond to genes that are more likely to be absent in high stress (and vice versa for positive coefficients).To determine how network and duplication traits influence the distribution of genes across the stress gradient, I performed a subsequent linear regression model where the gene’s z-scores was the response and gene traits as predictors. The environmental stress type (i.e. acidity, aridity, heat and salinity) was included as a categorical predictor, and the interaction between stress category and gene function traits were used to infer the influence of gene function traits on gene distributions in a given stress type (see Supplementary Methods S1 for z-score transformation).G. Quantifying molecular signals of natural selection on accessory and core genesTo examine molecular signatures of selection in accessory and core genes, I calculated dN/dS for a subsample of the total pool (n=74,089 genes), which estimates the efficiency of selection [34, 35]. Two major questions relevant to dN/dS that are addressed here require a different gene subsampling approach:(1) Do variable environmental stress responses lead to different dN/dS patterns among accessory genes?Here, I subsampled accessory genes (total accessory gene pool across 374 strains, 74,089) to generate and compare dN/dS among 3 categorical groups, each representing a different level of stress response based on their z-scores (accessory genes that either strongly increase, decrease or have no change in occurrence as stress increases; n = 1000 genes/category; see Supplementary Methods S2 for subsample stratification details).For each gene, sequences were aligned using codon-aware alignment tool, MACSE v2 [42]. dN/dS was estimated by codon within each gene using Genomegamap’s Bayesian model-based approach [43], which is a phylogeny-free method optimised for within bacterial species dN/dS calculation (see Supplementary Methods S3 for dN/dS calculation/summarisation; S9 for xml configuration). The proportion of codons with dN/dS that were credibly less than 1 (purifying selection) and those credibly greater than 1 (positive selection) were analysed, with respect to the genes’ occurrence response to stress. Specifically, I modelled the proportion of codons with dN/dS  1 was overall too low to analyse in this way, so the binary outcome (a gene with any codons with dN/dS  > 1 or not) was modelled using a binomial response model with the response categories as predictors (see Supplementary Methods S4 for details of both models).(2) Does dN/dS among microbial populations vary across environmental stress?Here, I compared the average change in dN/dS in core genes present across all environments at the population level (i.e. all isolates from one soil sample), which is used here to measure the change in the efficiency of selection across each stress gradient. Core genes were used since they occur in all soil samples, allowing a consistent set and sample size of genes to be used in the population-level dN/dS calculation. This contrasts with the previous section, which quantifies gene-level dN/dS on extant accessory genes that intrinsically have variable presence or absence across soil samples. For computational feasibility, 500 core genes were subsampled (total core 1015 genes) and, for each gene, individual strain variants were collated into a single fasta file based on soil sample membership, such that dN/dS could be calculated separately for each gene within each soil sample (i.e. 60 soil samples × 500 genes = 30,000 fasta files). Each fasta file was then aligned in MACSE and dN/dS calculated using the same methodology for accessory genes (Supplementary Method S3). This enabled the average dN/dS in a sample to be associated with soil-sample level environmental stress variables. Specifically, I modelled the mean proportion of codons with dN/dS  1 due to overall rarity of positive selection (average proportion of genes where at least 1 codon with dN/dS  > 1 was ~0.006). This low level of positive selection is expected for core genes which tend to be under strong selective constraint.H. Calculation and analysis of Fixation index (Fst) along stress gradientsUsing the core genome alignment (all SNPs among 1015 core genes) generated previously with ROARY, I computed pairwise environmentally-stratified Fst. Each soil sample (n = 60) was first placed into one of 5 bins based on their distances in total environmental stress space (using all four stress gradients), with the overall goal of generating roughly evenly sized bins such that the environmental similarity of stress was greater within bins than between (see Supplementary Methods S6 and Fig. S4 for clustering algorithm details). Next, fasta alignments were converted to binary SNPs using the adegenet package. Pairwise Fst between all strains (originating from a particular soil sample) within a single bin was calculated using StAMPP in R [44]. For each strain pair, the average of the two stress gradient values was assigned.The relationship between pairwise Fst and the average stress value was evaluated using a linear regression model with each of the four stress values as predictors. Since the analysis uses pairwise data (thus violating standard independence assumptions), the significance of the relationship was determined using a permutation test (see Supplementary Methods S7 for details).I. Chromosomal structure analysis of gene loss patternsTo gain insight into structural variation and test for regional hotspots in gene loss along the chromosome, I mapped each gene’s stress response (i.e. probability of loss or gain indicated by each genes z-score) onto a completed Bradyrhizobium genome (strain ‘36_1’ from the same set of 374 strains (Genbank CP067102.1; [45]). Putative CDS positions on the complete genome were determined using Prokka and annotated with SEED orthologue ID’s using eggNOG-mapper. Matching accessory genes derived from the full set of 374 incomplete draft genomes (n = 74,089) were mapped to positions on the complete genome (6274 matches). The magnitude of gene loss or gain (as measured by their standardised z-scores for each environment from the gene distribution models; see Method F) was then modelled across the genome using a one-dimensional spatial smoothing model. This model was implemented in R INLA [46] (www.r-inla.org), and models a response in a one-dimensional space using a Matern covariance-based random effect. The method uses an integrated nested Laplace approximation to a Bayesian posterior distribution, with a cyclical coordinate system to accommodate circular genomes. The model accounts for spatial non-independence among sites and produces a continuous posterior distribution of average z-score predictions along the entire genome, which was then used to visualise potential ‘hotspots’ of gene loss or gain. The modelling procedure was repeated, instead with gene network traits, such that model predictions of similarity and betweenness could be visualised on the reference chromosome. More

State Key Laboratory of Tree Genetics and Breeding, Chinese Academy of Forestry, Beijing, ChinaZhilin Yuan, Huanshen Wei & Long PengResearch Institute of Subtropical Forestry, Chinese Academy of Forestry, Hangzhou, ChinaZhilin Yuan, Xinyu Wang, Huanshen Wei & Long PengFungal Genomics Laboratory (FungiG), College of Resources and Environmental Sciences, Nanjing Agricultural University, Nanjing, ChinaIrina S. Druzhinina & Feng CaiDepartment of Food Science, University of Massachusetts, Amherst, MA, USAJohn G. GibbonsState Key Laboratory of Ecological Pest Control for Fujian and Taiwan Crops, College of Plant Protection, Fujian Agriculture and Forestry University, Fuzhou, ChinaZhenhui ZhongDepartment of Molecular, Cell and Developmental Biology, University of California, Los Angeles, CA, USAZhenhui ZhongDepartment of Plant Biotechnology and Bioinformatics, Ghent University, Ghent, BelgiumYves Van de PeerVIB Center for Plant Systems Biology, Ghent, BelgiumYves Van de PeerCentre for Microbial Ecology and Genomics, Department of Biochemistry, Genetics and Microbiology, University of Pretoria, Hatfield, South AfricaYves Van de PeerAdaptive Symbiotic Technologies, University of Washington, Seattle, WA, USARussell J. RodriguezKey Laboratory of National Forestry and Grassland Administration for Orchid Conservation and Utilization at College of Landscape Architecture, Fujian Agriculture and Forestry University, Fuzhou, ChinaZhongjian LiuState Key Laboratory of Mycology, Institute of Microbiology, Chinese Academy of Sciences, Beijing, ChinaQi Wu & Guohui ShiKey Laboratory of Plant Resources Conservation and Sustainable Utilization, South China Botanical Garden, Chinese Academy of Sciences, Guangzhou, ChinaJieyu WangBeijing Advanced Innovation Center for Tree Breeding by Molecular Design, Beijing Forestry University, Beijing, ChinaFrancis M. MartinUniversité de Lorraine, INRAE, UMR Interactions Arbres/Micro-Organismes, Centre INRAE Grand Est Nancy, Champenoux, FranceFrancis M. Martin More

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Detecting alternative attractors in ecosystem dynamicsWe use empirical dynamical modeling, a set of equation-free tools for analyzing non-linear time series (for a review and assumptions see25,26, respectively), to test if the temporal dynamics of alternative dynamical regimes are qualitatively different. Empirical dynamic modeling builds fundamentally on Takens embedding theorem, which shows that attractors of multi-dimensional dynamical systems can be reconstructed using higher order lags of its embedded time series27. However, if a dynamical system has gone through a bifurcation, or switched to an alternative basin of attraction, attractors are qualitative dissimilar in the two regimes. Theoretically, this infers that it should be possible to reconstruct the attractor of one regime using information from the same regime, but not from the other regime. In practice, this implies that if a model (attractor reconstruction) based on one dynamical regime is used to predict the dynamics of variables from the same dynamical regime predictions should be accurate (i.e. low prediction errors), whereas if an attractor reconstruction based on one dynamical regime is used to predict the dynamics of variables of another attractor predictions should be less accurate (i.e. high prediction errors). We make use of this idea by specifically testing if prediction errors of across and within regime predictions are different. As explained below this idea can be used for both univariate and multivariate time series data.Univariate approachUnivariate attractor reconstructions can be found using the simplex algorithm28,29. First, for a given dynamical regime, a time series can be split into a library of vectors, and each vector is described by$${underline{y}}_{A}(t)= < {Y}_{A}(t),{Y}_{A}(t-1),{Y}_{A}(t-2),ldots ,{Y}_{A}(t-(E-1)) > ,$$
(1)
where ({Y}_{A}(t)) is an observation of variable Y at time t in dynamical regime A and E is the reconstructed attractors embedding dimension. Using the simplex projection algorithm, a one-step ahead forecast is produced as follows:$${hat{Y}}_{A}(t+1)|{{{{{{boldsymbol{M}}}}}}}_{B}=mathop{sum}limits_{m=1ldots E+1}{w}_{m}{Y}_{B}({t}_{m}+1),$$
(2)
where tm is a time index of an observation in dynamical regime B, E is the embedding dimension of regime B, and wm is an exponential weighting described by:$${w}_{m}={u}_{m}/mathop{sum}limits_{n=1,ldots ,E+1}{u}_{n},$$
(3)
where n and m belongs to the set of the E+1 nearest neighbors of vector ({underline{y}}_{A}(t)) in the set of vectors ({{underline{y}}_{B}({t}_{m})}), ({u}_{m}=exp {-d[{underline{y}}_{A}(t),{underline{y}}_{B}({t}_{m})]/d[{underline{y}}_{A}(t),{underline{y}}_{B}({t}_{1})]}), and (d[{underline{y}}_{A}(t),{underline{y}}_{B}({t}_{1})],)is the Euclidean distance between the prediction vector ({underline{y}}_{A}(t)) and its nearest neighbor ({underline{y}}_{B}({t}_{1})) in the set ({{underline{y}}_{B}({t}_{m})}).The only parameter that is estimated using the simplex algorithm is the embedding dimension E. This parameter is estimated by optimizing the correlation between observations (({Y}_{A}(t+1))) and predictions (({hat{Y}}_{A}(t+1)|{{{{{{boldsymbol{M}}}}}}}_{A})) using a leave-one-out cross validation approach (See Supplementary Discussion). The embedding dimension E and its corresponding set of E-dimensional vectors (Eq. 1) constitutes the reconstructed attractor, MA, of a given dynamical regime A. This reconstructed attractor (MA) is then used to predict data for both the same dynamical regime (({hat{Y}}_{A}(t+1)|{{{{{{boldsymbol{M}}}}}}}_{A})), and the contrasting dynamical regime ({hat{Y}}_{B}(t+1)|{{{{{{boldsymbol{M}}}}}}}_{A}). Likewise, the reconstructed attractor MB can be used to predict time series dynamics from both dynamical regimes; that is, ({hat{Y}}_{A}(t+1)|{{{{{{boldsymbol{M}}}}}}}_{B}) and ({hat{Y}}_{B}(t+1)|{{{{{{boldsymbol{M}}}}}}}_{B}), respectively.Multivariate approachA multivariate time series describes a number of simultaneously evolving variables. For example, a bivariate time series can be described by variables X and Y. For such time series, Sugihara et al.30 developed an approach for testing if two variables (time series) are dynamically coupled. Their methodology builds on the fact that a reconstructed attractor should map 1:1 to the original attractor on which the reconstruction is based. This infers that two attractor reconstructions (based on two different variables) should also map 1:1 to each other30. Practically, this means that if two variables are dynamically coupled one-time series should be predictable based on an attractor reconstruction of another variable. However, if a dynamical system has gone through a bifurcation, or potentially switched to an alternative basin of attraction, a new set of rules will govern the dynamics of the system. Hence, a new attractor should have emerged. Now, since this new attractor is most likely governed by a new set of rules it should be difficult to predict the dynamics of this new alternative attractor based on information from the former attractor. Thus, if one variable in one dynamical regime is used to predict another variable in another dynamical regime, predictions should be biased. Yet, if one variable from one dynamical regime is used to predict another variable from the same regime predictions should be more accurate.The simplex algorithm can be used to make predictions of a variable Y using a time series of another variable X30. Predictions are produced as follows:$${hat{Y}}_{{{{{{boldsymbol{A}}}}}}}(t)|{{{{{{boldsymbol{M}}}}}}}_{B}=mathop{sum}limits_{m=1ldots E+1}{w}_{m}{Y}_{B}({t}_{m}),$$
(4)
where tm is the time series index of a vector of variable X of dynamical regime B, wm is an exponential weighting based on variable X:$${w}_{m}={u}_{m}/mathop{sum}limits_{n=1,ldots ,E+1}{u}_{n},$$
(5)
where n and m belongs to the set of the E+1 nearest neighbors of ({underline{x}}_{A}(t)) in ({{underline{x}}_{B}({t}_{m})}), ({u}_{m}=exp {-d[{underline{x}}_{A}(t),{underline{x}}_{B}({t}_{m})]/d[{underline{x}}_{A}(t),{underline{x}}_{B}({t}_{1})]}), and (d[{underline{x}}_{A}(t),{underline{x}}_{B}({t}_{1})],)is the Euclidean distance between the prediction vector(,{underline{x}}_{A}(t)) and its nearest neighbor ({underline{x}}_{B}({t}_{1})) in dynamical regime (B).The reconstructed attractors, MA and MB, for each variable and regime are found using the univariate simplex algorithm described above28,29,30. Similar to the univariate case, the reconstructed attractor (MA) is used to predict data from the same dynamical regime (({hat{Y}}_{{{{{{boldsymbol{A}}}}}}}(t)|{{{{{{boldsymbol{M}}}}}}}_{A})), and to predict time series of a contrasting dynamical regime (({hat{Y}}_{{{{{{boldsymbol{A}}}}}}}(t)|{{{{{{boldsymbol{M}}}}}}}_{B})). Yet, it is important to stress that MA here reflects an attractor reconstruction based on a variable that is not being predicted (that is, variable X is used to predict variable Y). This prediction approach thus infers that predictions are made on data that was not used to fit the model (X predicts Y and vice versa). Thus, neither across nor within regime predictions are made on data used to fit a model.Test statisticWe used mean absolute prediction errors to test for difference between across and within regime predictions. Alternative metrics, such as mean sum of square errors, can also be used. However, since our approach gives skewed prediction errors we used mean absolute prediction errors to reduce the impact of extreme values. Further, since the absolute prediction errors are non-normally distributed we used a permutation test. The null hypothesis that is tested reads:$$H0:{{{{{rm{MAP{E}}}}}}}_{A} < {{{{{rm{MAP{E}}}}}}}_{w},$$ (6) where MAPEA is the mean absolute prediction error for across regime predictions (that is, ({{{{{rm{MAP{E}}}}}}}_{A}=frac{1}{n}mathop{sum}limits_{t=1:n}{{{{{rm{abs}}}}}}({hat{Y}}_{{{{{{{boldsymbol{M}}}}}}}_{{{{{{boldsymbol{A}}}}}}}}(t)|{{{{{{boldsymbol{M}}}}}}}_{B}-{Y}_{{{{{{{boldsymbol{M}}}}}}}_{{{{{{boldsymbol{A}}}}}}}}(t))), and ({{{{{rm{MAP{E}}}}}}}_{w}) is the mean absolute prediction error for within regime predictions (that is, ({{{{{rm{MAP{E}}}}}}}_{w}=frac{1}{n}mathop{sum}limits_{t=1:n}{{{{{rm{abs}}}}}}({hat{Y}}_{{{{{{{boldsymbol{M}}}}}}}_{{{{{{boldsymbol{A}}}}}}}}(t)|{{{{{{boldsymbol{M}}}}}}}_{A}-{Y}_{{{{{{{boldsymbol{M}}}}}}}_{{{{{{boldsymbol{A}}}}}}}}(t))). A test is consider significant if observed difference in across and within regime mean prediction errors is larger than the 95th percentile of 1000 permuted data sets.Food-chain modelWe used a food-chain model parameterized as in McCann and Yodzis31 to simulate food-chain dynamics:$$frac{{{{{{rm{d}}}}}}R}{{{{{{rm{d}}}}}}t}=Rleft(1-frac{R}{K}right)-frac{{x}_{c}{y}_{c}CR}{R+{R}_{0}}$$ (7) $$frac{{{{{{rm{d}}}}}}C}{{{{{{rm{d}}}}}}t}={x}_{c}Cleft(-1+frac{{y}_{C}R}{R+{R}_{0}}right)-frac{{x}_{P}{y}_{P}PC}{C+{C}_{0}}$$$$,frac{{{{{{rm{d}}}}}}P}{{{{{{rm{d}}}}}}t}={x}_{P}Pleft(-1+frac{{y}_{P}C}{C+{C}_{0}}right),$$where R is the resource density, C consumer density, and P predator density. All parameters, except half-saturation constants R0 (here set to 0.16129) and C0 (here set to 0.5), and resource carrying capacity K, are derived from bioenergetics and body size allometry30 (xc = 0.4, yc = 2.009, yp = 2.876, R0, r = 1, xp = 0.08).This model can display a rich set of dynamics depending on parameter values31. Here we alter resource carrying capacity K in order to simulate the dynamics (using the deSolve package32 in R) of qualitatively different attractors (See Supplementary Fig. 1; K = 0.78, equilibrium; K = 0.85; two-point limit cycle; K = 0.92, four-point limit cycle; K = 0.997, chaotic dynamics). Every fifth time step of the simulated dynamics, corresponding to a sampling frequency of ≈10 samples per cycle for the 2-point limit cycle, was sampled. Observation noise was thereafter added to the deterministic dynamics produced by the model:$${N}_{l}(t)={N}_{l}^{prime}(t)+rho * e(t);e(t) sim N(0,{sigma }_{N^{prime_{l}}}),$$ (8) where (N_{l}^{prime}(t)) is the abundance of species l (P, C or R) simulated by the food-chain model at time point t, (rho) is the level of observation noise and ({sigma }_{N_{l}^{prime}}) is the standard deviation of the deterministic dynamics of species l produced by the food chain model.In order to investigate how time series length and observation noise affects the probability of detecting alternative attractors we derived probability landscapes. These were derived by testing the null-hypothesis (H0:(|{hat{{{{{{boldsymbol{Y}}}}}}}}_{{{{{{boldsymbol{i}}}}}}}|{{{{{{boldsymbol{M}}}}}}}_{{{{{{boldsymbol{i}}}}}}}-{{{{{{boldsymbol{Y}}}}}}}_{{{{{{boldsymbol{i}}}}}}}| > |{hat{{{{{{boldsymbol{Y}}}}}}}}_{{{{{{boldsymbol{i}}}}}}}|{{{{{{boldsymbol{M}}}}}}}_{{{{{{boldsymbol{j}}}}}}}-{{{{{{boldsymbol{Y}}}}}}}_{{{{{{boldsymbol{i}}}}}}}|); See Test statistic above) across 100 replicates for each combination of time series length and level of observation noise, (rho). Time-series length was varied from 10 to 100 in steps of 10, and observation noise, (rho), was varied from 0.01 to 0.3 in steps of 0.01, in total yielding 300 combinations of observation noise and time series length, for each combination of dynamical regimes i and j. Predator dynamics was used to predict consumer and resource dynamics using the multivariate approach described above (results for the cases where consumer or resource dynamics are used to predict the other species´ dynamics are presented in Supplementary Figs. 2, 3). All time series were standardized ((mu =0;sd=1)) prior testing for dynamical difference.Experimental data setThe experimental data set was given by Fussman et al.7. This data set contains 14 time series of a predator Brachionus calyciflorus and its prey Chlorella vulgaris derived from chemostat experiments. Time series for different dilution rates were produced by keeping the dilution rate fixed in different chemostats (Supplementary Figs. 3–11). Brachionus calyciflorus and Chlorella vulgaris time series were used to predict Chlorella vulgaris and Brachionus calyciflorus time series, respectively, using the multivariate approach described above. We tested for qualitative difference in the temporal dynamics across all time series, which were standardized ((mu =0;sd=1)) prior testing.Alternative stable state modelWe used a stochastic version of a well-known alternative stable state model4,33 to produce alternative stochastic dynamical regimes. The model is described by:$${{{{{rm{d}}}}}}x=left(xleft(1-frac{x}{{{{{{rm{K}}}}}}}right)+frac{c{x}^{2}}{1-{x}^{2}}right){{{{{rm{d}}}}}}t+sigma {{{{{rm{d}}}}}}w,$$
(9)
where K is the carrying capacity (here set to 11), c is a harvest rate, and σ (here set to 0.01) is the magnitude of noise which is described by a Wiener process (dw).The model was simulated for fixed harvest rates (c) assuming that the system state resides in either of its two basins of attraction. The initial value for the simulation was set to the equilibrium of the noise-free model skeleton for fixed harvest rates c, and σ is set low in order to avoid stochastic flips, so-called flickering, between alternative basins of attraction. Dynamics was integrated (Δt = 0.01) using the matlab-package SDE-Tools34.In order to investigate how time-series length and harvest rate, c, affects the probability of detecting alternative attractors in stochastic regimes we derived probability landscapes.These were derived by testing the null-hypothesis H0:(|{hat{{{{{{boldsymbol{Y}}}}}}}}_{{{{{{boldsymbol{i}}}}}}}|{{{{{{boldsymbol{M}}}}}}}_{{{{{{boldsymbol{i}}}}}}}-{{{{{{boldsymbol{Y}}}}}}}_{{{{{{boldsymbol{i}}}}}}}| > |{hat{{{{{{boldsymbol{Y}}}}}}}}_{{{{{{boldsymbol{i}}}}}}}|{{{{{{boldsymbol{M}}}}}}}_{{{{{{boldsymbol{j}}}}}}}-{{{{{{boldsymbol{Y}}}}}}}_{{{{{{boldsymbol{i}}}}}}}|) (permutation test p = 0.05) across 100 simulated data sets for each combination of time series length and harvest rate, c. Time-series length was varied between 50 and 150 in steps of 10, and c was varied between 1.83 and 2.73 in steps of 0.05, in total yielding 209 combinations of time series length and harvest rate. Each time series was standardized ((mu =0;sd=1)) prior testing for difference in temporal dynamics of contrasting regimes.Natural time-series dataIn a previous study on early warning signals of impending regime shifts, Gsell et al.18 used breakpoint analysis to identify two potential alternative dynamical regimes. We here test if these two-time series segments constitute alternative dynamical attractors. Prior analysis, we imputed a few missing observations (n = 24) using a kalman smoother35. The two time series segments, i.e. pre- and post-breakpoint time series, were standardized ((mu =0;sd=1)) prior testing for dynamical difference.Reporting summaryFurther information on research design is available in the Nature Research Reporting Summary linked to this article. More

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