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DatasetsImage acquisitionFor this study, we assembled three datasets: one for training of the RootDetector Convolutional Neural Network (Training-Set), one for a performance comparison between humans and RootDetector in segmenting roots in minirhizotron images (Comparison-Set), and one for the validation of the algorithm (Validation-Set). The Training-Set contained 129 images comprised of 17 randomly selected minirhizotron images sampled in a mesocosm experiment (see “Mesocosm sampling” Section), 47 randomly selected minirhizotron images sampled in a field study (see “Field sampling” Section) as well as the 65 minirhizotron images of soy roots published by Wang et al.15. The Comparison-Set contained 25 randomly selected minirhizotron images from the field-study which all were not part of the images included in the Training- and Validation-Sets. The Validation-Set contained 10 randomly selected minirhizotron images from the same field study, which had not been used in the Training-Set. All images were recorded with 2550 ✕ 2273 pixels at 300 dpi with a CI-600 In-Situ Root Imager (CID Bio-Science Inc., Camas, WA, USA) and stored as .tiff files to reduce compression loss. For all training and evaluation purposes we used raw, unprocessed output images from the CI-600.Mesocosm samplingThe mesocosm experiment was established in 2018 on the premises of the Institute for Botany and Landscape Ecology of the University of Greifswald (Fig. S1). It features 108 heavy duty plastic buckets of 100 l each, filled to two thirds of their height with moderately decomposed sedge fen peat. Each mesocosm contained one minirhizotron (inner diameter: 64 mm, outer diameter: 70 mm, length: 650 mm) installed at a 45°angle and capped in order to avoid penetration by light. The mesocosms were planted with varying compositions of plant species that typically occur in north-east German sedge fens (Carex rostrata, Carex acutiformis, Glyceria maxima, Equisetum fluviatile, Juncus inflexus, Mentha aquatica, Acorus calamus and Lycopus europaeus). The mesocosms were subjected to three different water table regimes: stable at soil surface level, stable at 20 cm below soil surface and fluctuating between the two levels every two weeks. The minirhizotrons were scanned weekly at two levels of soil depth (0–20 cm and 15–35 cm) between April 2019 and December 2021, resulting in roughly 9500 minirhizotron images of 216 × 196 mm. Manual quantification of root length would, based on own experience, take approximately three hours per image, resulting in approximately 28,500 h of manual processing for the complete dataset. Specimens planted were identified by author Dr. Blume-Werry, however no voucher specimen were deposited. All methods were carried out in accordance with relevant institutional, national, and international guidelines and legislation.Field samplingThe field study was established as part of the Wetscapes project in 201716. The study sites were located in Mecklenburg-Vorpommern, Germany, in three of the most common wetland types of the region: alder forest, percolation fen and coastal fen (Fig. S2). For each wetland type, a pair of drained versus rewetted study sites was established. A detailed description of the study sites and the experimental setup can be found in Jurasinski et al.16. At each site, 15 minirhizotrons (same diameter as above, length: 1500 mm) were installed at 45° angle along a central boardwalk. The minirhizotrons have been scanned biweekly since April 2018, then monthly since January 2019 at two to four levels of soil depth (0–20 cm, 20–40 cm, 40–60 cm and 60–80 cm), resulting in roughly 12,000 minirhizotron images of 216 × 196 cm, i.e. an estimated 36,000 h of manual processing for the complete dataset. Permission for the study was obtained from the all field owners. Figure 1Overview of the RootDetector system. The main component is a semantic segmentation network based on the U-Net architecture. The root length is estimated by skeletonizing the segmentation output and applying the formula introduced by Kimura et al.17. During training only, a weight map puts more emphasis on fine roots.Full size imageThe CNN RootDetectorImage annotationFor the generation of training data for the CNN, human analysts manually masked all root pixels in the 74 images of the Training-Set using GIMP 2.10.12. The resulting ground truth data are binary, black-and-white images in Portable Network Graphics (.png) format, where white pixels represent root structures and black pixels represent non-root objects and soil (Fig. 2b). All training data were checked and, if required, corrected by an expert (see “Selection of participants” for definition). The Validation-Set was created in the same way but exclusively by experts.Figure 2Example of segmentation and result of skeletonization. A 1000 by 1000 pixel input image (a), the manually annotated ground truth image (b), the RootDetector estimation image (c), the combined representation image (error map, d with green indicating true positives, red indicating false positive, blue indicating false negatives), the skeletonized RootDetector estimation image (e), and the skeletonized ground truth image (f).Full size imageArchitectureRootDetector’s core consists of a Deep Neural Network (DNN) based on the U-Net image segmentation architecture[27]nd is implemented in TensorFlow and Keras frameworks18. Although U-Net was originally developed for biomedical applications, it has since been successfully applied to other domains due to its generic design.RootDetector is built up of four down-sampling blocks, four up-sampling blocks and a final output block (Fig. 1). Every block contains two 3 × 3 convolutional layers, each followed by rectified linear units (ReLU). The last output layer instead utilizes Sigmoid activation. Starting from initial 64 feature channels, this number is doubled in every down-block and the resolution is halved via 2 × 2 max-pooling. Every up-block again doubles the resolution via bilinear interpolation and a 1 × 1 convolution which halves the number of channels. Importantly, after each up-sampling step, the feature map is concatenated with the corresponding feature map from the down-sampling path. This is crucial to preserve fine spatial details.Our modifications from the original architecture include BatchNormalization19 after each convolutional layer which significantly helps to speed up the training process and zero-padding instead of cropping as suggested by Ronneberger, Fischer, & Brox20 to preserve the original image size.In addition to the root segmentation network, we trained a second network to detect foreign objects, specifically the adhesive tape that is used as a light barrier on the aboveground part of the minirhizotrons. We used the same network architecture as above and trained in a supervised fashion with the binary cross-entropy loss. During inference, the result is thresholded (predefined threshold value: 0.5) and used without post-processing.TrainingWe pre-trained RootDetector on the COCO dataset21 to generate a starting point. Although the COCO dataset contains a wide variety of image types and classes not specifically related to minirhizotron images, Majurski et al.22 showed, that for small annotation counts, transfer-learning even from unrelated datasets may improve a CNNs performance by up to 20%. We fine-tuned for our dataset with the Adam optimizer23 for 15 epochs and trained on a total of 129 images from the Training-Set (17 mesocosm images, 47 field-experiment images, 65 soy root images). To enhance the dataset size and reduce over-fitting effects, we performed a series of augmentation operations as described by Shorten & Khoshgoftaar24. In many images, relatively coarse roots ( > 3 mm) occupied a major part of the positive (white) pixel space, which might have caused RootDetector to underestimate fine root details overall. Similarly, negative space (black pixels) between tightly packed, parallel roots was often very small and might have impacted the training process to a lesser extent when compared to large areas with few or no roots (Fig. 2). To mitigate both effects, we multiplied the result of the cross-entropy loss map with a weight map which emphasizes positive–negative transitions. This weight map is generated by applying the following formula to the annotated ground truth images:$$omega left( x right) = 1 – left( {tanh left( {2tilde{x} – 1} right)} right)^{2}$$
(1)
where ω(x) is the average pixel value of the annotated weight map in a 5 × 5 neighborhood around pixel x. Ronneberger, Fischer, & Brox20 implemented a similar weight map, however with stronger emphasis on space between objects. As this requires computation of distances between two comparatively large sets of points, we adapted and simplified their formula to be computable in a single 5 × 5 convolution.For the loss function we applied a combination of cross-entropy and Dice loss 25:$${mathcal{L}} = {mathcal{L}}_{CE} + lambda {mathcal{L}}_{Dice} = – frac{1}{N}sumnolimits_{i} {wleft( {x_{i} } right)y_{i} log left( {x_{i} } right) + lambda frac{{2sumnolimits_{i} {x_{i} y_{i} } }}{{sumnolimits_{i} {x_{i}^{2} sumnolimits_{i} {y_{i}^{2} } } }}}$$
(2)

where x are the predicted pixels, y the corresponding ground truth labels, N the number of pixels in an image and λ a balancing factor which we set to 0.01. This value was derived empirically. The Dice loss is applied per-image to counteract the usually high positive-to-negative pixel imbalance. Since this may produce overly confident outputs and restrict the application of weight maps, we used a relatively low value for λ.Output and post-processingRootDetector generates two types of output. The first type of output are greyscale .png files in which white pixels represent pixels associated with root structures and black pixels represent non-root structures and soil (Fig. 2c). The advantage of .png images is their lossless ad artifact-free compression at relatively small file sizes. RootDetector further skeletonizes the output images and reduces root-structures to single-pixel representations using the skeletonize function of scikit-image v. 0.17.1 (26; Fig. 2e,f). This helps to reduce the impact of large diameter roots or root-like structures such as rhizomes in subsequent analyses and is directly comparable to estimations of root length. The second type of output is a Comma-separated values (.csv) file, with numerical values indicating the number of identified root pixels, the number of root pixels after skeletonization, the number of orthogonal and diagonal connections between pixels after skeletonization and an estimation of the physical combined length of all roots for each processed image. The latter is a metric commonly used in root research as in many species, fine roots provide most vital functions such as nutrient and water transport3. Therefore, the combined length of all roots in a given space puts an emphasis on fine roots as they typically occupy a relatively smaller fraction of the area in a 2D image compared to often much thicker coarse roots. To derive physical length estimates from skeletonized images, RootDetector counts orthogonal- and diagonal connections between pixels of skeletonized images and employs the formula proposed by Kimura et al.17 (Eq. 3).$$L = left[ {N_{d}^{2} + left( {N_{d} + N_{o} /2} right)^{2} } right]^{{1/2}} + N_{o} /2$$
(3)
where Nd is the number of diagonally connected and No the number of orthogonally connected skeleton pixels. To compute Nd we convolve the skeletonized image with two 2 × 2 binary kernels, one for top-left-to-bottom-right connections and another for bottom-left-to-top-right connections and count the number of pixels with maximum response in the convolution result. Similarly, No is computed with a 1 × 2 and a 2 × 1 convolutional kernels.Performance comparisonSelection of participantsFor the performance comparison, we selected 10 human analysts and divided them into three groups of different expertise levels in plant physiology and with the usage of digital root measuring tools. The novice group consisted of 3 ecology students (2 bachelor’s, 1 master’s) who had taken or were taking courses in plant physiology but had no prior experience with minirhizotron images or digital root measuring tools. This group represents undergraduate students producing data for a Bachelor thesis or student assistants employed to process data. The advanced group consisted of 3 ecology students (1 bachelor’s, 2 master’s) who had already taken courses in plant physiology and had at least 100 h of experience with minirhizotron images and digital root measuring tools. The expert group consisted of 4 scientists (2 PhD, 2 PhD candidates) who had extensive experience in root science and at least 250 h of experience with digital root measuring tools. All methods were carried out in accordance with relevant institutional, national, and international guidelines and legislation and informed consent was obtained from all participants.Instruction and root tracingAll three groups were instructed by showing them a 60 min live demo of an expert tracing roots in minirhizotron images, during which commonly encountered challenges and pitfalls were thoroughly discussed. Additionally, all participants were provided with a previously generated, in-depth manual containing guidelines on the identification of root structures, the correct operation of the root tracing program and examples of often encountered challenges and suggested solutions. Before working on the Comparison-Set, all participants traced roots in one smaller-size sample image and received feedback from one expert.Image preparation and root tracingBecause the minirhizotron images acquired in the field covered a variety of different substrates, roots of different plant species, variance in image quality, and because tracing roots is very time consuming, we decided to maximize the number of images by tracing roots only in small sections, in order to cover the largest number of cases possible. To do this, we placed a box of 1000 × 1000 pixels (8.47 × 8.47 cm) at a random location in each of the images in the Comparison-Set and instructed participants to trace only roots within that box. Similarly, we provided RootDetector images where the parts of the image outside the rectangle were occluded. All groups used RootSnap! 1.3.2.25 (CID Bio-Science Inc., Camas, WA, USA;27), a vector based tool to manually trace roots in each of the 25 images in the comparison set. We decided on RootSnap! due to our previous good experience with the software and its’ relative ease of use. The combined length of all roots was then exported as a csv file for each person and image and compared to RootDetector’s output of the Kimura root length.ValidationWe tested the accuracy of RootDetector on a set of 10 image segments of 1000 by 1000 pixels cropped from random locations of the 10 images of the Validation-Set. These images were annotated by a human expert without knowledge of the estimations by the algorithm and were exempted from the training process. As commonly applied in binary classification, we use the F1 score as a metric to evaluate the performance RootDetector. F1 is calculated from precision (Eq. 4) and recall (Eq. 5) and represents their harmonic mean (Eq. 6). Ranging from 0 to 1, higher values indicate high classification (segmentation) performance. As one of the 10 image sections contained no roots and thus no F1 Score was calculable, it was excluded from the validation. We calculated the F1 score for each of the nine remaining image sections and averaged the values as a metric for overall segmentation performance.$$Precision;(P) = frac{{tp}}{{tp + fp}}$$
(4)
$$Recall;(R) = frac{{tp}}{{tp + fn}}$$
(5)
$$F1 = 2*frac{{P*R}}{{P + R}}$$
(6)
where P = precision, R = recall, tp = true positives; fp = false positives, fn = false negatives.Statistical analysisWe used R Version 4.1.2 (R Core Team, 2021) for all statistical analyses and R package ggplot2 Version 3.2.128 for visualizations. Pixel identification-performance comparisons were based on least-squares fit and the Pearson method. Root length estimation-performance comparisons between groups of human analysts (novice, advanced, expert) and RootDetector were based on the respective estimates of total root length plotted over the minirhizotron images in increasing order of total root length. Linear models were calculated using the lm function for each group of analysts. To determine significant differences between the groups and the algorithm, 95% CIs as well as 83% CIs were displayed and RootDetector root length outside the 95% CI were considered significantly different from the group estimate at α = 0.0529. The groups of human analysts were considered significantly different if their 83% CIs did not overlap, as the comparison of two 83% CIs approximates an alpha level of 5%30,31.This study is approved by Ethikkommission der Universitätsmedizin Greifswald, University of Greifswald, Germany. More

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Real-world dataThe dataset consisted of 2986 observations from 902 freshwater shallow lakes in Denmark and North America (data extracted from the LAGOSNE database on 22 February 2022 via R LAGOSNE package version 2.0.2)56 (Supplementary Fig. 9). The Danish lakes were sampled for one or several years from 1984 to 2020 (data extracted in October 2021 from https://odaforalle.au.dk/main.aspx) (Supplementary Fig. 10). Prerequisites for inclusion in the analysis were that lakes had been sampled for physical and chemical variables at least four times or at least three times over the growing season (May to September) for the Danish or North American lakes, respectively, had a mean depth of less than 3 m and were freshwater. Water chemistry samples were analysed using standard methods and data for total phosphorus (TP), total nitrogen (TN) and chlorophyll-a are included here57. The mean and range of TP, TN and chlorophyll-a for the combined sites is given in Table 1, along with the values for each region separately.To gain a longer-term perspective on the relationship between nutrients and chlorophyll-a, we calculated the across-year averages of the summer means of TP, TN and chlorophyll-a, sequentially increasing numbers of years included in the mean up to a total of a five-year mean, at which point there were only 99 lakes left in the dataset. In calculating the multi-year means we allowed a maximum gap of 2 years between observations (i.e. two observations could cover 3 years) to avoid including time series with too many missing years in between. Hence, only lakes with sufficient numbers of sequential data were included, resulting in a large drop in lake numbers as the length of the multi-year mean increased (Table 2).Numerical methodsDiagnostic tests or proxies of alternative equilibriaWe modelled the response of chlorophyll-a to TP and TN using generalised-linear models58 with Gamma distribution and an identity link on untransformed data for single-year and multiple-year means up to 5-year means. We used the Gamma distribution, as chlorophyll fit this significantly better than a normal or log-normal distribution. We used psuedo R2 of the model along with the patterns of residuals, and finally, we plotted the kernel density of the chlorophyll-a values as diagnostics of the presence, absence or prevalence of alternative equilibria in the simulated and real work data.To test how appropriate these diagnostics or proxies of alternative stable states in terms of how well they identify the existence of alternative stable states in randomly sampled multi-year data, we

1.

Simulated two scenarios for the main manuscript, with and without alternative stable states in the data, which were as close to the real-life data as possible. The results of these scenarios appear in the main text (please see details below in the “Data Simulation” section).

2.

We provide multiple scenarios with different degrees, or prevalence, of alternative stable states in the data, see simulations of alternative stable state scenarios. The results of these scenarios appear in Supplementary note 2.

Hierarchical bootstrap approachThere are a large number of permutations of data, both real-world and simulated, that can provide a mean of the two to five sequential years from each lake in the time series data. It was vital to have a method that selects the data for analysis that provides a valid and comparable representation of both real work and simulated data and the models’ errors. In order to provide this we used a non-parametric hierarchical bootstrap procedure38. The flowchart shows the data preparation and data analysis steps of the hierarchical bootstrap procedure (Fig. 4). In the first step (step 1 in Fig. 4), all possible longer-term means are calculated for each lake. To keep as much data as possible, we decided to allow for up to 2 years of gap in the data between years. Taking the 5-year mean data as an example, if data from a lake existed for the years 1991 and 1994−1997, a 5-year mean would be calculated for the years 1991, 1994, 1995, 1996 and 1997. However, if the time series would contain a larger gap, e.g. data would only exist for the years 1991 and 1995–1998, no 5-year mean could be calculated. After the selection procedure, all the 2-year, 3-year and 5-year means are transferred into a new table (step 2 in Fig. 4).Fig. 4Data preparation and analysis steps of the hierarchical bootstrap procedure.Full size imageThe procedure is the same for each temporal scale from 2-year means to 5-year means. For the example of 5 mean years, lakes are randomly sampled from the full 5-year mean dataset in step 2 (Fig. 4) with replacement up to the number of lakes as in the original dataset, for the 5-year means 99 (step 3a). Here, the same lake can appear multiple times or not at all. This step is common for every bootstrap procedure59. However, since we have nested data (5-year means within lakes), we need a second step, in which for every resampled lake in step 3a, one 5-year mean is chosen (step 3b in Fig. 4). Then the three GLM models are produced from the randomly selected data in step 3c (Fig. 4). These steps are then repeated 1000 times to get a good representation of the uncertainties of the model. To ensure a fair comparison between single-year data and their equivalent multi-year mean data, we repeated the bootstrap procedure with single years only using only the lakes for which we also calculated multi-year means. To take the five-year mean as an example, there were 99 lakes where we could calculate at least one 5-year mean observation. First, we ran the bootstrap procedure to calculate 5-year mean values of TP, TN and chlorophyll-a (1000 times) and then took single years’ values of TP, TN and chlorophyll-a (1000 times) from exactly the same 99 lakes. With this approach, exactly the same datasets with the same lakes and observations within lakes are used for the calculation of the multi-year means and their single-year counterparts, making for a robust analysis. The GLM models did not always converge. If either the TP, TN or TP*TN model with interaction did not converge, the iteration was not used in further analysis. The number of converging models equal for each iteration of random samples is given in the results.The described hierarchical approach is the best way to reflect the structure of the original data. A simple, non-hierarchical bootstrap would favour lakes with more five-year means over lakes with fewer five-year means, simply because these make up a larger part of the data. Furthermore, sampling without replacement at the lake level would result in five-year means from lakes with few data dominating the produced random dataset, as every lake would be sampled every time, which then would result in high model leverage of 5-year means from lakes with less data. In contrast, the hierarchical procedure ensures that every lake has the same chance to end up in the randomly sampled bootstrap, in the second step, it ensures that of each sampled lake, every 5-year mean has the same chance to end up in the random dataset. These notions are in agreement with the findings of an assessment on how to properly resample hierarchical data by non-parametric bootstrap38.Data simulationGeneral approach of simulation assumptions and proceduresWe generated random scatter for the generalised-linear model based on Gamma distributions for two different “populations” of lakes with two different intercepts and slopes. At first, we calculated the linear equations for the two populations:For each population i and j, 99 samples (equalling the number of lakes in real-life data with 5-year means, nlake) were generated with a specific number of data points depending on the scenario (nyear) each, hence nlake = 1−99 for each population of lakes, e.g. with 20 years (nyear = 20) each.We found the real nutrient data to be normally distributed, with total nitrogen (TN) having a range between 0.33 and 4.93 mg/L and a constant coefficient of variation (CV, with a mean CV of 0.35) across this range (the same is true for total phosphorus (TP) at a shorter range). Hence, for each nlake, the x for the nyear = 20 were generated based on the mean range (mean per lake of the real-life data) and CV (0.35) from the real-life TN concentration data, hence with a range of 0.33 to 4.33 mg/L. Therefore the values and random variability of x in the simulations are close to the true values of the TN concentrations. The x is then fed into the linear equations above.To the resulting yi and yj we added random noise based on the Gamma distribution (using the rgamma function in R). We used a Gamma distribution because the Chlorophyll-a concentration also follows a Gamma distribution. The variability of a Gamma distribution is expressed by the shape variable. The variability of chlorophyll-a, its shape value, equals 2.63. This shape value was used in the Gamma distribution of yi and yj. The final calculated yi and yj had therefore a random rate calculated as shape/yi or shape/yj. Hence, their variability in the y dimension was close to the true chlorophyll-a variability.The data from both lake populations were then pooled and randomly sampled using the same hierarchical bootstrap procedure with 500 iterations for the scenarios in the supplementary materials and with 1000 iterations for main text simulation scenarios, which is identical to what was done for the real-world data.Simulation scenarios based on characteristics of real-world dataThe real-world 5-year mean data consisted of 99 lakes with 5–20 years of data for each lake. For the simulation scenario in the main text, we therefore randomly sampled between 5 and 20 data points for each of the 99 simulated lakes based on the x distribution described above. Intercepts and slopes of the simulation, resembled the range of the true data (see scatter plots in Fig. 2 of the main manuscript).In the alternative stable state scenario, we chose two slopes and intercepts for different populations of lakes:

Population i: ai = 0, bi = 40

Population j: aj = 50, bj = 120

We based the slopes and intercepts of the ASS scenarios on the diagnostic combination defined by Scheffer and Carpenter7 which propose an abrupt shift in (a) the time series, (b) the multimodal distribution of states and (c) the dual relationship to a controlling factor. Here, the idea is that an ecosystem will jump from one state to the next at the same (nutrient) conditions (different intercept and/or slope, condition a within ref. 7), where any change in the nutrient will have different effects on algae or macrophytes (best represented by different slope, condition c), resulting in a multimodal distribution of the response (condition b). Hence, simulations are in line with what is predicted for ASS, but we took great lengths to also show other possibilities with the simulations in the Supplementary information to ensure we did not overlook any occasional occurrence of alternative equilibria.Here, the appearance of alternative stable states in the data could happen at any point in the time series of a single lake, or the entire time series could include only one of the two alternative stable states. To accommodate these alternative stable state constellations (for each of which we made a separate simulation scenario, (see Supplementary Note 2, “Simulations of alternative stable state constellations”), we forced the alternative stable state scenario to be constructed of 1/3 of data with one state, 1/3 of data with the second state and 1/3 of data where both alternative states could occur. In the latter case, the alternative stable state appeared after the first 20% but before the last 20% of the time series. Since the variability and range of x (nutrient) and y (chlorophyll -a response) is simulated as close as possible to the real-world data in all scenarios, the measures taken here (variable time series and combination of different alternative stable state scenario constellations) produce a simulation as close to the real-world data as possible. Specifically, we found the real-world nutrient data to be normally distributed, with total nitrogen (TN) having a range between 0.33 and 4.93 mg/L and a constant coefficient of variation (CV, with a mean CV of 0.35) across this range (the same is true for total phosphorus (TP) at a shorter range). Hence, for each simulated lake, the x were generated based on this mean range and CV. Furthermore, the resulting yi and yj were randomised by using a Gamma distribution (using the rgamma function in R). We used a Gamma distribution because the chlorophyll-a concentration also follows a Gamma distribution. The variability of a Gamma distribution is expressed by the shape variable. The variability of chlorophyll-a, its shape value, equals 2.63. This shape value was used in the Gamma distribution of yi and y. The final calculated yi and yj had, therefore a random rate calculated as shape/yi or shape/y. Hence, their variability in the y dimension was close to the true chlorophyll-a variability.For the scenario without alternative stable states, both populations of data had the same intercept and slope:

Population i: ai = 0, bi = 40

Population j: aj = 0, bj = 40.

Please see Supplementary Note 2 for further simulations of different potential constellations of alternative states. There we show that our approach finds alternative stable states in response to nutrient concentration, even if they appear in time series from different lakes.Assessment of diagnostic tests or proxies of alternative equilibriaWe modelled the response of chlorophyll-a to TP and TN using generalised-linear models3 with Gamma distribution and an identity link on untransformed data for single-year and multiple-year means up to 5-year means. We used the Gamma distribution, as chlorophyll fit this significantly better than a normal or log-normal distribution. We used R2 of the model along with the patterns of residuals, and finally, we plotted the kernel density of the chlorophyll-a values as diagnostics of the presence, absence or prevalence of alternative equilibria in the simulated and real work data.The comparison of how the diagnostics/proxies of alternative stable states respond to the variation in the prevalence of alternative equilibria in the simulated datasets provides a robust assessment of their ability to identify both the presence and absence of alternative equilibria. It is the response of these diagnostic tests over time, with the increase in the temporal perspective as more years are added to the mean values of TP, TN and chlorophyll-a, that are key to the identification of the presence and or absence of alternative equilibria in a given dataset. The simulations show that a dataset which contains alternative equilibria will show (1) no improvement in R2 as the temporal perspective of the data increases (more years in the multi-year mean); (2) an increased bimodality in the residuals of the models of nutrients predicting chlorophyll-a will increase as more years are added to the multi-year mean and (3) the kernel density function of chlorophyll-a will display increasingly bimodality as more years are added to the mean. In the absence of alternative equilibria, the patterns differ with an R2, and increase in unimodality of residuals and a consistent unimodal pattern in the kernel density function. Thus, the diagnostic tests provide a robust test of both the presence and absence of alternative equilibria in a given dataset.Alternative stable state assessment for real data with limited data rangeIt could be the case that alternative stable states do not appear in the full dataset but only in a limited TN and TP concentration range. We filtered and re-analyzed the data, only keeping data points within the following two ranges: – TN concentration = 0.5−2 mg/L–TP concentration = 0.05−0.4 mg/L. In the filtered data, 1329 out of the original 2876 single-year data points, 289 out of 1028 3-year mean data points and 212 out of the 864 five-mean year data points remained. The filtered data consisted of data points from 550, 48 and 27 lakes for the single-year data, 3-year means and 5-year means, respectively. The smaller range resulted in lower R² of the models, yet the pattern that multi-year means result in higher R² compared to single-year data was largely consistent, apart from the 5-year mean TN models for which both, the single-year and mean data resulted in very low R² (Supplementary Fig. 6). Furthermore, due to the lower number of samples, the errors of all proxies are higher, making conclusions more difficult than for the full data. Still, we do not see any clear indication of alternative stable states in the scatter plots (two groups of dots are not appearing (Supplementary Fig. 5), the Kernel density plots (or model residuals (Supplementary Fig. 6)). i.e. no signs of bimodality in residuals or Kernel density plots. Please see details on this analysis in the supplementary material.Details and the R code for the steps for the random multi-year sampling can be found in the supplementary materials.Reporting summaryFurther information on research design is available in the Nature Portfolio Reporting Summary linked to this article. More

Game theory is a framework for analysing the outcome of the strategic interaction between decision makers1. The fundamental concept is that of a Nash equilibrium where no player can improve her payoff by a unilateral strategy change. Typically, the Nash equilibrium is considered to be the optimal outcome of a game, however in social dilemmas the individual optimal outcome is at odds with the collective optimal outcome2. This means that one player can improve her payoff at the expense of the other by unilaterally deviating, but if both deviate, they end up with lower payoffs. In this type of games, the mutually beneficial, but non-Nash equilibrium strategy is called cooperation. However, in this context cooperation should not be interpreted as an interest in the welfare of others, as players only aim to secure a high payoff for themselves.In this framework, payoff maximisation is considered to be rational, but when such rational players then seize every opportunity to gain at the opponent’s expense, they may counterintuitively both end up with low payoffs. A game that clearly exhibits this contradiction is the Traveler’s Dilemma. Since its formulation in 1994 by the economist Kaushik Basu3, the game has become one of the most studied in the economics literature. Additionally, it has been discussed in theoretical biology in the context of evolutionary game theory.In general, the dilemma relies on the individuals’ incentive to undercut the opponent. To be more specific, players are motivated to claim a lower value than their opponent to reach a higher payoff at the opponent’s expense. Such incentive leads players to a systematic mutual undercutting until the lowest possible payoff is reached, which is the unique Nash equilibrium. It seems paradoxical that players defined as rational in a game theoretical sense end up with such a poor outcome. Therefore, the question that naturally arises is how can this poor outcome be prevented and how cooperation can be achieved.To address these questions, it can be helpful to better understand price wars, which consist in the mutual undercutting of prices to gain market share. In addition, it can provide information about human behaviour, because economic experiments have shown that individuals prefer to choose the cooperative high payoff action, instead of the Nash equilibrium4.Our analysis focuses on showing that the Traveler’s Dilemma can be decomposed into a local and a global game. If the payoff optimisation is constrained to the local game, then players will inevitably end up in the Nash equilibrium. However, if players escape the local maximisation and optimise their payoff for the global game, they can reach the cooperative high payoff equilibrium.Here, we show that the cooperative equilibrium can be reached in a game like the Traveler’s Dilemma due to diversity, which we define as the presence of suboptimal strategies. The appearance of strategies far from those of the residents allows for the local maximisation process to be escaped, such that an optimisation at a global level takes place. Overall this can lead to cooperation because by considering “suboptimal strategies” that play against each other it is possible to reach higher payoffs, both collectively and individually.GameThe Traveler’s Dilemma is a two-player game. Player i has to choose a claim, (n_i), from the action space, consisting of all integers on the interval [L, U], where (0 le L < U). The payoffs are determined as follows: If both players, i and j, choose the same value ((n_i = n_j)), both get paid that value. There is a reward parameter (R >1), such that if (n_i < n_j), then i receives (n_i + R) and j gets (n_i- R) Thus, the payoff of player i playing against player j is$$begin{aligned} pi _{ij} = {left{ begin{array}{ll} n_i& text { if } n_i = n_j\ n_i + R& text { if } n_i < n_j\ n_j - R& text { if } n_i > n_j end{array}right. } end{aligned}$$
(1)
Thus, a player is better off by choosing a slightly lower value than the opponent: when j plays (n_j), then it is best for i to play (n_j-1). The iteration of this reasoning, which we will call the stairway to hell, leads to the only Nash equilibrium of the game, ({L,L}), where both players choose the lowest possible claim. The classical game theory method to arrive to this equilibrium is called iterative elimination of dominated strategies5.The game can be visualised through its payoff matrix (Fig. 1). For simplicity, we use the values from the original formulation: (L=2), (U=100) and (R=2). The payoff matrix shows that the Traveler’s Dilemma can be decomposed into a local and a global game. Let us begin with the local game. When the action space of the game is reduced to two adjacent actions n and (n+1) (black boxes in Fig. 1), the Traveler’s Dilemma with (R=2) is equivalent to the Prisoner’s Dilemma6. In general, for any value of R, the Traveler’s Dilemma becomes a Prisoner’s Dilemma for any pair of actions n and (n+s), where ( 1 le s le R-1 ). For example, for (R=4) the pair of actions n and (n+1), n and (n+2), n and (n+3) follow the same game structure as the Prisoner’s Dilemma. Therefore, the Traveler’s Dilemma consists of many embedded Prisoners’ Dilemmas. This means that at a local level the game is a Prisoner’s Dilemma.If we now consider actions that are distant from each other in the action space, e.g. 2 and 100, we can observe a coordination game structure (gray boxes in Fig. 1), where ({100,100}) is payoff and risk dominant7,8. In general, any pair of actions n and (n+s), where ( R le s le U-n), construct a coordination game. As a result, the Traveler’s Dilemma becomes a coordination game at a global level, which has different equilibria than the local game.Figure 1Payoff matrix of the Traveler’s Dilemma. Visualisation of the payoff scheme described by Eq. (1). For simplicity, the action space is ( {n_i in {mathbb {N}} mid 2 le n_i le 100}) and the reward parameter is (R=2). The Traveler’s Dilemma can be decomposed into a local and a global game. At a local level the game is a Prisoner’s Dilemma (black boxes). This happens when the action space is reduced to any pair of actions n and (n+s), where ( 1 le s le R-1 ). While at the global level, we can observe a coordination game (gray boxes). This level is defined as any pair of actions n and (n+s), where ( R le s le U-n).Full size imageParadoxSocial dilemmas appear paradoxical in the sense that self-interested competing players, when rationally playing the Nash equilibrium, end up with a payoff that clearly goes against their self-interest. But with the Traveler’s Dilemma, the paradox goes further, as suggested in its original formulation3. Classical game theory proposes ({L,L}) as the Nash equilibrium of the game. However, it seems unlikely and implausible that, with R being moderately low, say (R=2), for individuals to play the Nash equilibrium. This has been confirmed in economic experiments where individuals rather choose values close to the upper bound of the interval. Such experiments have also shown that the chosen value depends on the reward parameter (R), where an increasing value of R shifts players’ decision towards ({L,L})4. Nonetheless, classical game theory states that the Nash equilibrium of the game is independent of R.Consequently, the aim of this paper is to seek and explore simple mechanisms through which the apparent non-rational cooperative behaviour can come about. We also examine the effect of the reward parameter on the game’s outcome. Given that the Traveler’s Dilemma paradox emerges in the classical game theory framework, we analyse the game using evolutionary game theory tools5,9,10. This dynamical approach allows us to explore adaptive behaviour outside of the stationary classical game theory framework. To be more precise, for this approach individuals dynamically adjust their actions according to their payoffs.The key point of course is to understand how the system can converge to high claims. We show that this behaviour is possible because the Traveler’s Dilemma can be decomposed into a local and a global game. If the payoff maximisation is constrained to the local level, then the stairway to hell leads the system to the Nash equilibrium; given that locally the game is a Prisoner’s Dilemma. On the other hand, at a global level the game follows a coordination game structure, where the high claim actions are payoff dominant. Thus, for the system to reach a high claim equilibrium the maximisation process needs to jump from the local to the global level.Our analysis led us to identify the mechanism of diversity as responsible for enabling this jump and preventing players from going down the stairway to hell. This mechanism works on the idea that to reach a high claim equilibrium, players have to benefit from playing a high claim. For a population setting, it means that players need to have the chance to encounter opponents also playing high. From a learning model point of view, it refers to the belief that the opponent will also play high, at least with a certain probability. If the belief is shared by both players, they should both play high and reach the cooperative equilibrium. Here, we explore these two types of models to unveil the mechanism leading to cooperation.Population based models unveil diversity as the cooperative mechanism via the effect of mutations on the game’s outcome. This is shown for the replicator-mutator equation and the Wright–Fisher model. Similarly, a two-player learning model approach, more in line with human reasoning, shows that if players are free to adopt a higher payoff action from a diverse action set during their introspection process, they can reach the cooperative equilibrium. This result is obtained using introspection dynamics.Finally, we explain how diversity is the underlying mechanism that enables the convergence to high claims in previously proposed models. To be more precise, we show that diversity is required because it allows for the maximisation process to jump from the local to the global level. More

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