Ecology

Sub-regionsThe study region focuses on 18 sub-regions within the Pacific mountain ranges of North American overlapping with the range of Pacific Salmon with >1.5% glacier cover (Figs. 1 and 2). The term “sub-region” here refers to either a single major salmon watershed or aggregates of small coastal watersheds, which range in area from ~13,000 to ~68,000 km2. For sub-regions within Alaska, USA, we accessed boundary data from the Watershed Boundary Database at the USGS (https://www.usgs.gov/). For sub-regions within British Columbia, Canada, we accessed boundary data from the Freshwater Atlas of British Columbia (https://catalogue.data.gov.bc.ca/). Pacific salmon range data were from the National Center for Ecological Analysis and Synthesis (Fig. 1). The study region covers ~623,000 km2 across British Columbia, Canada and Alaska, USA and ~20% of the total North American range of Pacific salmon.Glacier outlinesOutlines for the 45,963 glaciers within the study region were obtained from the Randolph Glacier Inventory v6.0 (https://www.glims.org/RGI/; RGI v6.0), which provides a globally complete data set of glacier outlines outside of Greenland and Antarctic ice sheets17. These glaciers cover a total area of ~81,000 km2, which corresponds to 80% of the total glacier area in the Pacific mountain ranges within North America. The glacier outlines refer roughly to the years 2009 ± 2 for Alaska, and 2004 ± 5 for Western Canada17,53. Glacierization for each of 18 sub-regions ranges from 1.5 to 52%.Present-day streamsSynthetic stream networks were constructed from Digital Elevation Models (DEMs) for each of the 18 sub-regions using Geographic Information Systems (GIS; ArcGIS 10.6 and QGIS 2.18) hydrology tools to represent present-day streams throughout the study region. Specifically, we used Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) global DEMs v2.0 with a spatial resolution of ~30 m54. Open access synthetic stream network datasets such as the National Hydrography Dataset (NHD) from the USGS and the Freshwater Atlas from the British Columbia government are available but were not used due to inconsistencies in spatial resolution across the study region. From our synthetic stream networks, we eliminated all stream segments that overlapped with the RGI glacier outlines because the ASTER global DEMs used to create the synthetic stream networks represent glacier surface elevation rather than estimated deglaciated terrain. All present-day streams within our study region are void of any major dams that inhibit salmon movement based on existing databases of dams55. To summarize present-day stream kms, and all subsequent analyses, we used rstudio: 1.4.1103-4, R: ‘Mirrors’.Identifying and verifying stream gradient thresholds for migrating salmon and for determining accessible glaciersWe used stream gradient-based thresholds the determine constraints in salmon migration and the number of glaciers that would be accessible and create future streams for migrating adult salmon. Based on the large body of literature suggesting stream gradients (e.g., ranging from More

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Our novel approach to bioacoustic source separation involves an end-to-end pipeline consisting of multiple discrete steps, including (1) synthesizing a dataset, (2) developing and training a separator network to disentangle the input mixture, and (3) constructing and training a classifier model to employ as a downstream evaluation task. This workflow requires few hyperparameter modifications to account for unique vocal behavior across different biological taxa but is otherwise general and makes no species-level assumptions about the spectrotemporal structure of the source calls. We develop a complete framework for bioacoustic source separation in a permutation-invariant mode using overlapping waveforms drawn from the same class of signals. We apply BioCPPNet to macaques, dolphins, and Egyptian fruit bats, and we consider two or three concurrent “speakers”. Note that we henceforth refer to non-human animal signalers as “speakers” for consistency with the human speech separation literature2. We address both the closed speaker regime in which the training and evaluation data subsets contain calls produced by individuals drawn from the same distribution as well the open speaker regime in which the model is tested on calls generated by individuals not present in the training dataset.Bioacoustic dataWe investigate a set of species with dissimilar vocal behaviors in terms of spectral and temporal properties. We apply BioCPPNet to a macaque coo call dataset47 consisting of 7285 coos produced by 8 unique individuals; a bottlenose dolphin signature whistle dataset26 comprised of 400 signature whistles generated by 20 individuals, of which we randomly select 8 for the purposes of this study; and an Egyptian fruit bat vocalization dataset48 containing a heterogeneous distribution of individuals, call types, and call contexts. In the case of the bat dataset, we extract the data (31399 calls) corresponding to the 15 most heavily represented individual bats, reserving 12 individuals (27586 calls) to address the closed speaker regime and the remaining 3 individuals (3813 calls) to evaluate model performance in the open speaker scenario.DatasetsThe mixture dataset is generated from a species-specific corpus of bioacoustic recordings containing signals annotated according to the known identity of the signaller. Motivated by WSJ0-2mix2, a preeminent reference dataset used for human single-channel acoustic source separation, we adopt a similar approach of constructing bioacoustic datasets by temporally overlapping and summing ground truth individual-specific signals to enable supervised training of our model. For macaques and dolphins, the mixture waveforms contain discrete source calls that overlap in the time domain, by design. For bats, mixtures are constructed by adding signal streams, each of which may exhibit one or more temporally separated sequential vocal elements. In all cases, the mixtures operate under the assumption that, without loss of generality, the constituent sources vary in the degree of spectral overlap due to differential spectrotemporal properties of sources, in accordance with the DUET principle (i.e, the mixtures contain approximately disjoint sources that rarely coincide in dominant frequency)19. The resultant dataset consists of an input array of the composite mixture waveforms, a target array containing the separated ground truth waveforms corresponding to the respective mixtures, and a class label array denoting the identities of the vocalizing animals responsible for generating the signals. In the case of macaques, we here consider closed speaker set mixtures of two and three simultaneous speakers, but our method is functionally not limited in the number of sources (N) it can handle. For dolphins, we consider the closed speaker regime with two overlapping calls, and for bats, we consider the closed and open speaker scenarios with two sources.We first extract the labeled waveforms either by truncating or zero-padding the waveforms to ensure that all the samples are of fixed duration. We select the number of frames either by computing the mean plus three-sigma of the durations of the calls contained in the corpus from which we draw the signals, by selecting the maximum duration of all calls, or by choosing a fixed value. For macaques, dolphins, and bats, we use 23156 frames (0.95s), 290680 frames (3.03s), and 250000 frames (1.0s), respectively. We then randomly select vocalizations from N different speakers drawn from the distribution of individuals used in the study (8 macaques, 8 dolphins, 12 bats for the closed speaker regime, 3 bats for the open speaker regime) and mix them additively, ensuring to randomly shift the overlaps to simulate a more plausible scenario and to provide for asynchronicity of start times, an important acoustic cue that has been suggested as a mechanism with which the animal brain can solve the CPP1. Despite higher computational and memory costs, we opt to use native sampling rates, since certain animal vocalizations may reach frequencies near the native Nyquist frequency. With this in mind, however, our method does provide for resampling when the vocalizations of the particular species of interest are amenable to downsampling. Explicitly, for the three species we consider including macaques, dolphins, and bats, we use sampling rates of 24414 Hz, 96 kHz, and 250 kHz, respectively. For the closed speaker regime, the training and evaluation subsets contain calls produced by the same distribution of individuals to ensure a closed speaker set. We segment the original nonoverlapping vocalizations into 80/20 training/validation subsets. We generate the mixture training waveforms using 80% of the calls, and we construct the mixture validation subset using the remaining 20% of calls held out from the training data. In the case of overlapping bat calls (for which the corpus of bioacoustic recordings contains (mathscr {O}(10text { hours})) of data as opposed to (mathscr {O}(10^{-1}text { hours})) for macaques and dolphins), we also address the open speaker source separation problem by constructing a further testing data subset of mixtures of calls of additional vocalizers not contained in the training distribution. For macaques, we construct a training data subset comprised of 12k samples and a validation subset with 3k samples, all of which contain calls drawn from 8 animals. For dolphins, we randomly select 8 individuals and construct training/validations subsets with 8k and 2k samples, respectively. For bats, we select 15 individuals, randomly reserving 12 for the closed speaker problem and the remaining 3 for the open speaker situation. We train the bat separator model on 24k mixtures. We evaluate performance in both the closed and open speaker scenarios using data subsets consisting of 6k mixtures containing unseen vocalizations produced by the appropriate distribution of individuals according to the regime under consideration. We repeat the bat training using a larger mixture dataset (denoted by +) containing 72k samples. We here report validation metrics to ensure that we are evaluating model performance on unseen mixtures of unseen calls in the closed speaker regime and on unseen mixtures of unseen calls of unseen individuals in the open speaker regime.For the downstream classification task, we extract vocalizations annotated according to the individual identity, and we segment the calls into an 80/20 training/testing split to ensure that we are evaluating model performance on unseen calls. For both the training and evaluation data subsets, we employ an augmentation scheme in which we apply random temporal shifts to call onsets to better reflect more plausible real-world scenarios.Classification modelsIn an effort to provide a more physically interpretable evaluation metric to supplement the commonly-implemented SI-SDR used in human speech separation studies, we develop CNN-based classifier models to label the individual identity of the separated vocalizations as a downstream task. This requires training classification networks to predict the speaker class label of the original unmixed waveforms. For each species we consider, we design and train custom simple and lightweight CNN-based architectures largely motivated by previous work24, tailored to accommodate the unique vocal behavior of the given species.The first layer in the model is an optional high pass filter constructed using a nontrainable 1D convolution (Conv1D) layer with frozen weights determined by a windowed sinc function49,50 to eliminate low-frequency background noise. We omit this computationally intensive layer for macaques and Egyptian fruit bats, but we implement a high pass filter for the dolphin dataset, selecting an arbitrary cutoff frequency of 4.7 kHz and transition bandwidth 0.08 to remove background without impinging on the region of support for dolphin whistles. After the optional filter is an encoder layer to compute on-the-fly feature extraction. We experimented with a fully learnable free Conv1D filterbank, a spectrogram, and a log-magnitude spectrogram and observed optimal performance using a non-decibel (dB)-scaled STFT layer computed with a nfft window width, a hop window shift, and a Hann window where nfft and hop are species-dependent variables. For macaques, we select nfft=1024 and hop=64 corresponding to temporal scales on the order of 40ms and frequency resolutions on the order of 20 Hz. We choose nfft=1024 and hop=256 for dolphins and nfft=2048 and hop=512 for bats, corresponding to temporal resolutions of ~ 10 ms and ~ 8 ms and frequency resolutions of ~ 90 Hz and ~ 120 Hz, respectively.Following the built-in feature engineering, the architecture includes 4 convolutional blocks, which consist of two sequential 2D convolution (Conv2D) layers with leaky ReLU activation and a max pooling layer with pool size 4. Next is a dense fully connected layer with leaky ReLU activation followed by another linear layer with log softmax activation to output the V log probabilities (i.e. confidences) where V is the number of individual vocalizers used in the study (8, 8, 12 for macaques, dolphins, and bats, respectively). We also include dropout regularization with p=0.25 for the macaque classifier and p=0.5 for the dolphin and bat classifiers to address potential overfitting. With these architectures, the macaque, dolphin, and bat classifier models have 230k, 279k, and 247k trainable parameters, respectively.For all species, we minimize the negative log-likelihood objective loss function using the Adam optimizer51 with learning rate lr = 3e−4. For macaques, dolphins, and bats, respectively, we train for 100, 50, and 100 epochs with batch sizes 32, 8, and 8. We serialize the model after each epoch and select the top-performing models. We opt not to carry out hyperparameter optimization since the classification task is of secondary importance and is used solely as a downstream task.Figure 1(a) Schematic overview of the BioCPPNet pipeline. Source vocalization waveforms are overlapped in time and mixed additively. BioCPPNet operates on the mixture waveform, yielding predictions for the separated waveforms, which are compared to the source ground truths, up to a permutation. The estimated waveforms are classified by the identity classification model24 (ID) to compute the downstream classification accuracy metric. (b) Block diagram of the BioCPPNet architecture. The input mixture waveform is transformed to a learnable or handcrafted representation (Rep), which then passes through a 2-dimensional U-Net52 composed of a contracting encoder path and an expanding decoder path with skip connnections. The encoder path consists of sequential downsampling convolutional blocks, each of which is constructed using two convolutional layers (Conv2D) with leaky ReLU activation and batch normalization (BatchNorm) followed by a max pooling. The decoder path employs upsampling convolutional blocks, consisting of an upsampling and skip connection concatenation followed again by the Conv2D layers with leaky ReLU and BatchNorm. The U-Net predicts masks (Mask 0 and Mask 1), the number of which is determined by the number of sources (N), that are multiplicatively applied to the original mixture representation. The predicted time-frequency representations of the separated waveforms are inverted with learnable or handcrafted inverse transforms (iRep) to output raw waveforms. All schematic diagrams were created using Affinity Designer (version 1.8.1) https://affinity.serif.com/en-us/designer/.Full size imageSeparation modelsBioCPPNet (Fig. 1) is a lightweight and modular architecture with a modifiable representation encoder, a 2D U-Net core, and an inverse transform decoder, which acts directly on raw audio via on-the-fly learnable or handcrafted transforms. The structure of the network is designed to provide for extensive experimentation, optimization, and enhancement across a range of species with variable vocal behavior. We construct and train a separation model for each species and each number N of sources contained in the input mixture.Figure 2Schematic diagram demonstrating the application of BioCPPNet to dolphin signature whistles using handcrafted STFT-based encoders and decoders. The source waveforms produced by N speakers of unique identity (e.g. T. truncatus 0 and T. truncatus 1) are overlapped in time, summed, and transformed to time-frequency space using an STFT layer, resulting in the mixture time-frequency representation (Mixture TFR). The U-Net predicts masks (Mask 0 and Mask 1) that are applied to the mixture representation. The separated spectrogram estimations (TFR 0 and TFR 1) are inverted using an iSTFT layer to yield the model’s predictions for the separated raw waveforms, which are compared to the ground truth waveforms and classified according to predicted identity using the classification model.Full size imageModel architectureAs with the classifier model, the network’s encoder consists of a feature engineering block, the initial layer of which is an optional high pass filter. This is followed by the representation transform, which includes several options including the Conv1D free encoder, the STFT filterbank, and the log-magnitude (dB) STFT filterbank. We choose the same kernel size (nfft) and stride (hop) parameters defined in the classifier model. Sequentially following the feature extraction encoder is a 2D U-Net core. This architecture consists of B (4 for macaques, 3 for dolphins, and 4 for bats) downsampling convolutional blocks, a middle convolutional block, and B upsampling convolutional blocks. The downsampling blocks consist of two 2D convolutional layers with filter number that increases with model depth with leaky ReLU activation followed by a max pooling with pool size 2, 6, and 3 for macaques, dolphins, and bats. The middle block contains two 2D convolutional layers with leaky ReLU activation. The upsampling blocks include an upsampling using the bilinear algorithm and a scale factor corresponding to the pool size used during downsampling, followed by skip connections in which the corresponding levels of the contracting and expanding paths are concatenated before passing through two 2D convolutional layers with leaky ReLU activation. All convolutional layers in the downsampling, middle, and upsampling blocks include batch normalization after the activation function to stabilize and expedite training and to promote regularization. Though our default implementation is phase-unaware, we also offer the option for a parallel U-Net pathway working directly on phase information, which has been shown to improve performance in other applications53,54,55. The final layer in the U-Net core is a 2D convolutional layer with N channels, which are then split prior to entering the inverse transform decoder. For the inverse transform, we again provide numerous choices including a free filterbank decoder based on a 1D convolutional transpose (ConvTranspose1D) layer, an iSTFT layer, an iSTFT layer accepting dB-scaled inputs, and a multi-head convolutional neural network (MCNN) for fast spectrogram inversion56. In detail, the U-Net returns N masks that are then multiplied by the original encoded representation of the mixture waveform. The separated representations are then passed into the inverse transform layer in order to yield the raw waveforms corresponding to the model’s predictions for the separated vocalizations. We initialize all trainable weights using the Xavier uniform initialization. In the case of macaques, we experiment across all combinations of representation encoders and inverse transform decoders, and we find optimal performance using the handcrafted non-dB STFT/iSTFT layers operating in the time-frequency domain. Since the model with the fully learnable Conv1D-based encoder/decoder uniquely operates in the time domain, we report evaluation metrics for this model, as well. For dolphins and bats, we here report metrics using exclusively the non-dB STFT/iSTFT technique.BioCPPNet (Fig. 1) is designed as a lightweight fully convolutional model in order to efficiently process large amounts of bioacoustic data sampled at high sampling rates while simultaneously minimizing computational costs and limitations and the likelihood of overfitting. For the macaque separators, the networks consist of 1.2M parameters (for the STFT, iSTFT combination), 2.5M parameters (for the STFT, iSTFT combination with parallel phase pathway), or 2.8M parameters (for the Conv1D free filterbanks). For the dolphin separator (Fig. 2), the model has 304k parameters, while the bat separator model has 1.2M parameters. This is to be contrasted with the comparatively heavyweight default implementations of models commonly used in human speech separation problems, such as Conv-TasNet3, which has 5.1M parameters; DPTNet4 with 2.7M parameters; or Wavesplit5 with 29M parameters. Regardless of the lower complexity of BioCPPNet, the model achieves comparable performance or even outperforms reference human speech separator models while still being lightweight enough to train on a single NVIDIA P100 GPU.Model training objectiveThe model training objective aims to optimize the reconstruction of separated waveforms from the aggregated composite input signal. We adopt a permutation-invariant training (PIT)57 scheme in which the model’s predicted outputs are compared with the ground truth sources by searching over the space of permutations of source orderings. This fundamental property of our training objective reflects that the order of estimations and their corresponding labels from a mixture waveform is not expressly germane to the task of acoustic source separation, i.e. separation is a set prediction problem independent of speaker identity ordering5.Source separation involves training a separator model f to reconstruct the source single-channel waveforms given a mixture (x=sum _{i=1}^N s^i) of N sources, where each source signal (s^i) for (i in [1, N]) is a real-valued continuous vector with fixed length T, i.e., (s^i in mathbb {R}^{1 times T}). The model outputs the predicted waveforms ({hat{s}^i}_{i=1}^N) where (forall i, hat{s}^i = f^i(x)), and a loss function is evaluated by comparing the predictions to the ground truth sources ({s^i}_{i=1}^N) up to a permutation. Explicitly, we consider a permutation-invariant objective function5,$$begin{aligned} mathscr {L}(hat{s}, s) = min _{sigma in S_N} frac{1}{N} sum _{i=1}^N ell (hat{s}^{sigma (i)}, s^i) qquad text {where} forall i, hat{s}^i = f^i(x) end{aligned}$$Here, (ell (cdot , cdot )) represents the loss function computed on an (output, target) pair, (sigma) indicates a permutation, and (S_N) is the space of permutations. In certain scenarios, we include the L2 regularization term,$$begin{aligned} mathscr {L} mapsto mathscr {L} + lambda sum _{j=1}^P beta _j^2 end{aligned}$$where (beta _j) represent the model parameters, P denotes the model complexity, and (lambda) is a hyperparameter empirically selected to minimize overfitting (i.e. enhance convergence of training and evaluation losses and metrics).For the single-channel loss function (ell), we consider a linear combination of several loss terms that compute the error in estimated waveform reconstructions ({hat{s}^i}_{i=1}^N) relative to the ground truth waveforms ({s^i}_{i=1}^N).

L1 Loss $$begin{aligned} |hat{s}^{sigma (i)} – s^{i}| end{aligned}$$ This represents the absolute error on raw time domain waveforms.

STFT L1 Loss $$begin{aligned} |text {STFT}(hat{s}^{sigma (i)}) – text {STFT}(s^{i})| end{aligned}$$ This term functions to minimize absolute error on time-frequency space representations. Empirically, the inclusion of this contribution enhances the reconstruction of signal harmonicity.

Spectral Convergence Loss $$begin{aligned} ||text {STFT}(hat{s}^{sigma (i)}) – text {STFT}(s^{i})||_F / ||text {STFT}(s^{i})||_F end{aligned}$$ where (||cdot ||_F) denotes the Frobenius norm over time and frequency. This term emphasizes high-magnitude spectral components56.

We also experimented with additional terms including L1 loss on log-magnitude spectrograms to address spectral valleys and negative SI-SDR (nSI-SDR), but the inclusion of these contributions did not yield empirical improvements in results.For macaques, we modify the training algorithm according to the representation transform and inverse transform built into the model. For the model with the fully learnable Conv1D encoder and decoder, we train using the AdamW58 optimizer with a learning rate 3e-4 and batch size 16 for 100 epochs. In order to stabilize training and avoid local minima when using handcrafted STFT and iSTFT filterbanks, we initially begin training the models for 3 epochs with batch size 16 using stochastic gradient descent (SGD) with Nesterov momentum 0.6 and learning rate 1e-3 before switching to the AdamW optimizer until reaching 100 epochs.For dolphins, we provide the model with the original mixture as input, but we use high pass-filtered source waveforms as the target, which means the separation model must additionally learn to denoise the input. We again initialize training with 3 epochs and batch size 8 using SGD with Nesterov momentum 0.6 and learning rate 1e-3 before switching to the AdamW optimizer with learning rate 3e-4 for the remaining 97 epochs. We use a similar training scheme for bats, initially training with SGD for 3 epochs before employing the optimizer switcher callback to switch to AdamW and to complete 100 epochs.Model evaluation metricsWe consider the reconstruction performance by computing evaluation metrics using an expression given by5,$$begin{aligned} mathscr {M}(hat{s}, s) = max _{sigma in S_N} frac{1}{N} sum _{i=1}^N m(hat{s}^{sigma (i)}, s^i) qquad text {where} forall i, hat{s}^i = f^i(x) end{aligned}$$where (m(cdot , cdot )) is the single-channel evaluation metric computed on permutations of (output, target) pairs.Specifically, we implement two evaluation metrics to assess reconstruction quality, including (1) SI-SDR and (2) downstream classification accuracy. We consider the signal-to-distortion ratio (SDR)2, defined as the negative log squared error normalized by reference signal energy5. However, as is commonly implemented in the human speech separation literature, we instead compute the scale-invariant SDR (SI-SDR), which disregards prediction scale by searching over gains5,40. Explicitly, SI-SDR((hat{s}, s) = -10log _{10}(|hat{s} – s|^2) + 10log _{10}(|alpha s|^2)) for optimal scaling factor (alpha = hat{s}^Ts / |s|^2).Additionally, to provide a physically interpretable metric, we evaluate the performance of the trained classifier models in labeling separated waveforms according to the predicted identity of the vocalizer. This metric assumes that the classification accuracy on a downstream task reflects the fidelity of the estimated signal relative to the ground truth source and thus serves as a proxy for reconstruction quality. More

Phylogenetic analysisFor both the mycobiont and photobiont molecular phylogenies from multi-locus sequence data (nrITS, mtSSU and RPB1 for the mycobiont (140 samples) and nrITS, psbJ-L and COX2 for the photobiont (139 samples) were inferred (Supplementary Figs. S1 and S3 online). Additionally, phylogenies based solely on the marker nrITS were calculated (Supplementary Figs. S2 and S4 online), to include samples where the additional markers were not available. Both analyses include only accessions from the study sites (Fig. 1; Table 1). The phylogenies based on the multi-locus data were congruent to the clades of the phylogenies based on the marker nrITS. Thus, in the following, the focus will be only on the latter.MycobiontThe final data matrix for the phylogeny based on the marker nrITS comprised 306 single sequences with a length of 550 bp. It included sequences of the families Lecanoraceae and Lecideaceae. The phylogenetic tree was midpoint rooted and shows a total of 19 strongly supported clades on species level, assigned to five genera. The backbone is not supported and therefore the topology will not be discussed. All genera are clearly assigned to their family level and are strongly supported. Only Lecanora physciella forms an extra clade as sister to the families Lecideaceae and Lecanoraeae, which is not the case at the multimarker phylogeny. L. physciella has still an uncertain status, because of morphological similarities to both sister families6. The clade of the genus Lecidea revealed seven species (L. andersonii, L. polypycnidophora, L. UCR1, L. sp. 5, L. lapicida, L. cancriformis and L. sp. 6), Lecanora five species (L. physciella, L. sp. 2, L. fuscobrunnea, L. cf. mons-nivis, L. sp. 3), Carbonea three species (C. sp. URm1, C. vorticosa, C. sp. 2), and Lecidella three species (L. greenii, L. siplei, L. sp. nov2). The samples allocated to the genus Rhizoplaca were monospecific (R. macleanii). The taxonomical assignment of the obtained sequences were based on the studies of Ruprecht et al.48 and Wagner et al.10.PhotobiontThe final data matrix for the phylogeny based on the marker nrITS comprised 281 single sequences with a length of 584 bp. The phylogenetic tree was midpoint rooted and shows six strongly supported clades, assigned to seven different OTU levels67, using the concept of Muggia et al.51 and Ruprecht et al.48. All of the OTUs belong to the genus Trebouxia (clades A, I, S), comprising Tr_A02, Tr_A04a, Tr_I01, Tr_I17, Tr_S02, Tr_S15 and Tr_S18. Photobiont sequences taken from Perez-Ortega et al.50, which were labelled only with numbers, were renamed to assign them to the appropriate OTUs48.Analysis of spatial distributionIn general, the most common mycobionts species were Lecidea cancriformis (94 of the 306 samples), Rhizoplaca macleanii (51 samples) and Lecidella greenii (37 samples), followed by Carbonea sp. 2 (13 samples), C. vorticosa (11 samples), Lecidea polypycnidophora (10 samples) and Lecidella siplei (10 samples; see Supplementary Fig. S5 online). Nine mycobiont species were found exclusively in area 5 (MDV, 78°S): Carbonea vorticosa, Lecanora cf. mons-nivis, L. sp. 2, Lecidea lapicida, L. polypycnidophora, L. sp. 5, L. sp. 6, L. UCR1 and Rhizoplaca macleanii. On the other hand, only Lecidea cancriformis was found in all the six areas; Lecanora fuscobrunnea was present in all the areas with the exception of area 2.The most common photobiont OTUs were Tr_A02 (165 of the 281 samples) and Tr_S02 (59 samples), both of them occurring in all the six different areas, followed by Tr_S18 (32 samples), Tr_S15 (10 samples, confined to area 5) and Tr_I01 (10 samples). However, of the 149 photobiont accessions of area 5, 134 (89.93%) were assigned to Tr_A02. This percentage is much higher than in the other areas (area 1: 44.44%, area 2: 69.23%, area 3: 21.74%, area 4a: 7.69%, area 4b: 6.67%), even if those samples with mycobionts occurring exclusively in area 5 (see above) were excluded (76.56% of the 64 remaining samples are assigned to Tr_A02).The alpha, beta and gamma diversity values are given in Table 2. For the mycobionts, the alpha diversity of the communities was the highest in area 5 (8.93, which results in nine species) and the lowest in area 4b (two species, 1.88). In contrast, for the photobionts, the lowest alpha diversity value was found in area 5 (two OTUs, 1.50) and the highest in area 4a (four OTUs, 4.06). Thus, referring to this, area 5 plays a remarkable role: compared to the other areas, it shows the highest diversity of mycobiont species on the one hand and the lowest diversity of photobiont OTUs on the other hand.Table 2 Number of lichen samples, number of identified mycobiont species and photobiont OTUs, as well as alpha, beta and gamma diversity values of mycobiont species/photobiont OTUs for the different areas.Full size tableThe beta diversity values (diversity of local assemblages) for mycobiont species and photobiont OTUs are quite similar (1.69 and 1.64, respectively). This is in contrast to gamma diversity values: the overall diversity for the different areas within the whole region is much higher for the mycobionts (ten species, 9.92) than for the photobionts (three OTUs, 3.35).For mycobionts, the overall sample coverage equals to 0.993. That means that the probability for an individual of the community to belong to a sampled species is 99.3%, or, from another point of view, the probability for an individual of the whole community to belong to a species that has not been sampled is 0.7%. The sample coverage is highest for area 4b (1.000) and lowest for area 2 (0.771). Sample coverage values of the other areas are in between (area 1: 0.895, area 3: 0.931, area 4a: 0.939, area 5: 0.981). The rarefaction/extrapolation curves for the mycobiont species (see Supplementary Fig. S6a) suggest that for any sample size up to the specified level of sample coverage of 0.95, alpha diversity within area 4b is significantly lower than alpha diversity within any other area, and alpha diversity within area 5 is significantly greater than that of area 4a and 4b (based on 95% confidence intervals).For photobionts, the overall sample coverage as well as the sample coverages of area 1, area 2, area 3, area 4b as well as area 5 is equal 1.000. Only the sample coverage of area 4a (0.951) differs. The rarefaction/extrapolation curves for the photobiont OTUs (see Supplementary Fig. S6b) suggest that for any sample size up to the specified level of sample coverage of 0.95, alpha diversity within area 1 is significantly lower than alpha diversity of area 3 and 4a and significantly greater than that of area 5. Alpha diversity of area 5 is significantly lower than that of area 1, area 3 and area 4a.Influence of environmental factors (elevation, precipitation and temperature)First, the proportion of the OTU Tr_A02 samples was significantly correlated to BIO10 means of the areas (R = 0.87, p = 0.022; see Supplementary Fig. S7 online): the higher the temperature mean values of the warmest quarter of an area, the higher the proportion of samples containing photobionts that are assigned to Tr_A02.The alpha diversity values of mycobiont species significantly positively correlated with BIO10 (R = 0.88, p = 0.021; see Supplementary Fig. S8 online): the higher the temperature mean values of the warmest quarter, the higher the mycobiont diversity within this particular area.Furthermore, the differences in mycobiont species community composition were significantly related to BIO10 (constrained principal coordinate analysis: F = 14.7137, p = 0.001, see Supplementary Fig. S9 online), BIO12 (F = 2.7535, p = 0.012), elevation (F = 2.5108, p = 0.025) and the geographic separation of the samples (Mantel statistic r = 0.1288, p = 0.0002).The differences in community composition of photobiont OTUs were related significantly to BIO10 (constrained principal coordinate analysis: F = 48.5952, p = 0.001, see Supplementary Fig. S10 online), BIO12 (F = 4.4848, p = 0.008), elevation (F = 6.8608, p = 0.002), and physical distance (Mantel statistic r = 0.4472, p = 0.0001).Haplotype analysisHaplotype networks were computed for the mycobiont species and photobiont OTUs with h ≥ 2 and at least one haplotype with n ≥ 3 (Carbonea sp. 2, Lecanora fuscobrunnea, Lecidea cancriformis, Lecidella greenii, L. siplei, L. sp. nov2 and Rhizoplaca macleanii, as well as Tr_A02, Tr_I01 and Tr_S02), in both cases based on nrITS sequence data (Figs. 2, 3). The samples of Carbonea vorticosa (11) were all assigned to a single haplotype, which was also true for Lecidea polypycnidophora (10 samples), Tr_S15 (10 samples) and Tr_S18 (32 samples). Figure 3b, c illustrate the subdivision of Tr_I0151 into Tr_I01j35,48 and Tr_I01k (in this study), and the subdivision of Tr_S02 into Tr_S0235, and Tr_S02b and Tr_S02c48.Figure 2Haplotype networks of mycobiont species with h ≥ 2 and at least one haplotype with n ≥ 3, showing the spatial distribution within the different areas, based on nrITS data. (a) Carbonea sp. 2, (b) Lecanora fuscobrunnea, (c) Lecidea cancriformis, (d) Lecidella greenii, (e) Lecidella siplei, (f) Lecidella sp. nov2, (g) Rhizoplaca macleanii. Roman numerals at the center of the pie charts refer to the haplotype IDs; the italic numbers next to the pie charts give the total number of samples per haplotype. The circle sizes reflect relative frequency within the species; the frequencies were clustered in ten (e.g. the circles of all haplotypes making up between 20 and 30% have the same size). Note: only complete sequences were included.Full size imageFigure 3Haplotype networks of photobiont OTUs with h ≥ 2 and at least one haplotype with n ≥ 3, showing the spatial distribution within the different areas, based on nrITS data. (a) Tr_A02, (b) Tr_I01, (c) Tr_S02. Roman numerals at the center of the pie charts refer to the haplotype IDs; the italic numbers next to the pie charts give the total number of samples per haplotype. The circle sizes reflect relative frequency within the species; the frequencies were clustered in ten (e.g. the circles of all haplotypes making up between 20 and 30% have the same size). Note: only complete sequences were included.Full size imageThe haplotype networks include pie charts showing the occurrence of the different haplotypes within the different areas. All haplotypes of Rhizoplaca macleanii are restricted to area 5, as well as Lecidella greenii mainly to area 5 and areas 1 and 4a, and Lecidella sp. 2 to areas 2 and 3. However, all other species do not suggest a spatial pattern with different haplotypes being specific for different areas. Moreover, the distribution turned out to be rather unspecific, with a great part of the haplotypes found in multiple areas. For the sake of completeness, additionally, haplotype networks based on multi-locus sequence data were computed for the most abundant mycobiont species and photobiont OTU with multi-locus data available (Lecidea cancriformis and Tr_S02). Not surprisingly, those networks show a greater number of different haplotypes, but they also do not allow conclusions concerning spatial patterns of area specific haplotypes (see Supplementary Fig. S11 online).Diversity and specificity indices of mycobiont species and photobiont OTUsThe diversity and specificity indices for the different mycobiont species and photobiont OTUs are given in Supplementary Table S8 online.For the sample locations of mycobiont species with n ≥ 10, BIO10 was strongly correlated to the specificity indices NRI (net relatedness index) and significantly correlated to PSR (phylogenetic species richness) and 1 – J′ (Pielou evenness index). BIO12 was significantly correlated to NRI, PSR and 1 – J′. Figure 4 illustrates these correlations: the higher the BIO10 and BIO12 mean values, the higher was the NRI (phylogenetic clustering of the photobiont symbiotic partners), the lower was the PSR (increased phylogenetically relatedness of photobiont symbiotic partners) and the higher was 1 – J′ (less numerically evenness of the photobiont symbiotic partners). Thus, for the mean values of the sample locations of a mycobiont species, a comparatively high temperature of the warmest quarter and high annual precipitation occurs with associated photobionts that are phylogenetically clustered and closer related to each other. The lowest values of NRI and the highest values of PSR were developed by Lecidea cancriformis and Lecanora fuscobrunnea, which also showed the lowest BIO10 and BIO12 mean values at their sample sites. On the contrary, the highest values of NRI and PSR were developed by Rhizoplaca macleanii, which also had the highest BIO10 and BIO12 means.Figure 4Correlation plots. Specificity indices NRI (net relatedness index), PSR (phylogenetic species richness and 1 – J′ (Pielou evenness index) against mean values of BIO10 (mean temperature of warmest quarter) and BIO12 (annual precipitation) for mycobiont species with n ≥ 10.Full size imageFor the sample locations of photobiont OTUs with n ≥ 10, elevation significantly negatively correlated with h (number of haplotypes) and Hd (haplotype diversity): the higher the mean elevation of sample sites, the lower the number of haplotypes and the lower the probability that two randomly chosen haplotypes are different (Fig. 5). The highest values of h and Hd were shown by Tr_A02, Tr_I01 and Tr_S02, which occurred at sample sites with comparatively low elevations. In contrast, Tr_S15 and Tr_S18 occurred at very high elevations and showed very low values of h and Hd.Figure 5Correlation plots. Diversity indices h (number of haplotypes) and Hd (haplotype diversity) against mean elevation of sample sites for photobiont OTUs with n ≥ 10.Full size imageAnalysis of mycobiont–photobiont associationsBipartite networks were calculated for all associations between mycobiont species (lower level) and the respective photobiont OTUs (higher level) for all areas (Fig. 6). The H2′ value (overall level of complementary specialization of all interacting species) was highest in area 2 (0.921), indicating a network with mostly specialized interactions: within this network, with the exception of Lecidea andersonii, the mycobiont species are associated exclusively with one single photobiont OTU. The second highest H2′ value was developed by area 4b (0.710); in contrast, area 4a showed the lowest H2′ value (0.260), with the most abundant mycobiont species Lecidea cancriformis showing associations with five different photobiont OTUs. The H2′ values of area 1, area 3 and area 5 indicate medium specification.Figure 6Bipartite networks showing the associations between mycobiont species and photobiont OTUs for the different areas. Rectangles represent species/OTUs, and the width is proportional to the number of samples. Associated species/OTUs are linked by lines whose width is proportional to the number of associations.Full size imageIn addition, the bipartite networks illustrate the different occurrence of mycobiont species and photobiont OTUs within the different areas: For example, in area 1 (and area 2), five (seven) different mycobiont species are associated with only three different photobiont OTUs. In contrast, in area 4b, only two different mycobiont species are associated with four different photobiont OTUs. In area 5, the number of associated photobiont OTUs is also four, but those four OTUs are associated with 16 different mycobiont species.The network matrix giving all the associations between the mycobiont species and photobiont OTUs is presented in Supplementary Table S9 online. More

Bigg’s killer whale photo-identification datasetThe dataset of this study includes photos of Bigg’s killer whale individuals accumulated over a period of 8 years (2011–2018), from the coastal waters of southeastern Alaska down to central California15. None of these animals were directly approached explicitly for this study. All photo-identification data was collected under federally authorized research licenses or from beyond mandated minimum viewing distances.Supplementary Figure S1 visualizes a series of example images of this dataset. Each image contains one or more individuals. In addition to the identification name of the individual(s), further metadata such as photographer, GPS-coordinates, date, and time are provided. Every identification label is an alphanumeric sequence based on the animals’ ecotype (T—Transient), order of original documentation (e.g. T109), and order of birth (e.g. T109A2—the second offspring of the first offspring of T109)15.A parsing procedure was designed to verify, analyze, and prepare the image data, guaranteeing adequate preparation for subsequent machine (deep) learning methods. Results of the entire data parsing procedure are presented in Fig. 2 and Supplementary Table S1. Figure 2 visualizes the number of identified individuals, together with the total amount of occurrences in descending order, considering (1) all images, and (2) only photos including a single label. General statistics with respect to the entire dataset are reported in the caption of Fig. 2. Supplementary Table S1 illustrates the 10 most commonly occurring individuals across all 8 years of data, considering all images including single and multiple labels, compared to photos only containing a single label.The dataset exhibits a substantial class imbalance, as evidenced by the exponential decline in frequencies per killer whale individual (see Fig. 2). Especially for real-world datasets, such unbalanced data partitioning is a common and well-known phenomenon, also referred to as long-tailed data distribution79. Such long-tailed data distributions are divided into two sections79: (1) the Head region—representing the most commonly identified killer whale individuals, and (2) the Long-Tail region—visualizing a significantly larger number of killer whale individuals, however, with considerably less occurrences. For the purpose of this pilot study, the top-100 most commonly occurring killer whale individuals were selected for supervised classification and as boundary between the head and long-tail area (see Fig. 2). The defined boundary of the top-100 killer whales (head region) represents approximately 1/4 (100 out of 367) of the individuals, however, covering about 2/3 (55,305 out of 86,789) of the entire dataset of single-labeled images.Figure 2Bigg’s killer whale image long-tailed data distribution (2011–2018), summing up a total of 121,095 identification images, with 86,789 containing single labels, as well as 34,306 photos including multiple labels, resulting in 367 identified individuals (average number of images per individual (approx)456, standard deviation (approx)442). The two colored graphs visualize the number of identification images per whale in descending order w.r.t. all images, including single and multiple labels (purple curve) and those only containing a single label (green curve). Furthermore, an exemplary data point is visualized for both curves, presenting the number of identification images in relation to a selected number of whales, here for the top-100, clearly describing the exponential decline. Moreover, the number of animals at which the total amount of identification images is More

Selection-switch experimentThe dataset used for this article is previously published14, but we include a brief summary for completeness: Natural seawater was collected and used to inoculate microcosms in a 2 × 2 factorial crossover design with 3 replicates conducted for 50 days, which were sampled 18 times during the experiment. Half of the microcosms were given high (H) resource supply, whereas the other half were given low (L) resource supply. The factor of resource supply level was constant throughout the experiment. The other factor was the selection regime, which meant that the microcosms were either given continuous supply of nutrients (favouring K-selection, and hence the designation K) or being pulse-fed with nutrients after diluting the contents of the microcosms with growth medium (favouring r-selection, designated R). The active selection regime was switched at the experimental halfway point (between days 28 and 29), yielding two selection groups designated as RK and KR.DNA was extracted from the collected samples, and the V3-V4 region of the bacterial 16S-rRNA gene was amplified with PCR using broad-coverage primers and the index sequences were ligated. The amplicon library was pooled and sequenced with two runs on an Illumina MiSeq machine. The reads are available at the European Nucleotide Archive with accession number ERS7182426-ERS7182513.The USEARCH pipeline47 (v11) was used to remove low-quality reads and cluster the reads into OTUs at 97% similarity level. Finally, the taxonomy of the OTUs was determined by the Sintax classifier using data from the RPD training set (v.16) where the confidence threshold was set to 80%.Quantification of bacterial densityFor each sample, the bacterial density was quantified using flow cytometry (BC Accuri C6)14. In brief, the bacterial communities were diluted in 0.1x TE buffer, mixed with 2x SYBR Green II RNA gel stain (ThermoFisher Scientific) and incubated in the dark at room temperature for 15 minutes. Then, each sample was measured for 2.5 minutes at 35 μL min−1 with an FL1-H (533/30 nm) threshold of 3000. We gated the bacterial population as those events with an FL1-A ( > 10^4) and FSC-A (< 10^5). The raw flow cytometry data files are available at https://doi.org/10.6084/m9.figshare.15104409.Alignment and phylogentic treeThe selection-switch dataset was acquired directly from the authors14. This dataset consists of a total of 206 samples. Two of these samples were taken from the communities from which the reactors were inoculated, whereas the other samples were taken from the microcosms with 17 time points x 4 regimes x 3 replicates. We discarded the inoculum samples for further analysis. The OTU reference sequences were aligned with SINA version 1.6.148 using the SILVA Release 138 NR 99 SSU dataset49. Using this aligment, the phylogentic tree was constructed by neighbour-joining using MEGA X50 with default parameters.Filtering and preprocessingThe mean number of reads per sample was 63,460 with standard deviation 31,411. For our analysis, we wanted to estimate the abundance of each OTU as accurately as possible and therefore skipped any correction for unequal sequencing depth. Read counts for each OTU in each sample were divided by the total number of reads for the sample, generating relative abundances. Thereafter, all OTUs having a maximum abundance (across all samples) below a certain threshold, were removed. Three levels of filtering thresholds (as count proportions) were applied: High level at ( 5cdot 10^{-3} ), medium level at ( 1cdot 10^{-3} ) and low level at ( 5cdot 10^{-4}). The purpose of the filtering was to remove rare OTUs in order to avoid noise and spurious correlations11. For obtaining estimates of absolute abundances, the relative abundances were scaled by the estimate of total bacterial cell density for each sample. The phyloseq package (version 1.36.0)51 and the R programming language (version 4.1.1)52 facilitated this procedure. In addition, we wrote an R-package named micInt (version 0.18.0, available at https://github.com/AlmaasLab/micInt) to facilitate and provide a pipeline for the analysis.Similarity measures and addition of noiseFor this study, we used two similarity measures, the Pearson correlation and the Spearman correlation. A similarity measure, as referred to in this article, can be thought of as a function (f: mathbb {R}^ntimes mathbb {R}^n rightarrow D) where ( D = [-1,1] ). In this regard, (fleft( {mathbf {x}},{mathbf {y}}right) ) is the similarity of two abundance vectors ( {mathbf {x}} ) and ({mathbf {y}}) belonging to different OTUs, where (fleft( {mathbf {x}},{mathbf {y}}right) = 1) indicates perfect correlation, (fleft( {mathbf {x}},{mathbf {y}}right) = 0) indicates no correlation and (fleft( {mathbf {x}},{mathbf {y}}right) = -1) indicates perfect negative correlation. Noise was added to distort patterns of double zeros, which otherwise could result in spurious correlations. Given two vectors ( {mathbf {x}} ) and ( {mathbf {y}} ) of abundances, normally distributed noise was added to each of the abundance vectors, and the similarity measure has invoked thereafter: Given a similarity measure f, the similarity between the abundance vectors after adding noise is given by:$$begin{aligned} f^*left( {mathbf {x}},{mathbf {y}}right) =fleft( {mathbf {x}} +varvec{varepsilon _x},{mathbf {y}}+varvec{varepsilon _y }right) , end{aligned}$$ (1) where (varvec{varepsilon _x}) and ( varvec{varepsilon _y} ) are random vector where all components are independent and normally distributed with mean zero and variance ( gamma ^2 ). The level of noise ( gamma ) was determined by the smallest non-zero relative abundance ( x_{mathrm {min}} ) in the dataset and a fixed constant s called the magnitude factor, such that ( gamma = scdot x_{mathrm {min}}). For no noise, ( s=0 ), for low noise ( s=1 ), for middle noise ( s=10 ) and for high noise ( s=100 ).Network creationSignificance of the pairwise OTU associations were determined by the ReBoot procedure introduced by Faust et al.22 and shares the underlying algorithm used in the CoNet Cytoscape package53. This approach accepts a dataset of microbial abundances and a similarity measure, and evaluates for each pair of OTUs in the dataset the null hypothesis ( H_0 ): “The association between the OTUs is caused by chance”. By bootstrapping over the samples, the similarity score of each pair of OTUs is estimated, forming a bootstrap distribution. By randomly permuting the pairwise abundances of OTUs and finding the pairwise similarity scores, a bootstrap distribution is formed. The bootstrap and permutation distribution are then compared with a two-sided Z-test (based on the normal distribution) to evaluate whether the difference is statistically significant. For this, the z-value, p-value and q-value (calculated by the Benjamini-Hochberg-Yekutieli procedure54) are provided for each pair of OTUs in the dataset. Our ReBoot approach is based on the R-package ccrepe (version 1.28.0)55, but is integrated into the micInt package with the following major changes: The original ReBoot uses renormalization of the permuted abundances to keep the sum-to-constant constraint. Whereas this is reasonable to do with relative abundances, our modified version enables turning this feature off when we analyse data with absolute abundances. Optimizations have been made to memory use and CPU consumption to enable analyses of large datasets. In contrast to the usual ReBoot procedure, networks generated by the different similarity measures are not merged by p-value, but kept as they are. For our analysis the number of bootstrap and permutation iterations was set to 1000. All OTUs being absent in more than ( ncdot 10^{-frac{4}{n}} ) samples, where n is the total number of samples, were excluded through the errthresh argument but still kept for renormalization (if turned on). The associations were made across all samples, even the ones belonging to a different selection group or resource supply.Dynamic PCoA visualizationAll samples in the dataset were used for PCoA ordination, where the Bray-Curtis distance metric between the samples was applied before creating the decomposition. After the ordination was computed, the samples were divided into four facets based on their combination of current selection regime and resource supply. Finally, all samples belonging to the same microcosm were connected by a line in chronological order and the line was given a separate style based on the resource supply and coloured to visually distinguish it from the two other replicate microcosm within the same facet.Permutational multivariate analysis of varianceSequential PERmutational Multivariate Analysis of VAriance (PERMANOVA) of the samples was conducted on the absolute abundances, where only the samples from day 28 and 50 were included. These sample points correspond to time just before the experimental selection-regime crossover and a point at the end of the experiment. These days were selected because they were the most likely to capture the composition of stable communities in contrast to transient ones. The procedure was carried out by the function adonis from the R package vegan (version 2.5-7) with ( 10^6 ) permutations. The dependent data given to the function was the matrix of one minus the Spearman correlation of the samples (in order to resample dissimilarity), while the independent variables were the selection group (first variable) and the current selection regime (second variable).Network visualizationThe networks were plotted by the R package igraph (version 1.2.6)56. Network modules were found by the walktrap25 algorithm implemented in igraph with the setting steps=20, including the positive edges only. Later, the negative edges were added and the networks plotted with the community labelling.The time dynamics of the networks were visualised by taking the former network and adjusting the node colour and size, as well as the edge colour. For this, a certain combination of selection group (i.e RK) and resource supply (i.e H) was chosen. Further, let (x_{i,j,k} ) be the abundance of OTU k at sampling day i in microcosm j. As there are three replicates, we have that ( j= 1,2,3). If the underlying network was created by Pearson correlation, we denote the day mean ( x_{i,.,k} ) as the average over the replicates, this is:$$begin{aligned} x_{i,.,k}= frac{x_{i,1,k}+x_{i,2,k}+x_{i,3,k}}{3}. end{aligned}$$ (2) The time series mean of OTU k, (x_{.,.,k} ) is the mean of these daily means over all sampling days,$$begin{aligned} x_{.,.,k} = frac{sum _{i=1}^{N}x_{i,.,k}}{N}, end{aligned}$$ (3) where N denotes the number of sampling days. Furthermore, we have the associated standard deviation (sigma _k) as given by:$$begin{aligned} sigma _k =sqrt{ frac{1}{N}sum _{i=1}^{N}left( x_{i,.,k}-x_{.,.,k}right) ^2}. end{aligned}$$ (4) The z-value of the abundance of OTU k at day i is then:$$begin{aligned} z_{i,k} = frac{x_{i,.,k}-x_{.,.,k}}{sigma _k}. end{aligned}$$ (5) This value is used in the mapping of the node sizes and colours. The node for OTU k at sampling day i has the size ( a+bcdot left| z_{i,k}right| ), where a and b are constants. Furthermore, the same node is coloured: Black if ( z_{i,k} < -1 ). This indicates that the OTU that day had a lower abundance than the average. Grey if (-1 le z_{i,k} le 1 ). This indicates that the OTU that day had about the same abundance as the average. Orange if ( z_{i,k} > 1 ). This indicates that the OTU that day had a higher abundance than the average.

Furthermore, the edge colour are dependent on the product of the two participating nodes. Hence, the edge between OTU k and OTU l at day i will have the colour:

Red if ( z_{i,k}cdot z_{i,l} < -0.3 ). This shows a contribution to a negative interaction. Gray if (-0.3 le z_{i,k}cdot z_{i,l} le 0.3 ). This shows no major contribution of neither a positive nor negative interaction. Blue if (z_{i,k}cdot z_{i,l} > 0.3 ). This shows a contribution to a positive interaction.

Our approach is motivated by the fact that the Pearson correlation ( rho _{k,l} ) of the day means of OTU k and OTU l is given by:$$begin{aligned} rho _{k,l} = frac{1}{N} sum _{i=1}^{N} z_{i,k}cdot z_{i,l}. end{aligned}$$
(6)
For the Spearman correlation, the visualization is based on the rank of each of the OTU abundance values in a sample. Hence, instead of using the raw abundances ( x_{i,j,k} ) in the calculation of the day mean, the ranks ( r_{i,j,k} ) are used instead, and all subsequent calculations and mappings are the same. In a scenario when there is only one replicate, the quantity ( rho _{k,l} ) would then be the Spearman correlation of the abundances of OTU k and OTU l. More