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AS systems display many novel and a high fraction of shared virusesWe used metagenomic sequencing of 30 Gb of sequences per sample to characterize the composition of viral concentrates across six WWTPs (see Methods). After assembly and mapping of reads, 24–34% of the total sequence information could be classified as viral using two different identification pipelines (see Methods). By combining the results of these two pipelines, the final data set consisted of 50,037 viral contigs with an N50 >20 kb. To evaluate whether our sequencing effort sufficiently sampled the viromes, all six WWTP samples were subsampled iteratively to evaluate the saturation dynamics. Rarefaction curves of the number of viral contigs, reached a plateau ~15 Gb of sequencing data for all six samples (Fig. 1a), indicating adequate recovery of prokaryotic viruses in these AS systems at the sequencing depth (30 Gb per sample) in this study.Fig. 1: Compositional variation in viromes among wastewater treatment plants.a Rarefaction curve of each sample. b Rank-abundance curve of each sample. c PCoA analysis of viromes based on the relative abundance of viral genera. d Relative abundance and appearance of dominant viral genera in each WWTP. White color denotes no appearance in this WWTP. Source data are provided as a Source Data file.Full size imageViral contigs were further classified into 8756 viral clusters (VC, equivalent to viral genera) using vConTACT2 by calculating the gene-content-based distance between viral contigs (~40% proteome similarity)11, and each VC was assigned an ID for identification (mean length = 15.2 kb, mean genera size = 3). Compared with the current number of viral sequences from AS systems, our sequencing data increase the AS virome database (N = 2103 in the IMG/VR database v.2.0)10 by 12-fold at the genus level and by 23-fold at the species-level (95% identity, 80% coverage). Comparison with NCBI RefSeq viral genome database showed that across the six AS systems, only 0.4–1.6% of total viral contigs (coverage percentage) could be assigned to a known viral family. Similar to previously described viral metagenomes from the soil, freshwater, and marine system7,12,13, this limited annotation highlights substantial uncharacterized viral diversity in AS communities. Among these recognizable viruses, members of the family Podoviridae (short-tailed phages from the Caudovirales order) were the most prevalent, comprising on average 41.3% of these viral contigs (coverage percentage) across the six WWTPs.All samples displayed high but variable diversity of viral genera with Shannon’s diversity index H’ ranging from 5.22 to 7.14 and Pielou’s evenness index J’ ranging from 0.71 to 0.86 (Supplementary Data 1). These differences are evident in rank-abundance curves, which show that each sample has different viral frequency patterns (Fig. 1b). Most viruses occurred at low frequency with the relative abundance of individual genera diminishing below 0.1% after counting the top 138 viral genera.Principal coordinate analysis (PCoA) of the Bray–Curtis dissimilarity based on the relative abundance of viral genera suggested that most samples are divergent from each other (Fig. 1c), with only two pairs of AS viromes, ST and STL as well as SK and SWH, displaying higher similarity to each other.The overall variability in the viromes is also reflected in different dominance patterns. Each AS sample yielded a different dominant viral genus, and while these were also abundant in some WWTPs, they were below the detection limit in others (Fig. 1d). Although high relative abundance across all WWTPs indicates linkage to consistently abundant hosts, highly variable occurrence suggests that host populations are also more dynamic.Although the viromes appear overall variable in rank abundance, many viral genera were shared across the WWTPs. Fifty-three viral genera were detected in all samples and were thus considered to be common members of the AS viromes, accounting for 1.7–5.4% of viral contigs (coverage percentage) in each WWTP (Fig. 2). Thirteen of these common viral genera were also present in AS viromes in the IMG/VR database v.2.010. Of the total of 8756 unique viral genera collected across samples, STL and ST contained the largest fraction (5245 and 5149 genera, respectively) and shared most viral genera (N = 2885) with each other, far exceeding the number of all the other shared or unique viral genera (Fig. 2). These two WWTPs also had only 45 and 64 site-specific viral genera, consistent with the pattern in the PCoA virome profile. On the other hand, SWH possessed the most unique viral genera (N = 323), making up 11.0% of the total (Fig. 2). In fact, only relatively few viral genera were found exclusively in one of the WWTPs.Fig. 2: Shared viral genera in each WWTP.The UpSet55 chart shows the total number of viral genera and their sharedness in each WWTP. The bar chart on the top right shows the distribution of predicted hosts for common viral genera in all WWTPs. Shared viral genera between ST and STL were labeled orange and shared viral genera in all WWTPs were labeled red. Source data are provided as a Source Data file.Full size imageOverall, these results suggest that the virome across WWTPs consists of many shared genera. The lack of detection of some viral genera in the AS virome of one WWTP may be primarily due to the biological variation in the grab samples and/or the technical variation. Hence, if such technical and biological variations are taken into account, the virome shared among all AS maybe even more diverse.Viruses infect a broad spectrum of bacteria and archaeaTo examine putative host associations of all 50,037 viral contigs in the six WWTPs, we amassed a database of approximately three million CRISPR-Cas spacers from the NCBI prokaryotic complete genomes and metagenomes database (https://www.ncbi.nlm.nih.gov/assembly/). Host prediction was performed by matching CRISPR-Cas spacers at a sequence identity above 97%, sequence coverage over 90% in length, and mismatches 1000 ORFs in each sample were found in the following categories, L: replication, recombination, and repair, M: cell wall/membrane/envelope biogenesis, and K: transcription, confirming that the main functions sustain viral reproduction and transcription. It is noteworthy that viruses encode on average 541 ORFs (1.4% of annotated viral ORFs) in each sample in category G: carbohydrate transport and metabolism. After removing redundant viral ORFs, most unique ORFs (N = 1610) were classified into the glycoside hydrolases (GH) module24 (Fig. 5b). These GHs may be involved in the digestion of capsules to allow the viral tail to reach its membrane receptor on the host. The high representation of this function may be explained by the prevalence of biofilm formation among AS microbes and has previously also been noted among mangrove sediment viruses25.Fig. 5: Distribution of auxiliary metabolic genes (AMGs) relevant to the carbon cycle and nutrient removal.a Boxplot of the overall gene profile for six WWTPs was summarized both as viral ORF hits and viral ORF relative abundance (ORF hits to each COG function class/total ORFs that have hits to eggNOG database). Data in a are presented as mean values (center) and 25%, 75% percentiles (bounds of box). The minima and maxima represent the range of the data. b Number of unique viral ORFs related to each CAZy function class was shown in a descendant order. Source data are provided as a Source Data file.Full size imagePrevious work has shown that many prokaryotic viruses carry auxiliary metabolic genes (AMGs), which can modulate host energy metabolism to provide an energetic advantage during viral genome and protein synthesis26. Broadly speaking, AMGs refer to all metabolic genes in lytic phages27, i.e., all genes in categories C, E, F, G, H, I, and P. Though only about a quarter of the viral ORFs have annotations in the eggNOG database, our data reveal a large repertoire of potential AMGs in the AS viromes (Fig. 5a). For example, 72 ORFs in total are potentially involved with carbon fixation pathways and 35 of them annotate as photosynthetic carbon fixation pathways in KEGG28. Moreover, 10 viral ORFs belong to CobS genes, which are essential for the biosynthesis of cobalamin. Cobalamin biosynthesis pathway is usually not complete in bacterial genomes29, and these viral encoded CobS genes could possibly assist the host metabolic capability. Seven viral genes encode adenylyl-sulfate kinase (CysC), which could facilitate host’s assimilatory sulfate reduction. By modulating host metabolism during infection, AMGs could alter the specific functions of their hosts in WWTPs and therefore influence carbon cycling and the removal of nutrients.Viromes are shared between WWTPs and the water environmentTo investigate whether the WWTP viral genera also occur in other habitats, we compared all viral sequences recovered here with the IMG/VR database v.2.0, which consists of 735,112 viral contigs predicted from metagenomic data10. Viral sequences from five ecosystems (AS, AD, solid waste, freshwater, marine) were included for comparison. For AS (N = 2103), AD (N = 8580), and solid waste (N = 5760), all viral sequences were subjected to our viral clustering pipeline, whereas for freshwater and marine environments, we each randomly selected 50,037 viral sequences to match the number in our samples (N = 50,037).Results showed that viral genera in our samples were shared among multiple environments. There was considerable overlap between WWTPs and freshwater (N = 402) and marine (N = 172) environments (Fig. 6). Moreover, 200, 273, and 244 viral genera were shared with AS, AD, and solid waste, respectively (Fig. 6). This represents a higher number than with marine viromes, and if normalized against the dataset size, there is also a larger fraction of connections than with freshwater viromes. When hosts were predicted for these shared viral genera, Proteobacteria was the most shared host phylum, followed by Firmicutes, Actinobacteria, Bacteroidetes, and Cyanobacteria (Supplementary Data 2). At the genus level, Bacillus was the most abundant between our samples and marine, AD, and AS environments, while Streptomyces displayed a higher prevalence between our samples and freshwater and solid waste environments (Supplementary Data 3). Although it is difficult to identify the source and sink dynamics, AS and AD are typical processes in WWTPs, and solid waste viral sequences mainly stem from compost and leachate microbial communities. The considerable sharing of viromes between WWTPs and marine water may be caused by Hong Kong’s extensive application of marine water for toilet flushing, causing the influent sewage of WWTPs to contain a sizable amount of marine water. These data thus suggest that viruses are extensively shared and that the same viral genera may manipulate microbial communities in these different environments.Fig. 6: Connections between viral genera in AS samples and in five ecosystems from IMG/VR database.Pairwise connections were shown in a Circos56 plot between samples and different ecosystems, including AS, AD, solid waste, freshwater, and marine water. Source data are provided as a Source Data file.Full size imageHi-C validation of virus–host interactions in AS systemHigh throughput chromosome conformation capture (Hi-C) method was used to validate the virus–host connections predicted by our CRISPR-based methods using an additional sample in December 2020 at ST WWTP, by referring to viral contigs and host genome bins obtained from direct sequencing using Illumina and Nanopore metagenomic sequencing (Fig. 7).Fig. 7: General workflow to validate the precision of CRISPR-based methods in the present study.For Illumina data, 91% precision was observed. For Nanopore data, 94% precision was observed. Source data are provided as a Source Data file.Full size imageAs for Illumina metagenomic sequencing, 4578 viral contigs were identified and 1695 of them were deconvoluted in Hi-C data to have virus–host interactions with 197 host bins (Supplementary Data 9). To compare the Hi-C results with the CRISPR-based methods, 21 viruses were predicted by BLASTn-short to link with spacers in eight bins (Supplementary Data 10).As for hybrid assembly using both Nanopore and Illumina reads, 2593 viral contigs were identified and 989 of them were deconvoluted in Hi-C data to have virus–host interactions with 144 host bins (Supplementary Data 11). To compare the Hi-C results with the CRISPR-based methods, 28 viruses were predicted by BLASTn-short to link with spacers in 10 bins (Supplementary Data 12).Results show that CRISPR-based results have very high accuracy. For Illumina data, of the 21 virus–host connections predicted using CRISPR spacers, 11 are simultaneously found in Hi-C data and 10 are not detected in Hi-C data. Of 11 detected connections, only 1 is different in Hi-C data and 10 are the same (91% precision) (Supplementary Data 13). Also for the Nanopore/Illumina hybrid data, of the 28 virus–host connections predicted using CRISPR spacers, 16 are simultaneously found in Hi-C data and 12 are not detected in Hi-C data. Of 16 detected connections, only 1 is different in Hi-C data, 15 are the same (94% precision) (Supplementary Data 14).It should be noticed that some of the predicted CRISPR-based virus–host interactions are undetected in Hi-C data. CRISPR spacers represent a collection of memories regarding past virus invasions, whereas Hi-C data provide a snapshot of ongoing virus–host interactions. Also, Hi-C crosslinking may not be 100% efficient and might miss some of the virus–host interactions. More

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The model is organized as follows. We simulated water striders as an array of the discrete “objects” which interact one with another according to more or less natural rules. The interaction includes strong short-range repulsion between the animals. Mathematically, short range repulsion means that one animal cannot penetrate inside a private area of other animal. Normally such repulsion appears at relatively short distances corresponding to a radius (R^{repuls}) of their private territory. Besides, there is mutual attraction at longer distances (R^{attract} > R^{repuls}), which biologically corresponds to a tendency to aggregation4. The tendency to aggregation often gives some competitive advantage due to possible collective reactions on the external challenges (for example, on the attacks of predators).The simplest way to simulate an interaction with regulated characteristic distance is to use Gaussian effective potential with some characteristic radius (R_{0}), since the Gaussian potential guaranty the stability in dynamics simulations with relatively large (Delta t)4. Repulsion force in this case looks as follows:$$f_{j}^{rep} = A_{0} left( {vec{r} – vec{r}_{j} } right)expleft[ { – left( {frac{{vec{r} – vec{r}_{j} }}{{R_{0} }}} right)^{2} } right],$$
(2)
where factor A0  R^{repuls}) attraction normally causes a minimum of the interaction energy at some intermediate distance (R^{min }). In infinite empty space, being used in the equations of motion, such a combination of the forces leads to an ordering of the objects with the equilibrium distance corresponding to the position (R_{min }) of the energy minimum. But, if the area ({ [0,Lx],[0,Ly]}) is limited, the equations of motion must be supplied by appropriate boundary conditions which do not allow the animals to leave this area.The simplest way to introduce the boundary conditions is to apply mathematically “soft” but extremely high and narrow walls around the system, which repulse the animals back to the internal space with exponentially growing force$$f_{j}^{bound} = B_{j}^{bound} exp left[ { – left| {frac{{overrightarrow {r}_{j} – overrightarrow {r}_{bound} }}{{R_{{}}^{bound} }}} right|} right]$$
(5)
acting in the direction opposite to that in which the animal occasionally crosses any of the boundaries. The words “extremely high” mean that the amplitude of this force should be supplied by the pre-factor (B_{j}^{bound}) which is much bigger than the amplitude of the repulsion force between the animals (regulated by the amplitude (B_{{_{jk} }}^{repuls})) to be able to overpower their mutual repulsion (B_{j}^{bound} > > B_{j}^{repuls}) pushing them out the boundaries. Therefore also the characteristic distance of the repulsion from the wall should be much shorter than the typical distance between the animals in the population (R^{bond} < < R^{min }). In this sense the boundary should be as narrow as possible.One can expect that mean density of the population is not extremely high and does not force the animals to stay exactly on the minimal distance (R^{min }) between them. In this case total combination of the attracting and repulsing interactions combined with the rejecting boundary conditions normally leads to the specific patterns where relatively dense groups of the animals spread on the distances close to the equilibrium ones are accompanied by almost empty voids between them5. We have checked this hypothesis numerically many times. Below, such patterns will be seen in all the particular visualizations of the numerical results.It should be noted that mathematically, if number of the animals increases inside of the same limited and already populated area, the individuals tend to fill all the voids with the equilibrium (sometimes, even higher than equilibrium) density. It happens in real population as well in that cases when the number of the animals grows quickly for a self-regulation of the population and they become simply forced to occupy every empty space inside the area.However, normally when the population grows too quickly the growth should be regulated by a number of factors. In particular, it will be restricted by a competition for the food and space accompanied by death of some of the participants of the process. From mathematical point of view it means that the equations of motion, which will be written below, must be accompanied by natural generation of the new individuals as well as by their (reasonable) disappearance from the system.It is almost impossible to write such a generation in analytical form, but in numerical model it can be formally presented as a set of natural rules. First of all the length of the array is supposed to be a variable integer. Let us suppose that at every step of calculation (Delta t) the length of the array can potentially increase to the new one (N to N + 1). The potential act of the generation becomes real if the numerically generated random number (varsigma) uniformly distributed in interval [0;1] is less than some threshold (varsigma_{thres} < 1). In the numerical procedure the rate of the generation is certainly regulated by (varsigma_{{{text{thres}}}}) and can be extremely low for example, if (varsigma_{thres} < < 1).New member of the array is generated with random coordinates within the prescribed area ({ [0,Lx],[0,Ly]}) with the mass ((m_{N + 1})) randomly distributed around a starting (basically small) one (m_{s}). Qualitatively it means that at every time moment the array can be supplemented with some probability by a new member which is physically placed in some arbitrary place inside the area ({ [0,Lx],[0,Ly]}). At the same time, we have to introduce a process of disappearance of some of the array members. Natural criterion for this will be established below from the checking of the accumulation or loss of the mass (e.g. reserve fat mass) (m_{j}) by every individual. It is supposed that, if an animal gets critical size (which can be both: maximal (m_{max }) or minimal (m_{min })) it disappears from the system and the length of the array decreases (N to N - 1).Both of such events must be adjusted numerically to some rate natural for a particular biological system. It is quite expected from the very beginning, that there should be a kind of balance between the creation and disappearance rates. If new animas statistically appear too often, the overpopulation will take place. In opposite limit the array will quickly shrink (N to 0) and it will cause a distinction of the population.Obviously both these rates should be naturally regulated by the available food resources. For the particular problem under consideration the resources are generated by a random deposition of the potentially available food onto the water surface. It means that we introduce a new array for the food with coordinates (overrightarrow {r}_{n}^{food}). The length of the array (N^{food}) is also variable and index (n) running in the interval (n = 0,..,N^{food}). In principle, it is possible, and very often happened in our simulations, that at some particular moment available food inside the area can completely disappear. In this case the length of the food array reduces to zero (N^{food} = 0).The food is generated by the random deposition of the food portions distributed inside the area of the system ({ [0,Lx],[0,Ly]}). Generally, it is organized in the same manner as the deposition of the new individuals. If the random number (zeta) uniformly distributed in the interval [0;1] is less than some threshold (varsigma_{thres} < 1) the food portion is deposed at an arbitrary place of the area, in principle at any given time step (Delta t).It is obvious that if the threshold is much smaller than unit: (zeta_{thres}^{food} < < 1) the food portions are produced very rarely. Certainly, the rate of the “food production” in the frames of the model must be properly regulated to make its behaviour natural. Of course, it relates to the size of the portions, but also to the intervals between the food depositions. In any case, these intervals are expected to be much longer than discrete time steps (Delta t) used to solve numerically the equations of motion. At the same time, it must be much shorter than other (biologically reasonable) time-scales of the problem. To control the stability and reasonability of the simulations we have varied this rate in very wide intervals. It was found that at all the reasonable rates food balance in the system tends to the stationary scenario, which corresponds to the expected scenarios in nature.When a portion of the food falls onto the surface the animals which occasionally appear relatively close to it are attracted to this food portion and “eat” it with some characteristic rate. In the particular simulation it was simulated by the additional term of the attraction force:$$f_{{_{jn} }}^{food} (overrightarrow {r}_{j} ,overrightarrow {r}_{n} ) = B_{jn}^{food} (overrightarrow {r}_{j} - overrightarrow {r}_{n} )exp left[ { - left( {frac{{overrightarrow {r}_{j} - overrightarrow {r}_{n} }}{{R_{{_{j} }}^{food} }}} right)^{2} } right]$$ (6) As it is seen from the interactions in the model the animals compete for the food, repulsing one another and reacting faster on the force (f_{{_{jk} }}^{food} (overrightarrow {r}_{j} ,overrightarrow {r}_{k} )). It’s why we apply here non-uniform coefficient (B_{{_{jk} }}^{food}), which is different for different index (j) and in fact depends on some power of mass (B_{jk}^{food} sim m_{j}^{alpha }) with some exponent (alpha). Scaling estimation gives the value of the exponent (alpha) = 2/3. Another form of the competition is related to their simultaneous consumption the food with different rate depending on the size of the different individuals. We suppose that the consumption is proportional to already accumulated mass of the individual: (partial m_{j} /partial t = mu m_{j}). Effective dumping (eta_{j}) on the water surface also depends on the mass of animal (eta_{j} sim m_{j}^{beta }), where estimated exponent (beta) = 1/3.Let us remind that the consumption of the food portion with index (n) is possible only when the animal is close enough to this portion. In other words, one more threshold has to be incorporated: (left| {overrightarrow {r}_{j} - overrightarrow {r}_{n} } right| < R_{thres}^{food}). As we see, the model consists of the dynamic equations of motion and a number of the procedures, which work in parallel and essentially affect both: the dynamic behavior and the results.The equations of motion can be formally written accumulating all the above mentioned forces of the problem:$$m_{j} partial^{2} r_{j} /partial t^{2} + eta_{j} partial r_{j} /partial t = sumlimits_{k} {left[ {f_{{_{jk} }}^{repuls} (overrightarrow {r}_{j} ,overrightarrow {r}_{k} ) + f_{{_{jk} }}^{attract} (overrightarrow {r}_{j} ,overrightarrow {r}_{k} )} right]} + sumlimits_{n} {f_{{_{jn} }}^{food} (overrightarrow {r}_{j} ,overrightarrow {r}_{n} )} + sumlimits_{Boundaries} {f_{j}^{bound} }$$ (7) with their solution combined with the procedures described above. All the initial velocities are zero. This combination makes the solution nontrivial because it is performed for the arrays with changeable lengths and with the varied sizes and forces of the participants. Nevertheless, expected scenarios can be generally predicted and verified later by the large set of numerical experiments at different combinations of the parameters.The longer the time an individual spends near the portion of food and the larger the size it has to the current moment the faster it eats up and faster accumulates additional mass. In general, the larger animals consume more food. However, one should remember that large animals may also have disadvantages. For example, due to inertia of motion faster animals can occasionally jump over a good position near the food. We have observed many such local events during the simulations. Such events partially can be observed in the movies presented below. It is also possible that small animals can appear near the randomly deposed food earlier than the big ones and start consuming it.In some cases such possibilities even emerges due to not just probabilistic but quite regular reason. For example, it can happen due to a further described effect of the “wave of fear”. By influence of such a wave (being scared by somebody moving along the shore), the animals almost synchronically start to run away from one of the boundaries. Big, strong and fast animals escape far away, while the small ones still remain near to the shore. As result, the food deposed into this region will be partially (or even completely) consumed by the weak individuals before the strong ones will return. This effect will be reported further on.Basically, dynamic balance of the population is determined by the relation between a set of the time constants of the problem: how often the food appears, how quick the animals consume it and how efficiently they grow. However, the population structure is also important for the total balance. The more an animal consumes the bigger and stronger it becomes, than faster moves to the new portion of the food and more eats. The individuals are born small and more or less equal. They grow initially depending on their “luck” only. However, population quickly forms some non-uniform distribution of the sizes.In terms of distributions one can say that with the time the individual members of the array shift along the distribution to the larger or smaller size depending on the personal balance of accumulation or losing the mass. At every instant moment the distribution actually demonstrates two opposite fluxes: to the limit of the small and large sizes. Strongest animals evolve to the predefined critical large size (m_{max }), at which they leave the population. Weakest ones tend to the critically small weight mmin and also leave the population. The balance is maintained by the fact that at every time moment the population is incorporated by new young members who either grow or die.In this respect, it is interesting to observe the dynamics of the animal’s sizes distribution. Obviously, the initial population consisting of young animals has histogram localized around small sizes. Later the distribution becomes wider and its edge moves to the larger sizes. The asymptotic histogram shape is determined by the two opposite processes described above. This shape will be compared further with a real distribution found from the field observations.Typical behavior of the model evolution for a population inside a limited area is presented in illustrative movie (video_S2). For a convenience, the population is formally divided into 3 subgroups, which are plotted by the circles having different sizes (small, medium, and big) and colors (blue, green, and red, respectively). The randomly deposed portions of food are shown by large black circles.We start from relatively young population, which contains 100 individuals and allow them to move, grow and disappear according to the rules of game. This behavior demonstrates correlative motions, as well as variation in the population composition, which look self-consistent and rather natural. The interactions between the animals with other animals and with food cause quite predictable motions of the individuals and the changes of their sizes (and colors, respectively) when they leave one of the three groups and join to another. The process stabilizes with the time and seems to become stationary. Below we will study this process by the quantitative time-dependencies and statistical histograms.The bigger an animal the faster it moves in average. Therefore, it is convenient to sort the animals according to the masses and velocities. Such representation is reproduced in the third movie “video_S3.mp4”. The separation between subgroups, their evolution from the initial to a stationary state is clearly seen from the movie in dynamics. Moreover, one can observe even how group of the small animals divides by itself into two subgroups with quite well pronounced gap between them. This separation is caused by the mentioned above two fluxes of the sub-populations, which either grow from the initially small sizes to the medium ones or “in unlucky case” decrease and disappear.Actually, the rate of evolution with fast changes of the masses after very few acts of the interaction with deposed food represented in both movies is strongly overestimated in contrast to the reality. Therefore, as a next step we reduced the rates of accumulation and loss of the mass and proportionally prolonged the simulation time. This process was also recorded in two movies “video_S4.mp4” and “video_S5.mp4”. To reduce the lengths of the videos the time intervals between the frames were made 10 times longer. Because of this reduction the movies look almost stroboscopic. Nevertheless, the movies illustrate quite well long-time dynamics of the system at realistic rate of food deposition and consumption.Information about the system accumulated during long runs including typical pattern formed by the moving animals at some intermediate stage and other plots is presented in Figs. 1, 2, 3 and 4. The spatial pattern in Fig. 1 is taken at some intermediate stage of evolution. Blue, green and red circles correspond to the small medium and large sizes of the individuals. Large black circles mark the food portions, which are recently deposed and not eaten yet.Figure 1Typical spatial pattern formed by the population at intermediate stage of evolution. Blue, green and red circles correspond to the small medium and large sizes of the individuals. One can see the groups packed by the individuals with the distances close to the equilibrium Rmin(R^{min }) and voids. Black circles show food portions remaining to the current moment.Full size imageFigure 2Histograms of the distances between nearest neighbors. Blue line shows some instant distribution. Black line presents histogram accumulated during long run. Maximum of the histogram corresponds to the distance Rmin close to the equilibrium. Long tail of the black curve corresponds to the presence of some individuals inside the voids.Full size imageFigure 3Mass depending values: (a) instant configuration of the velocities of different subgroups marked by the same symbols as in Fig. 1; (b) instant and long run averaged histograms of the masses for complete population; (c) instant and accumulated distributions of the velocities monotonously increasing with masses (volumes) of the individuals.Full size imageFigure 4The comparison between the numerically (curves with points) and experimentally (bars) obtained data. The subplot (a) shows the histogram of the distances between nearest neighbors and the subplot (b) presents the histograms of the volumes of the animals (which are supposed to be approximately proportional to their masses).Full size imageOne can well distinguish in Fig. 1 relatively dense domains packed with the distances between the individuals close to the equilibrium radius (R^{min }) determined by the relation between repulsive (f_{{_{jk} }}^{repuls} (overrightarrow {r}_{j} ,overrightarrow {r}_{k} )) and attraction (f_{{_{jk} }}^{attract} (overrightarrow {r}_{j} ,overrightarrow {r}_{k} )) forces. These domains are separated by voids. To characterize such patterns quantitatively the histograms of the distances between nearest neighbors6 were accumulated during a long simulation run. These histograms are shown in Fig. 2. Thin blue line here reproduces some arbitrary instant distribution. Black line with the circles presents the histogram accumulated during a long run. Maximum of the histogram is close to the equilibrium distance (R^{min }) determined by the balance of interactions. Long tail of the black curve, which extends to the few times longer distances than (R^{min }), exists, since some individuals are located inside voids far-away from all the neighbors.Typical size distributions to which the population evolves with the time are reproduced in Fig. 3. It shows three important size depending values. Figure 3a demonstrates an instant configuration of the velocities. Three different subgroups are marked by the same symbols as in Fig. 1. Figure 3b shows instant histogram of the animal’s masses in complete population and the histogram averaged over long run. They are plotted by the blue and black lines respectively. How mean velocities of the animals depend on their mass is shown in Fig. 3c. As above, the instant and accumulated distributions are plotted by the blue and black lines respectively.It can be noticed from the subplot (c), Fig. 3, that the averaged velocities correlate with the mass of the individuals. In principle, it could be expected according to the model rules, namely due to the nonlinear dependence of strength (f_{{_{jk} }}^{repuls} (overrightarrow {r}_{j} ,overrightarrow {r}_{k} )) as well as effective damping (eta_{j}) for the individual animals on their size (mass (m_{j})). However, taken into account multiple interactions in the system the result was not obvious in advance. The velocity increase is especially pronounced when the animals become large and are close to the final stage of its growth.Further, we would like to compare the numerical and experimentally obtained data. This comparison is presented in Fig. 4 by the curves and bar-plots respectively. From the field videos it is difficult to determine the masses (m_{j}) of the individuals, but we can estimate their volumes (V_{j}) which are approximately proportional to their masses (V_{j} sim m_{j}). Therefore, to compare numerical results with the measurements from videos we have plotted the distributions along the volume coordinate. The subplot (a) in Fig. 4 shows the histograms of the distances between nearest neighbors, and the subplot (b) presents the histograms of the volumes of the animals. The histograms obtained from simulations match very well to the histograms calculated using experimental data. Two sample Kolmogorov–Smirnov test does not reject the hypothesis, that the distributions obtained in experiment and in numerical simulations are from the same continues distribution for both the animal volumes (p = 0.59, D55,500 = 0.105) and for the distances between nearest neighbors (p = 0.84, D55,500 = 0.083).It is important to note that due to stability of the system the distributions independent on their initial shape converge to the same quasi-stationary histograms. To check this convergence, we started from two extreme populations. The first one consisted almost from the large individuals only and another one contained only small individuals. Time depending volume histograms were accumulated into the volume distribution over time gray-scale maps. In Fig. 5 the results for the two cases are shown in the subplots (a) and (b) respectively. Lighter gray color corresponds to the higher density. Both distributions attract to the practically the same final one with the time. The shapes of distributions, which have started from the two extreme initial distributions, almost coincide after long runs, while the distributions flanks shift in different directions as marked by the red and blue arrows in Fig. 6 and visualized in the supplementary movies “video_S6.mp4” and “video_S7.mp4” respectively. As in the static figures the instant and time averaged histograms are shown by the blue and black lines respectively. For the supplementary video “video_S6.mp4” the distribution is initially localized mainly near the right side of the interval and after some quick transient period “jumps” to the distribution close to the final histogram. Comparison with second video “video_S7.mp4” shows how both averaged distributions are attracted after a long simulation run to the similar distribution.Figure 5Gray-scale maps of time-volume histograms accumulated for two limit cases: starting from large and small animals, are shown in the subplots (a,b) respectively. Lighter gray color corresponds to the higher density. Dashed line marks common position of distribution maximums to which they converge during extremely long runs.Full size imageFigure 6Final volume distributions after long runs. It is seen that the maximums coincide already. Remaining trends of the histogram alterations for the distribution started from the large (open circles) and small (closed circles) individuals are marked by the red and blue arrows respectively.Full size imageDuring the system evolution some of the animals leave the population and some new appear. One can accumulate the numbers of died and survived animals for each given time moment. Accumulated numbers of died and survived animals are plotted in Fig. 7a by black and red curves respectively. The balance between these numbers depends on the relation of all the model parameters. Here the same set of the parameters was used as for simulations presented in Figs. 1, 2 and 3.Figure 7Time depending integral populations: (a) accumulated numbers of died (N1) and survived (N2) animals (black and red curves); (b) variations of the numbers of animals in subpopulations of small (Nc1), medium (Nc2) and large (Nc3) animals shown by blue, green and red curves respectively.Full size imageThe curves presenting the integral number of animals are relatively smooth, Fig. 7a, yet the instant numbers of animals in small, medium and large subpopulations shown by blue, green and red curves respectively in Fig. 7b vary much stronger, because at every particular time moment, their numbers are relatively small and the fluctuations are strong. From the figure, it is also obvious that during initial transient time interval the number of small individuals do not exceeds the number of two other populations. It is natural and was expected, because all newborn animals are small. Therefore, subpopulation with small size prevails at stationary stage.As we already announced nontrivial transformation of the spatial pattern and modification of all the resulting distributions can appear due to the “waves of fear”. Such a wave can appear when the animals almost synchronously try to escape from the shoreline if they are being scared by somebody moving along. Mathematically such a wave can be provoked by an additional repulsion force acting from one side of the simulation area to the distance much longer than already incorporated short range reflection from the boundary (Eq. 6). This force influences all the members of the array but appears for relatively limited time. It can be added to the model interaction occurring either randomly or periodically. Second variant is more regular and preferable and allows accumulate statistics faster.Analytically this force has the same form as (f_{j}^{bound}) with the same boundary position (overrightarrow {r}_{bound}):$$f_{j}^{wave} = B_{j}^{wave} exp left[ { - left| {frac{{overrightarrow {r}_{j} - overrightarrow {r}_{bound} }}{{R_{{}}^{wave} }}} right|} right],$$ (8) but it has larger amplitude (B_{j}^{wave} > B_{j}^{bound}) and much longer distance of the exponential decay (R_{{}}^{wave} > > R_{{}}^{bound}). So, it influences not only the individuals occasionally attempting to cross the boundary but almost all others too, even if they are currently relatively far from the boundary.The video in supplementary material “video_S8.mp4” illustrates typical behavior of the system at presence of the “waves of fear” occasionally observed in original sequences (video_S1.avi). One can see how bigger, stronger and faster animals escape quicker and further from the dangerous boundary, while some smallest ones do not react so quickly and remain almost near to it. Food deposition is not correlated with the “waves of fear”. As result, the food deposed near the dangerous boundary is consumed mostly by small animals, since that boundary region is practically depopulated by bigger animals. While stronger individuals return to the shore, such food may be already consumed by the weaker individuals.Static image of such “shifted” to one side pattern for a moment when food and weak members of the array are presented in one side of the area while absolute majority of the population is collected in a center or closer to another side of the area is reproduced in Fig. 8.Figure 8Shifted pattern at the presence of the “wave of fear”. Some food was just deposed inside the depopulated area near to the dangerous boundary. Some small individuals are in the vicinity to the food, while absolute majority of the population is localized in a center or near opposite boundary. The symbols have the same meaning as in Fig. 1.Full size imageBeing regularly applied the “waves of fear” in principle leads to a redistribution of the food and animals in the space. It is quite expected that the density of the animals will spend some time in the areas distanced from the dangerous wall. As result, the food will longer remain available in the places close to the dangerous wall. We have accumulated these densities during long runs at different parameters and obtained such shifted distributions. Typical distributions of the food and animals for the set of parameters which were used in simulations presented in the previous figures are shown in Fig. 9.Figure 9Density distributions of the food and animals at presence of regularly applied “waves of fear”. The densities of the food and animals are shown by the solid black and dashed magenta curves respectively. The curves for the densities of large, middle and small individuals are plotted by the same colors as above. The density shift with increasing size is marked by the arrow.Full size imageMean density of the food is shown by the solid black curve. Total density of the animals is presented by the dashed magenta curve. These densities are anti-correlated. Besides, there is a fine structure of the partial densities of the animals. The curves for the densities of large, middle and small individuals are plotted by the same colors as above. Strongest correlation between the size of the animals and their spatial density is observed in interval between two dash-dotted lines. The density shift with increasing size is marked by the arrow. In close proximity to the dangerous wall, where mean density of the food is abnormally high, the density shift dependence on the individuals size change the direction. Number of the small individuals reduces in average, because they quickly move to the category of middle or even large individuals. More