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Study area
The field work was carried out at the Sturt Meadows calcrete aquifer (28°41′ S 120°58′ E) located on Sturt Meadows pastoral station, Western Australia, ~ 42 km from the settlement of Leonora (833 km northeast of Perth, see Fig. 1a). The study area is a calcrete aquifer lying in the Raeside paleodrainages in the Yilgarn region of Western Australia (Fig. 1a). The vegetation of the area is Acacia woodlands, primarily Acacia aneura (F.Muell. ex Benth.), and is subjected to combined grazing pressure from domestic stock, feral animals and macropods. The aquifer is accessible through a bore grid comprising 115 bore holes of between 5 and 11 m depth (Fig. 1b).
Figure 1

Map of the Sturt Meadows calcrete. (a) Location within the Yilgarn craton region and detailed paleodrainages/cacretes in the area and (b) the grid map showing the location of the boreholes sampled for stygofauna together with probe samples, water samples (in light blue) and the combination of both. Map was produced in ArcGIS Desktop 10.679 and edited in Adobe Illustrator 25.080.

Full size image

Three sampling campaigns were carried out, two of them (LR1: 26/07/2017 and LR2: 07/11/2017) corresponding to low rainfall periods24 and one during the wet season (high rainfall, HR; two consecutive days of sampling collection: 17-18/03/2018) (Supplementary Fig. 1). The well-studied stygofaunal community of the area is composed of 11 main stygofaunal taxa belonging to five Classes: Oligochaeta (family Tubificidae (Vejdovský 1884)), subcohort Hydrachnidia, Maxillopoda (two species of harpacticoids: Novanitocrella cf. aboriginesi (Karanovic, 2004), Schizopera cf. austindownsi (Karanovic, 2004) and four species of cyclopoids: Halicyclops kieferi (Karanovic, 2004), Halicyclops cf. ambiguous (Kiefer, 1967), Schizopera slenderfurca (Karanovic & Cooper, 2012) and Fierscyclops fiersi (De Laurentiis et al., 2001)), Malacostraca: Amphipoda (species Scutachiltonia axfordi (King, 2012), Yilgarniella sturtensis (King, 2012) and Stygochiltonia bradfordae (King, 2012)) and Insecta: Coleoptera: Dytiscidae (species Paroster macrosturtensis (Watts & Humphreys 2006), Paroster mesosturtensis (Watts & Humphreys 2006) and Paroster microsturtensis (Watts & Humphreys 2006) and respective larvae).
Field work procedures and sample preparation
Given the sensitivity of the hydrological dynamics in shallow calretes23,25, extensive water extraction along the bores was avoided, and preliminary tests on the bores with the highest water depth were carried out to quantify potential risk of dewatering the calcrete. During the field campaigns LR2 and HR, 20 water samples in total (two samples for stable isotope analysis on DOC (Dissolved Organic Carbon) and DIC (Dissolved Inorganic Carbon), three samples for radiocarbon analysis on DOC, one sample for radiocarbon analysis on DIC, and two samples for stable isotope and radiocarbon analyses on POC (Particulate Organic Carbon)) were collected from bores D13 and W4 (Fig. 1b), which are representative of the two main geological conformations of the area—calcrete (W4) and clay (D13) (Supplementary Fig. 2)—and host stable hydrological and biotic conditions7. Water samples were collected using a submersible centrifugal pump (GEOSub 12 V Purging Pump) after wells were purged of three well-volumes and stabilisation of in-field parameters was observed, according to the methodology in Bryan et al.26.
Samples for 14CDIC analysis were filtered through 0.45 μm filters and collected in 1 L high density poly-ethylene (HDPE) bottles. δ13CDIC samples were filtered through 0.2 μm filters, collected in 12 mL glass vials (Exetainers) and refrigerated after sampling. δ 13CDOC samples were filtered through 0.2 μm filters, collected in 60 mL HDPE bottles and frozen after sampling.14CDOC samples were filtered through 0.2 μm filters, collected in 3 L HDPE bottles and frozen after sampling.
In order to investigate 14C and δ 13C content of POC, two extra liters were collected from the same bores (D13, W4) and kept frozen (− 20 °C) until further analyses. 14CPOC δ13CPOC samples were then filtered through pre-combusted GF/F filters (12 h at 450 °C), washed with 1.2 N HCl to remove any inorganic carbon, and subsequently dried at 60 °C for 24 h. All samples were closed with sealing tape after collection to limit atmospheric exchange and kept in darkness.
Temperature, pH, ORP, salinity, DO and depth were measured in situ using a portable Hydrolab Quanta Multi‐Probe Meter across 30 bores during LR1, LR2 and HR23 (presented in Supplementary Table 8). Adult and larval stygofaunal specimens were collected from the same bores by hauling a weighted plankton net (mesh 100 μm27) five times through the water column (Fig. 1b). All biological samples were kept frozen (− 20 °C) in darkness until laboratory processing. Individual organisms were counted and identified (and consequently separated) to the lowest taxonomic level via optical microscopy and reference to specific taxonomic keys. Plant material, sediment samples and fauna were each separated during the sorting in the laboratory and each taxon pooled according to sampling campaign (LR1, LR2 or HR) and subsequently washed with Milli-Q water to remove surface impurities from their bodies. Sediment samples were soaked in acid (0.1 N HCl) to remove inorganic carbon, and together with the other samples were then oven dried at 60 °C overnight and ground until obtaining a homogeneous fine powder and stored at − 20 °C until further analyses.
Previous investigations on the ecological niche trends at Sturt Meadows indicated that all stygofauna characterize similar niche occupations under low rainfall regimes (LR1 and LR2)23. Stygofaunal specimens from the two low rainfall sampling events were combined to form sample LR to address the competing requirements between isotopic detection limits, analytical replicates and cost, while maintaining the main taxonomic and biological classifications7.
Bulk isotope and 14C analyses
Water δ13CDIC and δ13CPOC isotopic ratios were analysed by Isotope Ratio Mass Spectrometer—WABC at The University of Western Australia using a GasBench II coupled with a Delta XL Mass Spectrometer (Thermo-Fisher Scientific)—and the results, with a precision of ± 0.10‰, were reported as ‰ deviation from the NBS19 and NSB18 international carbonate standard28.
δ13CDOC isotopic ratios of waters were analysed via Liquid Chromatography Isotope Ratio Mass Spectrometer (LC-IRMS) at the La Trobe Institute for Molecular Sciences (LIMS, La Trobe University, Melbourne, Australia) comprising an Accela 600 pump connected to a Delta V Plus Isotope Ratio Mass Spectrometer via a Thermo Scientific LC Isolink (Thermo Scientific).
C and N bulk SIA on homogenised samples of sediment, roots, stygofauna and copepods (cyclopoids and harpacticoids) were performed at the Australian Nuclear Science and Technology Organisation (ANSTO, Sydney, Australia). Samples were loaded into tin capsules and analysed with a continuous flow isotope ratio mass spectrometer (CF-IRMS), model Delta V Plus (Thermo Scientific Corporation, U.S.A.), interfaced with an elemental analyser (Thermo Fisher Flash 2000 HT EA, Thermo Electron Corporation, USA) following the procedure published by Mazumder et al.29.
For radiocarbon analyses, samples (sediment, roots, copepods, ants, stygofauna) were treated with 1 M HCl for 2 h to remove all possible carbonate contamination. These pre-treated samples together with 14CPOC, 14CDOC and 14CDIC samples were subjected to CO2 extraction and graphitization following the methodology published by Hua et al.30. 14C content of samples was determined by means of the Accelerator Mass Spectrometry (AMS) at ANSTO.
Carbon CSIA
Carbon CSIA followed the procedure described in Saccò et al.7. Samples of roots and stygofaunal specimens were hydrolysed under vacuum with 0.5 to 1 mL of amino acid-free 6 M HCl (Sigma-Aldrich) at 110 °C for 24 h. The protein hydrolysates were dried overnight in a rotary vacuum concentrator and stored in a freezer. Prior to analysis, the samples were dissolved in Milli-Q water and 10 μL of 1-mmol solution of 2-aminoisobutyric acid (Sigma-Aldrich) as internal standard. The sample stock had a concentration of approximately 8 to 10 mg/mL, which was further diluted as needed. Single amino acid carbon isotope analysis was carried out at the La Trobe Institute for Molecular Sciences (LIMS, La Trobe University, Melbourne, Australia) using an Accela 600 pump connected to a Delta V Plus Isotope Ratio Mass Spectrometer via a Thermo Scientific LC Isolink (Thermo Scientific).
The amino acids were separated using a mixed mode (reverse phase/ion exchange) Primesep A column (2.1 × 250 mm, 100 °C, 5 μm, SIELC Technologies) following the chromatographic method described in Mora et al.31 after Smith et al.32. Mobile phases are those described in Mora et al.33. Percentage of Phases B and C in the conditioning run, as well as flow rate of the analytical run and timing of onset of 100% Phase C were adjusted as needed. Samples were injected onto the column in the 15 μL—partial loop or no waste—injection mode, and measured in duplicate or triplicate.
To elucidate carbon flows through the stygofaunal community we focused on the essential amino acids Valine (Val), Phenylalanine (Phe) and Arginine (Arg), as these compounds must be integrated through diet and cannot be synthetised internally by the fauna14,34. In addition, to distinguish between terrestrial and aquatic carbon sources, the ratio between Val and Phe signals (δ13CVal-Phe), a widely employed index in archaeology and freshwater biology35, was calculated for roots, water mites, aquatic worms, amphipods and beetles (larvae and adults).
Microbial taxonomic and functional gene analyses
Consumer amphipods (Scutachiltonia axfordi (AM1), Yilgarniella sturtensis (AM2), S. bradfordae (AM3)), cyclopoids and harpacticoids, together with predator stygobiotic beetles (Paroster macrosturtensis (B), P. mesosturtensis (M) and P. microsturtensis (S)) (see Saccò el at 7. for further details on the trophic characterisation of the stygofaunal community at Sturt Meadows), were used for gut microbiome bacterial 16S metabarcoding and microbial functional analysis. A total of 16 AM1, 16 AM2, 16 AM3, 20 cyclopoids and 20 harpaticoids and 20 of each one of the three Paroster species (B, M and S), were sorted into duplicates of stygobiotic pools of 3–5 individuals from both LR and HR events for DNA extraction. Prior to DNA extraction stygobitic animals (3–5 individuals per pool; n = 40) were placed in a petri dish containing ultrapure water and UV sterilized in a UV oven for 10 min to eliminate any bacterial species that may be contained on the exoskeleton as this study targeted the gut microbiome. Immediately post-UV treatment, the animals were placed into tissue lysis tubes with 180 μL tissue lysis buffer (ATL) and 20 μL Proteinase K and homogenised using Minilys tissue homogeniser (ThermoFisher Scientific, Australia) at high speed for 30 s. Lysis tubes, inclusive of two laboratory controls, were incubated at 56 °C using an agitating heat block (Eppendorf ThermoStat C, VWR, Australia) for 5 h.
Following the incubation, the analytical procedure was adapted from Saccò et al.22 and DNA extraction was carried out using DNeasy Blood and Tissue Kit (Qiagen; Venlo, Netherlands) and eluted off the silica column in 30–50 μL AE buffer. The quality and quantity of DNA extracted from each stygobitic pool was measured using quantitative PCR (qPCR), targeting the bacterial 16S gene. PCR reactions were used to assess the quality and quantity of the DNA target of interest via qPCR (Applied Biosystems [ABI], USA) in 25 μL reaction volumes consisting of 2 mM MgCl2 (Fisher Biotec, Australia), 1 × PCR Gold Buffer (Fisher Biotec, Australia), 0.4 μM dNTPs (Astral Scientific, Australia), 0.1 mg bovine serum albumin (Fisher Biotec, Australia), 0.4 μM of each primer (Bact16S_515F and Bact16S_806R36,37), and 0.2 μL of AmpliTaq Gold (AmpliTaq Gold, ABI, USA), and 2 μL of template DNA (Neat, 1/10, 1/100 dilutions). The cycling conditions were: initial denaturation at 95 °C for 5 min, followed by 40 cycles of 95 °C for 30 s, 52 °C for 30 s, 72 °C for 30 s, and a final extension at 72 °C for 10 min.
DNA extracts that successfully yielded DNA of sufficient quality, free of inhibition, as determined by the initial qPCR screen (detailed above), were assigned a unique 6–8 bp multiplex identifier tag (MID-tag) with the bacterial 16S primer set. Independent MID-tag qPCR for each stygobiotic pool were carried out in 25 μL reactions containing 1 × PCR Gold Buffer, 2.5 mM MgCl2, 0.4 mg/mL BSA, 0.25 mM of each dNTP, 0.4 μM of each primer, 0.2 μL AmpliTaq Gold and 2–4 μL of DNA as determined by the initial qPCR screen. The cycling conditions for qPCR using the MID-tag primer sets were as described above. MID-tag PCR amplicons were generated in duplicate and the library was pooled in equimolar ratio post-PCR for DNA sequencing. The final library was size selected (160–600 bp) using Pippin Prep (Sage Sciences, USA) to remove any MID-tag primer-dimer products that may have formed during amplification. The final library concentration was determined using a QuBitTM 4 Fluorometer (Thermofischer, Australia) and sequenced using a 300 cycle V2 kit on an Illumina MiSeq platform (Illumina, USA).
MID-tag bacterial 16S sequence reads obtained from the MiSeq were sorted (filtered) back to the stygobitic pool based on the MID-tags assigned to each DNA extract using Geneious v10.2.538. MID-tag and primer sequences were trimmed from the sequence reads allowing for no mismatch in length or base composition.
Filtered reads were then input into a containerised workflow comprising USEARCH39 and BLASTN40, which was run on a high-throughput HPC cluster at Pawsey supercomputing centre. The fastx-uniques, unoise3 (with minimum abundance of 8) and otutab commands of USEARCH were applied to generate unique sequences, ZOTUs (zero-radius Operational Taxonomic Units) and abundance table, respectively. The ZOTUs were compared against the nucleotide database using the following parameters in BLASTN: perc_identity ≥ 94, evalue ≤ 1e−3, best_hit_score_edge 0.05, best_hit_overhang 0.25, qcov_hsp_perc 100, max_target_seqs = 5. An in-house Python script was used to assign the ZOTUs to their lowest common ancestor (LCA)41. The threshold for dropping a taxonomic assignment to LCA was set to perc_identity ≥ 96 and the difference between the % of identity of the two hits when their query coverage is equal was set to 1. Results on the microbial taxonomic diversity detected in ground water samples from a previous study on carbon inputs in the aquifer22 were incorporated in this work to allow qualitative comparison with the stygofaunal gut diversity.
To investigate functional activity involved in carbon cycling, the 16S metabarcoding data were fed to the Phylogenetic Investigation of Communities by Reconstruction of Unobserved States 2 (PICRUSt2) software package to generate predicted metagenome profiles42. These profiles were clustered into Kyoto Encyclopedia of Genes and Genomes (KEGG) Orthologs (KOs)43 and MetaCyc pathway abundances44 focusing on the relative abundances of four carbon metabolisms: carbon fixation in prokaryotes, carbohydrates, lipids and amino acid metabolisms. Relative abundance of pathways linked with methane, nitrogen and sulfur metabolisms were also investigated.
Statistical analyses
The Phyloseq package in R45,46 was used to plot the ZOTU abundance for the styfofaunal specimens at the order level under LR and HR periods. The Statistical Analysis of Metagenomic Profiles (STAMP) bioinformatics software package was used to carry out Principal Components Analysis (PCA) to assess the differences between functional genomic fingerprints based on the KEGG orthologs function prediction between copepods (C and H) and amphipods (AM1, AM2 and AM3), and determine statistically significant results from the PICRUSt2 output47. Clustering patterns according to rainfall periods (LR and HR) and major consumers taxonomic groups (cyclopoids, harpacticoids and amphipods) were assessed through Permutational multivariate analysis of variance (PERMANOVA, R-package46 ‘vegan’) and pairwise post hoc pairwise multilevel comparisons48.
For comparison of potential shifts in abundances of microbial metabolic pathways within groundwater samples, copepods and amphipods across rainfall periods, analysis of variance (ANOVA) was performed on the abundance data (two replicates per each group) on the predicted pathways depicting carbon fixation, carbohydrate, lipid, amino acid, methane, nitrogen and sulfur metabolisms. ANOVAs coupled with Tukey’s HSD pairwise comparisons (R-package46 ‘stats’) were employed to inspect significant differences between bulk SIA (δ13C and δ15N) and essential amino acid (δ13CPhe, δ13CArg, δ13CVal and δ13CVal-Phe) data from the stygofaunal taxa within the different rainfall conditions (LR and HR). PERMANOVAs (R-package46 ‘vegan’) were also performed to investigate the potential clustering trends within the stygofaunal taxa across rainfall periods from the combination of radiocarbon (Δ14C) and carbon SIA (δ13C) isotopic fingerprints.
SIMM (Stable Isotope Mixing Models, R-package46 ‘simmr’) were used to estimate dietary proportions of copepods and amphipods within a Bayesian framework. Due to the lack of species-specific trophic discrimination factors for stygofauna, we employed the widely accepted values of 3.4 ± 2‰ for nitrogen and 0.5 ± 1‰ for carbon49. Markov chain Monte Carlo (MCMC) algorithms were used for simulating posterior distributions in SIMM, and MCMC convergence was evaluated using the Gelman-Rubin diagnostic by using 1.1 as a threshold value for analysis validation. More

Interseasonal VGC dominates peak-to-late-season growth
We first examined the VGC effect at the seasonal scale, using satellite-derived Normalized Difference Vegetation Index (NDVI, see “Methods”) for the 1982–2016 period. We defined the dormancy season (DS) and three periods of the growing season, i.e., EGS, peak growing season (PGS), and late growing season (LGS), based on phenological metrics (see “Methods”). The partial autocorrelation calculated for NDVI time series of two consecutive seasons, after factoring out concurrent and preceding climatic impacts, provides an estimate of the interseasonal VGC effects (see “Methods”). At the hemispheric scale, our analyses show a significant (p  More

State variables
The first step in describing the macroscopic properties of the swarm is to define a set of state variables that fully characterizes the state of the system. The equation of state then links these state variables in a functional relation. In classical thermodynamics, a complete set of state variables is given by the conjugate pairs of pressure P and volume V, temperature T and entropy S, and, if the number of particles is not fixed, chemical potential μ and number of particles N. We use an analogous set of state variables here to characterize swarms. The most straightforward state variable to define is the number of individuals N, which is given simply by the number of midges in the swarm at a given time (note that midges that are not swarming simply sit on the walls or floor of the laboratory enclosure). The volume V of the swarm can be straightforwardly defined and computed as the volume of the convex hull enclosing all the midges. Note that while N and V are not independently controllable quantities, the ratio N/V is empirically approximately constant in large swarms24, meaning that the “thermodynamic” limit (that is, (N to infty) and (V to infty) with (frac{N}{V} to rho)) is approached in our swarms25. In typical swarming events, N changes on a time scale that is very slow compared to the swarm dynamics; thus, a chemical potential is not needed to describe the instantaneous state of the swarm. Note, though, that since the number of midges varies between measurements that may be separated by many days, N remains a relevant state variable for capturing swarm-to-swarm variability.
The remaining three state variables are somewhat more subtle, but can be defined by building on previous work. It has been explicitly shown26 that a virial relation based on the kinetic energy and an effective potential energy holds for laboratory swarms of Chironomus riparius. For particles moving in a potential, this virial relation can be used to define a pressure26. As we have shown previously, swarming midges behave as if they are trapped in a harmonic potential well that binds them to the swarm, with a spring constant k(N) that depends on the swarm size24,26 (Fig. 1b). The difference between the kinetic energy and this harmonic potential energy thus allows us to compute a pressure4,26,27, which is conceptually similar to the swim pressure defined in other active systems28. The virial theorem thus provides a link between kinetic energy, potential energy, and a field that plays the role of a pressure, when coupled with the observation that individual midges to a good approximation behave as if they are moving in a harmonic potential24,26. We can write this virial pressure P (per unit mass, assuming a constant mass per midge) as

$$P = frac{1}{3NV}mathop sum limits_{i = 1}^{N} left( {v_{i}^{2} – frac{1}{2}langle krangle r_{i}^{2} } right),$$

where N is the number of midges in the swarm, V is the swarm volume, vi is the velocity of midge i, ri is its distance from the swarm centre of mass, and (langle krangle = langle – {mathbf{a}}_{i} cdot hat{mathbf{r}}_{i} / r_{i} rangle) is the effective spring constant of the emergent potential well that binds midges to the swarm. In this expression, ai is the acceleration of midge i, (hat{mathbf{r}}_{i}) is the unit vector pointing from a midge towards the instantaneous centre of mass of the swarm (defined as (1/Nsumnolimits_{i = 1}^{N} {{mathbf{r}}_i })) and averages are taken over the individuals in the swarm. This spring constant depends on the swarm size N (Fig. 1b). We note that we have previously simply used the directly computed potential energy (- langle {mathbf{a}}_{i} cdot {mathbf{r}}_{i} rangle) to define the pressure4,27; here, we instead average the potential terms and fit them to a power law in N (Fig. 1b) to mitigate the contribution of spurious instantaneous noise in the individual positions that would be enhanced by differentiating them twice to compute accelerations. We use this power law to determine the spring constant k instantaneously at each time step.
The results from the two methods for computing the pressure are similar and consistent, but the method we use here is less prone to noise. Physically, this pressure P can be interpreted as the additional spatially variable energy density required to keep the midges bound to the swarm given that their potential energy varies in space but their mean velocity (and therefore kinetic energy) does not. Thus, compared to a simple passive particle moving in a harmonic well, midges have more kinetic energy than expected at the swarm edges; this pressure compensates for the excess kinetic energy. This pressure should be viewed as a manifestation of the active nature of the midges (similar to a swim pressure28), since the kinetic energy is an active property of each individual midge and the potential energy is an emergent property of the swarm.
We can define a Shannon-like entropy S via its definition in terms of the joint probability distributions of position and velocity. This entropy is defined as

$$S = – mathop smallint limits_{ – infty }^{infty } p(x,;v)log_{2} p(x,;v)dxdv,$$

where p(x,v) is the joint probability density function (PDF) of midge position and velocity. S here is measured in bits, as it is naturally an information entropy. Empirically, we find that the position and velocity PDFs are nearly statistically independent for all components and close to Gaussian, aside from the vertical component of the position (Fig. 1c–f). However, the deviation from Gaussianity in this component (which occurs because of the symmetry breaking due to the ground) does not significantly affect the estimate of the entropy; thus, we approximate it as Gaussian as well. Making these approximations, we can thus analytically write the (extensive) entropy as

$$S = frac{3N}{{ln 2}}ln left( {2pi esigma_{x} sigma_{v} } right),$$

where (sigma_{x}) and (sigma_{v}) are the standard deviations of the midge positions and velocities, respectively. In practice, we calculated (sigma_{v}) by averaging the instantaneous root-mean-square values of all three velocity components rather than a time-averaged value; the difference between these components was always less than 10%. This expression makes it more clear why the Gaussian approximation for the vertical component of the position is reasonable here: only the mean and variance of the PDFs are required to compute the entropy, and these low moments are very similar for the true data and the Gaussian estimate.
Although there is no obvious definition of temperature for a swarm, we can define one starting from the entropy, since temperature (when scaled by a Boltzmann constant) can be defined as the increase in the total physical energy of the system due to the addition of a single bit of entropy. Given our definitions, adding a single bit of entropy (that is, setting (S to S + 1)) for constant (sigma_{x}) and N (that is, a swarm of fixed number and spatial size) is equivalent to setting (sigma_{v} to 2^{1/(3N)} sigma_{v} .) Adding this entropy changes the total energy of the system by an amount

$$frac{3}{2}sigma_{v}^{2} Nleft( {2^{frac{2}{3N}} – 1} right) equiv k_{B}^{*} T,$$

which we thus define as the temperature (k_{B}^{*} T). Even though this temperature is nominally a function of the swarm size N, it correctly yields an intensive temperature as expected in the limit of large N, as the explicit N-dependence vanishes in that limit since (lim_{n to infty } k_{B}^{*} T = sigma_{v}^{2} ln 2). In practice, this limit is achieved very rapidly: we find that this temperature is nearly independent of N for N larger than about 20, consistent with our earlier results on the effective “thermodynamic limit” for swarms25. The effective Boltzmann constant (k_{B}^{*}) is included here to convert between temperature and energy, though we note that we cannot set its value, as there is no intrinsically preferred temperature scale.
Equipartition
With these definitions in hand, we can evaluate the suitability of these quantities for describing the macroscopic state of midge swarms. First, we note that proper state variables ought to be independent of the swarm history; that is, they ought to describe only the current state of the system rather than the protocol by which that state was prepared. Although this property is difficult to prove incontrovertibly, none of the definitions of our state variables have history dependence. We further find that when these state variables are modulated (see below), their correlation times are very short, lending support to their interpretation as true state variables. We can also compare the relationships between these state variables and the swarm behaviour to what would be expected classically. In equilibrium thermodynamics, for example, temperature is connected to the number of degrees of freedom (d.o.f.) in a system via equipartition, such that each d.o.f. contributes an energy of (frac{1}{2}k_{B}^{*} T). We can write the total energy E of a swarm as the sum of the kinetic energy (E_{k} (t) = frac{1}{2}v^{2}) and potential energy (E_{p} (t) = frac{1}{2}k(N)r(t)^{2}) for all the individuals, where r is the distance of a midge to the swarm centre of mass, v is the velocity of a midge, and k(N) is the effective spring constant. Surprisingly, even though individual midges are certainly not in equilibrium due to their active nature, we find that the total energy is linear in both T and N (Fig. 2a), and that there is no apparent anisotropy, suggesting that equipartition holds for our swarms. This result is highly nontrivial, especially given that our definition of T does not contain the spring constant k(N), which is only determined empirically from our data. Moreover, the slope of the (E/k_{B}^{*} T) curve is well approximated as (9/2)N, implying that each midge has 9 effective d.o.f. (or 6 after discounting the factor of ({text{ln}}2) in our definition of (k_{B}^{*} T)) These d.o.f. can be identified as 3 translational and 3 potential modes, given that the potential well in which the midges reside is three-dimensional. These results demonstrate the surprising applicability of equilibrium thermodynamics for describing the macroscopic state of swarms29.
Figure 2

Equipartition and the equation of state. (a) The total energy of the system (E) normalized by (k_{B}^{*} T) as a function of swarm size (blue) along with the kinetic energy (E_{k}) (yellow) and potential energy (E_{p}) (blue). The total normalized energy of the system is well approximated by (9/2)N (black dashed line), indicating that each individual midge contributes ((9/2)k_{B}^{*} T) to E and thus has 9 degrees of freedom (6 after discounting the factor of ({text{ln}}2) in our definition of (k_{B}^{*} T)). The deviations from that behaviour for the largest swarms can be attributed to a growing uncertainty in the energy due to the smaller number of experiments with such large swarms. (b) A portion of our ensemble of data of the measured pressure (blue). The yellow line is the reconstruction of the pressure from our equation of state. The inset shows a zoomed-in portion of the data to highlight the quality of the reconstruction. (c) PDF of the pressure for our entire data ensemble23. The statistics of the directly measured pressure (blue) and reconstructed pressure from the equation of state have nearly identical statistics for the full dynamic range of the signal.

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Equation of state
The fundamental relation in any thermodynamic system is the equation of state that expresses how the state variables co-vary. Equations of state are thus the foundation for the design and control of thermodynamic systems, because they describe how the system will respond when a subset of the state variables are modulated. Any equation of state can be written in the form (P = f(V, ;T,;N)) for some function f. Although the form of f is a priori unknown, it can typically be written as a power series in V, T, and N, in the spirit of a virial expansion. We fit the equation of state to our data assuming the functional form

$$P = f(V, ;k_{B}^{*} T,;N) = c_{4} V^{{c_{1} }} (k_{B}^{*} T)^{{c_{2} }} N^{{c_{3} }},$$

and using nonlinear least-squares regression. We chose to fit to the pressure for convenient analogy with a thermodynamic framework, but any other variable would have been an equivalent possibility. We note that when fitting, we normalized all the state variables by their root-mean-square values so that they were all of the same order of magnitude. These normalization pre-factors do not change the exponents, but are instead simply absorbed into (c_{4}). Thus, to leading order, we assume (P = f(V,;k_{B}^{*} T,;N) propto V^{{c_{1} }} (k_{B}^{*} T)^{{c_{2} }} N^{{c_{3} }}) and fit this relation to the swarm pressure (Fig. 2b,c), obtaining c1 = − 1.7, c2 = 2, and c3 = 1, with uncertainties on the order of 1%. Although the expression for the pressure does depend on three parameters in a nonlinear fashion, the resulting estimates for these parameters are remarkably stable and consistent across all measurements. Hence, we arrive at the equation of state (PV^{1.7} propto N(k_{B}^{*} T)^{2} .)
This equation of state reveals aspects of the nature of swarms, particularly when compared with the linear equation of state for an ideal gas (where (PV = Nk_{B} T)). In both cases, for example, to maintain a fixed pressure and volume, smaller systems need to be hotter; but this requirement is less severe for swarms since the temperature is squared, meaning that midges have to speed up less than ideal gas molecules do. Likewise, to maintain a fixed temperature, volume expansion must be counteracted by a reduction in pressure; but midges must lower the pressure more than a corresponding ideal gas, which is reflective of the decrease of the swarm spring constant with size.
Thermodynamic cycling
Beyond such reasoning, however, the true power of an equation of state in thermodynamics lies in specifying how the state variables will change when some are varied but the system remains in the same state, such as in an engine. To demonstrate that our equation of state similarly describes swarms, it is thus necessary to drive them away from their natural state. Although it is impossible to manipulate the state variables directly in this system of living organisms as one would do with a mechanical system, we have shown previously that time-varying acoustic30 and illumination27 stimulation lead to macroscopic changes in swarm behaviour. Here we therefore build on these findings and use interlaced illumination changes and acoustic signals to drive swarms along four distinct paths in pressure–volume space, analogous to a thermodynamic engine cycle. The stimulation protocol is sketched in Fig. 3a. The “on” state of the acoustic signal is telegraph noise (see Experimental details), while the “off” state is completely quiet. The illumination signal simply switches between two different steady light levels. Switching between the four states of “light-high and sound-on,” “light-high and sound-off,” “light-low and sound-off,” and “light-low and sound-on” with a 40-s period (Fig. 3a) produces the pressure–volume cycle shown in Fig. 3b. We suspect that the loops in the cycle stem from the swarm’s typical “startle” response after abrupt changes in environmental conditions, followed by a rapid relaxation to a steady state27,30.
Figure 3

Thermodynamic cycling of a midge swarm with (langle Nrangle = 27). Schematic of the perturbation cycle showing the illumination (solid) and sound (dashed) signal timings. The symbols indicate the switching points identified in (b). (b,c) Phase-averaged swarm behaviour during the perturbation cycle plotted in the pressure–volume phase plane for (b) the pressure signal as measured and (c) as reconstructed using our equation of state. (leftlangle {,} rightrangle_{phi }) denotes a phase average of a quantity over a full cycle. The four different states of the perturbation signal are indicated. The data has been averaged using a moving 3.5-s window for clarity. The swarm behaviour moves in a closed loop in this phase plane during this cycling, as would be expected for an engine in equilibrium thermodynamics, and the equation of state holds throughout even though it was developed only for unperturbed swarms. (d) Phase-averaged pressure (langle Prangle_{phi }) of the swarm during a continuous cycle through the four light and sound states. The blue line shows the directly measured pressure and the yellow line shows the reconstruction using the equation of state.

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In addition to the pressure and volume, we can also measure the other state variables as we perturb the swarms. Given that we do not observe any evidence of a phase transition, we would expect that our equation of state, if valid, should hold throughout this cycle. To check this hypothesis, we used the measured V, T, and N values during unperturbed experiments along with the equation of state to predict the scaling exponents, and in turn the pressure P. We then use these baseline, unperturbed exponents and V, T, and N during the interlaced perturbations to predict a pressure P. This pressure prediction matches the measured signal exceptionally well (Fig. 3c,d) even though the equation of state was formulated only using data from unperturbed swarms, highlighting the quality of this thermodynamic analogy. Although we might expect that a strong enough perturbation might lead to qualitatively different behaviour (if the swarm went through the analog of a phase transition31), our results give strong support to the hypothesis that our equation of state should hold for any perturbation that does not drive such a transition. More

The knifetooth sawfish (Anoxypristis cuspidata) is one of five sawfish species, all of which are facing extinction. Credit: Norbert Wu/Minden/Nature Picture Library

Conservation biology
12 February 2021

The strange-looking sawfish, itself a predator, falls prey to overfishing and habitat destruction.

The long-snouted, shark-like predators called sawfish have vanished from nearly 60% of their historical habitat and are nearing extinction, according to a mathematical method of counting marine species.
Sawfish (family Pristidae) are rays that live near seashores and in mangrove areas: habitats that are rapidly disappearing. To study what this means for the five species in the family, Helen Yan at Simon Fraser University in Burnaby, Canada, and her colleagues reviewed sawfish research conducted between 2014 and 2019 in 64 countries. From this, they estimated the distribution patterns of these fishes, which are known for their large size — some species routinely reach 5 metres in length — and their chainsaw-like noses.
The team found the creatures’ distribution patterns had changed in places affected by fishing pressures and habitat loss. These pressures have driven the fishes to extinction in 55 of the 90 countries whose waters they once occupied.
Saving the sawfish, the researchers say, will require countries to protect the creatures’ offshore habitats and to ban fishers from keeping sawfish that are caught in their nets. More

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