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Study systemWe conducted our research at the Jornada Caves, New Mexico, USA from 8 to 29 June 2018. This remote cave site on private land in the Chihuahuan Desert occupies an elevated volcanic plateau at approximately 1500 m altitude, with the remains of collapsed lava tubes forming a deep canyon with cave and arch features. The site was chosen because of the presence of a population of Swainson’s Hawks (Buteo swainsoni) that predates the population of Mexican Free-tailed Bats (Tadarida brasiliensis) that emerge from the caves en masse daily throughout the summer39. The bats migrate to the site during their breeding season from May to September40, and use the caves as a day roost before flying to their feeding grounds at dusk. The population consists of a maternal colony of approximately 700,000 to 900,000 bats which inhabit two connected caves named North and South. The largest and most reliable emergence was from the South cave, occurring every evening without exception. Emergence from the North cave was less reliable, with no bats emerging at all on some nights during the first week of observations. The numbers of bats were topped up in the second week by new arrivals, and emergence from the North cave was reliable thereafter. Emergence began at a variable time between approximately 18:30 and 20:00 MDT and lasted from 10 to 25 min depending on the number of bats emerging. Sunset was between 20:16 and 20:21 MDT, so the bats usually emerged in broad daylight. During the third week of observations, a substantial second emergence usually occurred at each cave, beginning around 0.5 h after the end of the first emergence, when fewer hawks were present. No ethical issues were identified by the Animal Welfare and Ethical Review Board of the University of Oxford’s Department of Zoology. We attended only as observers, and never entered the caves, so the risk of causing disturbance as the bats emerged was low41.Video observationsWe recorded video of the hawks attacking the bats every evening from 8 to 29 June 2018, except for one evening that had to be missed due to bad weather. We used three pairs of high-definition video cameras (Lumix DMC-FZ1000/2500, Panasonic Corporation, Osaka, Japan) to enable reconstruction of the three-dimensional flight trajectories of the hawks and bats, setting the camera lens to its widest zoom setting. We recorded 25 Hz video at 3840 × 2160 pixels for the first three days and 50 Hz video at 1920 × 1080 pixels for the remainder of the study (Movie S1). This higher frame rate proved necessary to facilitate tracking of the bats’ erratic movements but was traded off against lower spatial resolution. Each camera pair was set in widely spaced stereo configuration to enable three-dimensional reconstruction of the attacks, with a baseline distance of 16 to 27 m. The cameras were mounted on tripods which were adjusted to the same height using an optical level kit (GOL20D/BT160/GR500, Robert Bosch GmbH, Gerlingen, Germany). We used the same optical level kit to measure the baseline distance between the cameras.We set up two camera pairs facing approximately north and south across the South cave for the duration of the study. As the swarm’s overall flight direction was variable and influenced by the wind, we positioned the north- and south-facing camera pairs to allow them to be panned from northeast to northwest and from southeast to southwest, respectively. This enabled us to cover most flight directions, except due east (where the bats rarely flew) and due west (which was subject to glare). We set up a third camera pair to view the emergence that occurred from the North cave from the second week onward. When leaving the North cave, the bats usually flew along the lava tube and beneath a rock arch before climbing out of the canyon. We therefore positioned the cameras close to where the swarm began climbing out above the canyon rim, aiming to capture attacks as the hawks swooped low over the canyon.The hawks consistently appeared within a few minutes of the start of emergence, which enabled us to observe the general direction in which the bat swarm was emerging, and to reorient the cameras to view the swarm before the attacks began. As soon as the bats began emerging, the cameras were turned on and left to record. To begin with, all fieldworkers retreated into make-shift hides, but these were gradually phased out for reasons of practicality. The birds quickly became habituated to our presence, venturing close to the cave even when fieldworkers were present. Each attack began with the hawk approaching the swarm in level flight or stooping in from above. This was followed by fast flight through the stream of bats, with one or more attempts made to grab a bat using a pitch-up, pitch-down, or rolling grab manoeuvre with the legs and talons extended (Movie S1). If the first attack was unsuccessful, then the hawks would usually perform further short-range swoops through the stream until they made a catch. Once a bat was caught, the hawk would drift away from the swarm, to consume its prey on the wing.VideogrammetryWe synchronized the videos using the DLTdv5 video tracking toolbox42 in MATLAB R2020a (MathWorks Inc., Natick, MA). To do so, we matched the complex motions involved in the hawks’ attack manoeuvres visually between videos, and applied the relevant frame offset to synchronize them to the nearest frame. To verify the accuracy of this method, we compared the position of the hawk’s wings between the two videos for the three pairs of frames used for synchronization, and again for the three pairs of frames recorded 50 frames later (Fig. S3). This comparison shows that the frame synchronization remains stable as expected over this 1 s time interval, for the randomly selected flight displayed in Fig. S3. Nevertheless, because the cameras’ shutters were not electronically synchronized, this post hoc procedure can only guarantee synchronization of the frames to within ±0.01 s at the 50 Hz frame rate (see Fig. S3). To assess the sensitivity of our trajectory reconstructions to this remaining synchronization error, we compared the flight trajectories that we had already reconstructed with those that would have been reconstructed had the videos been shifted ±1 frame (Fig. S4). This comparison shows that the displacement of the trajectories resulting from a synchronization error of ±1 frame is small in comparison to their path length, and that their shape remains approximately the same, even for the two stooping flight trajectories plotted in Fig. S4.We used the DLTdv5 toolbox to identify the pixel coordinates of the hawk in both videos within a pair, manually tracking the visual centre of the subject’s body from the point at which it appeared in both cameras up to the point of interception. We used the same method to track the bat that the hawk caught or attempted to catch during the terminal attack sequences that we recorded at close range. The bats were too distant to be tracked individually in recordings of the hawks’ long-range approaches, but the point of actual or attempted capture was nevertheless obvious from the hawks’ flight behaviour. We aimed to reconstruct all attack trajectories that were captured by both cameras within a pair. We were able to reconstruct n = 62 terminal attack trajectories, drawn from n = 50 separate attack flights (i.e. n = 12 of these comprised follow-on attack passes, up to a maximum of four consecutive passes made in cases where the first attack pass was unsuccessful; see Supporting Data and Code for details). We were also able to reconstruct n = 26 long-range approaches. Hence, as the population of hawks peaked at approximately 20 birds, there will have been repeated sampling within individuals in both cases.We calibrated the cameras by matching 15 points across both frames, including background features and points on the hawk, which we selected with the objective of covering as much of the capture volume as possible. The image coordinates of these calibration points were exported from the DLTdv5 toolbox into custom-written software in MATLAB, which solved the camera collinearity equations43 using a nonlinear least squares bundle adjustment implemented using the MATLAB Optimization Toolbox R2020a (see Supporting Data and Code). The bundle adjustment routine identifies jointly optimal estimates of the camera calibration parameters and unknown spatial coordinates of the calibration points, by minimizing the sum of the squared reprojection error of the associated image points. The reprojection error of an image point matched across camera views is defined as the difference between its measured image coordinates and those expected under the camera calibration model given its estimated spatial coordinates. This nonlinear approach enabled us to self-calibrate the cameras using identified features of the environment, whilst also incorporating prior knowledge of the intrinsic and extrinsic camera parameters. This in turn avoided the need to move a known calibration object through the very large imaging volume.We set the calibrated baseline distance between the cameras equal to the measurement that we made of this in the field using the optical level. We fixed the focal length of each camera at 1468.9 pixels for the 1920 × 1080 recordings and at 3918.5 pixels for the 3840 × 2160 recordings. These values were estimated using the Camera Calibrator toolbox in MATLAB, from a set of 20 calibration images of a checkerboard pattern held in front of the camera. Lens distortions were found to be minimal, and we therefore assumed a central perspective projection43 in which we assumed no lens distortion and no principal point offset with respect to the camera sensor. The resulting stereo camera calibration was used to solve for the spatial coordinates of the tracked hawk and bat in MATLAB. This is a least squares solution, in the sense that it minimizes the sum of the squared reprojection error for each image point matched across stereo video frames. We therefore report the root mean square (RMS) reprojection error as a check on the accuracy of the calibrations and reconstructions.For the terminal attack trajectories filmed at close range, the mean RMS reprojection error of the 16 calibrations was 0.73 ± 0.35 pixels, whilst for the reconstructed flight trajectories it was 1.22 ± 1.18 pixels for the hawks and 1.87 ± 2.39 pixels for the bats over all n = 62 flights (mean ± SD). For the long-range approaches filmed at a distance, the RMS reprojection error of the 18 calibrations was 0.53 ± 0.61 pixels, whilst for the reconstructed flight trajectories it was 1.08 ± 1.07 pixels for the hawks over all n = 28 flights (mean ± SD). The sub-pixel reprojection error that we achieved in the calibrations is appropriate to the method. The higher reprojection error of the reconstructions is also to be expected, because whereas the bundle adjustment optimizes the camera calibration parameters jointly with the estimated spatial coordinates of the calibration points, the calibration is held fixed in the reconstructions. In addition, any spatiotemporal error in the matching of points across camera frames will manifest itself as reprojection error in the reconstructions.The foregoing calibration reconstructs the spatial coordinates of the matched image points in a Cartesian coordinate system aligned with the sensor axes of one of the cameras. To aid visualization and interpretation of the flight trajectories, we therefore transformed the spatial coordinates of the hawks and bats into an Earth axis system in which the z axis was vertical. To do so, we filmed and reconstructed the ballistic trajectory of a small rock thrown high into the air through the volume of stereo overlap. We identified the image coordinates of the peak of its parabolic path, together with the image coordinates of two flanking points located ±20 or 25 frames to either side. We took the line dropped from the peak of the parabola perpendicular to the line connecting the two flanking points to define the direction of gravitational acceleration. We then used this to identify the rotation needed to transform the spatial coordinates of the hawks and bats into Earth axes with the z axis as vertical. Finally, we made use of the fact that the two cameras in each pair were fixed at the same height to verify the transformation to Earth axes. For the 16 calibrations used to reconstruct the terminal attack trajectories, the inclination of the baseline between the cameras in Earth axes had a median absolute value of just 1.2˚ (1st, 3rd quartiles: 0.8˚, 2.2˚), providing independent validation of the calibration method that we used.Trajectory analysisAll trajectory analysis was done using custom-written software in MATLAB R2020a (see Supporting Data and Code). We used piecewise cubic Hermite interpolation of the reconstructed trajectories to estimate the spatial coordinates of the hawk or bat for any occasional frames in which this was obscured. We then smoothed the trajectories using quintic spline fitting. For the long-range approaches, we used a spline tolerance designed to remove an RMS spatial position error of 0.5 m, corresponding approximately to the wing length of a hawk. For the terminal attack trajectories, we used a tolerance designed to remove an RMS position error of 0.12 m, corresponding approximately to the wing length of a bat. These values were chosen as representative estimates of the accuracy with which it was possible to match points across frames at long and close range, respectively. Finally, we differentiated and evaluated the splines analytically to estimate the velocity and acceleration of the bird and bat at an up-sampled frequency of 2 kHz. This ensured a suitably small integration step size for the subsequent numerical simulations. On average, the hawks flew faster than the bats (Fig. S5A), so were tracked over longer distances (Fig. S5B), but with considerable overlap in their respective distributions.We simulated the hawk’s attack trajectory in the Earth axes using a guidance law of the form:$${{{{{bf{a}}}}}}(t){{{{{boldsymbol{=}}}}}}N{{{{{boldsymbol{omega }}}}}}(t-tau )times {{{{{bf{v}}}}}}(t){{{{{boldsymbol{-}}}}}}K{{{{{boldsymbol{delta }}}}}}(t-tau )times {{{{{bf{v}}}}}}(t)$$
(1)
where a is the hawk’s commanded centripetal acceleration, v is its velocity, ω is the angular velocity of the line-of-sight r from the hawk to its target, and δ is the deviation angle between r and v, written in vector form with δ mutually perpendicular to r and v. Here, t is time, τ is a fixed time delay, and N and K are guidance constants. With K = 0, Eq. 1 describes proportional navigation (PN), whereas with N = 0, Eq. 1 describes pure proportional pursuit (PP). In the case that K ≠ 0 and N ≠ 0, Eq. 1 describes mixed PN + PP guidance. Dividing through by the hawk’s speed (v=left|{{{{{bf{v}}}}}}right|) converts the commanded centripetal acceleration to the commanded angular velocity. It can therefore be seen that Eq. 1 generalizes, in vector form, the PN + PP guidance law that is written as (dot{gamma }(t)=Ndot{lambda }(t-tau )-Kdelta (t-tau )) in the main text, where the magnitudes of the scalar turn rate, scalar line-of-sight rate, and scalar deviation angle are given respectively as (left|dot{gamma }right vert=left|{{{{{bf{a}}}}}}right|/left|{{{{{bf{v}}}}}}right|), (left|dot{lambda }right vert=left|{{{{{boldsymbol{omega }}}}}}right|), and (left|deltaright vert=left|{{{{{boldsymbol{delta }}}}}}right|).Our simulations make use of the kinematic equations:$${{{{{bf{r}}}}}}={hat{{{{{{bf{x}}}}}}}}_{{{{{{rm{T}}}}}}}-{{{{{bf{x}}}}}}$$
(2)
$${{{{{boldsymbol{omega }}}}}}=frac{{{{{{bf{r}}}}}},times left({hat{{{{{{bf{v}}}}}}}}_{{{{{{rm{T}}}}}}}-{{{{{bf{v}}}}}}right)}{{left|{{{{{bf{r}}}}}}right|}^{{{{{{bf{2}}}}}}}}$$
(3)
$${{{{{boldsymbol{delta }}}}}}=left({{{cos }}}^{-1}frac{{{{{{bf{r}}}}}},cdot, {{{{{bf{v}}}}}}}{left|{{{{{bf{r}}}}}}right|,left|{{{{{bf{v}}}}}}right|}right)left(frac{{{{{{bf{r}}}}}},times {{{{{bf{v}}}}}}}{left|{{{{{bf{r}}}}}},times {{{{{bf{v}}}}}}right|}right)$$
(4)
where x is the simulated position of the hawk, and where ({hat{{{{{{bf{x}}}}}}}}_{{{{{{rm{T}}}}}}}) and ({hat{{{{{{bf{v}}}}}}}}_{{{{{{rm{T}}}}}}}) are the measured position and velocity of the target with respect to the Earth axes. Our simulations are implemented in discrete time by coupling the guidance law (Eq. 1) with the kinematic equations (Eqs. 2–4) using the difference equations:$${{{{{{bf{x}}}}}}}_{n+1}={{{{{{bf{x}}}}}}}_{n}+Delta t,{{{{{{bf{v}}}}}}}_{n}.$$
(5)
$${{{{{{bf{v}}}}}}}_{n+1}={hat{v}}_{n+1},frac{{{{{{{bf{v}}}}}}}_{n}+Delta t,{{{{{{bf{a}}}}}}}_{n}}{left|{{{{{{bf{v}}}}}}}_{n}+Delta t,{{{{{{bf{a}}}}}}}_{n}right|}$$
(6)
where the subscript notation indicates the values of the variables at successive time steps, such that ({t}_{n+1}={t}_{n}+Delta t), and where (hat{v}) is the hawk’s measured groundspeed. The simulations were initiated given the hawk’s measured initial position ({{{{{{bf{x}}}}}}}_{0}={hat{{{{{{bf{x}}}}}}}}_{0}) and velocity ({{{{{{bf{v}}}}}}}_{0}={hat{{{{{{bf{v}}}}}}}}_{0}), and were used to predict the trajectory that it would follow under the guidance law (Eq. 1) parameterized by the guidance constants N and K, and time delay τ. Note that Eq. 6 matches the hawk’s simulated groundspeed (v=left|{{{{{bf{v}}}}}}right|) to its measured groundspeed (hat{v}) at all times, such that the guidance law is only used to command turning. We verified that the step size of our simulations ((Delta t=5times {10}^{-4}) s) was small enough to guarantee the numerical accuracy of the fitted guidance parameters and prediction error to the level of precision at which they are reported in the Results.We defined the prediction error η of each simulation as the mean absolute distance between the measured and simulated flight trajectories:$$eta=frac{1}{k}mathop{sum }limits_{n=1}^{k}left|{{{{{{bf{x}}}}}}}_{n}-{hat{{{{{{bf{x}}}}}}}}_{n}right|$$
(7)
where (hat{{{{{{bf{x}}}}}}}) is the hawk’s simulated position, and k is the number of time steps in the simulation. We fitted the guidance constants K and/or N under the various combinations of guidance law (i.e. PN, PP or PN + PP) and target definition (i.e. measured bat position, final bat position, final hawk position) for delays of 0 ≤ τ ≤ 0.1 s at 0.02 s spacing corresponding to the inter-frame interval. In each case, we used a Nelder–Mead simplex algorithm in MATLAB to find the value of K and/or N that minimised the prediction error η for each flight at the given time delay τ. To ensure that we fitted the same section of flight for all time delays 0 ≤ τ ≤ 0.1 s, we began each simulation from 0.1 s after the first point on the trajectory, and ended the simulation at the time of intercept or near-miss. However, as we found the best-fitting delay to be τ = 0, we subsequently re-fitted the simulations with no delay to begin from the first point on the trajectory and report these simulations in the Results. For the terminal attack trajectories, we took the first point on the trajectory to be the earliest point from which it was possible to track the bat that the hawk caught or attempted to catch, and took the time of intercept or near-miss to be the time at which the measured distance between the hawk and bat was minimal. For the long-range approaches, we tested a range of alternative start points from 1.0 s up to a maximum of 20.0 s before the observed grab manoeuvre, in 0.2 s intervals, to accommodate the fact that the hawk could sometimes be tracked for longer than it appeared to be engaged in directed attack behaviour.Statistical analysisAll statistics were computed using MATLAB R2020a. As the hawks could not be individually identified, we were unable to control for repeated measures from the same individual, and therefore treated each attack trajectory as an independent sample. Because the distributions of the model parameters and errors are skewed (Fig. 2), we report their median, denoted using tilde notation, together with a bias-corrected and accelerated bootstrap 95% confidence interval (CI) computed using 100,000 resamples44. For robustness, we use two-tailed sign tests to compare their distributions between different guidance models and target definitions. We state sample proportions together with a 95% confidence interval (CI) computed using the Clopper–Pearson method. We used a two-tailed Fisher’s exact test to compare the odds of success in attacks on lone bats versus attacks on the swarm. Following our previous observational study18, bats classified as lone bats were judged to be flying >5 body lengths from their nearest neighbours and/or appeared to be flying in a different direction to the coordinated members of the swarm (Table S3).Reporting summaryFurther information on research design is available in the Nature Research Reporting Summary linked to this article. More

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(1)
where (x_{i} left( t right)) = (T_{i} left( t right)), (M_{i} left( t right)), (R_{i} left( t right)). We obtained a Pearson correlation matrix for each time series as follows:$$R_{S} = left[ {begin{array}{*{20}c} {r_{11} } & cdots & {r_{1N} } \ vdots & {r_{ij} } & vdots \ {r_{N1} } & cdots & {r_{NN} } \ end{array} } right]$$
(2)
where (S = T, M, R).We calculated the average value, (overline{r }), and the standard deviation, (sigma), for the correlation coefficients of the matrix. We applied the threshold method to generate a sparse network from the correlation matrix. Two countries are connected in the correlation network if and only if the value of the correlation coefficient is greater than, or equal to, the threshold value:$$r_{{ij}} = left{ {begin{array}{*{20}c} 1 & {{text{if}};r_{{ij}} ge bar{r}{text{ + n}}sigma } \ 0 & {{text{otherwise}}} \ end{array} } right.$$
(3)
where (r_{ij}) is the correlation coefficient between two countries, and (n) is an element of real numbers ((n in {mathbb{R}})). The value of (n) determines whether the network is sparsely or densely connected.We use Python programming language, packages, numpy for mathematical functions and random number generator, pandas for data analysis and manipulations, networkx for creation, manipulation, and studying the structure of the complex network, matplotlib for visualization and plotting graph and base map for map projection and visualization of geographic information. More

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