Falcons pursue prey using visual motion cues: new perspectives from animal-borne cameras

How predators track and capture their prey poses a fundamental problem in animal behavior that combines sensory perception, neural computation and locomotion. Empirical studies in combination with computational modeling (Pais and Leonard, 2010; Reddy et al., 2006; Srinivasan and Davey, 1995) have identified pursuit strategies used by insects (Olberg, 2012), bats (Ghose et al., 2006), dogs (Shaffer et al., 2004), fish (Lanchester and Mark, 1975) and humans (Fajen and Warren, 2004; McBeath et al., 1995). However, no empirical studies have addressed how falcons and other birds pursue flying prey, mostly due to the difficulty of recording their 3D flight trajectories in the field. The pursuit strategies used by birds have evolved in the context of their unique flight capabilities, as well as their need to pursue rapid, erratically moving prey in complex environments. Understanding their methods for tracking and following rapidly moving objects should provide inspiration for the design of unmanned aerial vehicles (UAVs) and other biomimetic robots.

Stereoscopic video methods successfully used to study flocking (Ballerini et al., 2008a; Cavagna et al., 2010; Cavagna et al., 2008) are challenging to apply to this problem because of the wide geographic areas covered and the rapid, unpredictable motion of both predator and prey. However, bird-mounted sensors offer new opportunities for this field. While miniaturized, bird-mounted GPS sensors have been used to study navigation, flocking energetics and decision making in bird flocks (Biro et al., 2006; Nagy et al., 2010; Steiner et al., 2000; Usherwood et al., 2011), the 10 Hz data collection rate and 0.3 m spatial resolution of the GPS are of limited use in the present context. By contrast, miniaturized bird-mounted cameras have proved effective in studying the aerodynamics of avian flight (Carruthers et al., 2007; Gillies et al., 2011) and bird behavior in other contexts (Bluff and Rutz, 2008; Moll et al., 2007; Rutz and Bluff, 2008).

Here, we consider how animal-borne video data can distinguish between different proposed pursuit strategies that use perceptual cues from the environment. The simplest strategy is classical pursuit, in which the pursuer (the predator) always flies directly toward the evading prey (Nahin, 2012) (Fig. 1A), so the evader's image is centered on the pursuer's visual field. Chases consistent with classical pursuit have been observed for honeybees (Zhang et al., 1990), flies (Land, 1973; Land, 1993; Trischler et al., 2010) and tiger beetles (Gilbert, 1997).

In another strategy, termed motion camouflage (Justh and Krishnaprasad, 2006), the pursuer maneuvers to reduce parallax-based cues on the evader's visual field (Reddy et al., 2006), a behavior observed in dragonflies (Olberg, 2012), hoverflies (Srinivasan and Davey, 1995), bats (Ghose et al., 2006) and humans (Anderson and McOwan, 2003). The pursuer can achieve motion camouflage by maneuvering to keep the evader's apparent position on the pursuer's visual field at a fixed angle (Fig. 1C); here, we assume that the pursuer's head is maintained at a constant angle with respect to its velocity. For an evader that moves in approximately linear trajectories with occasional changes in speed or direction, this angle agrees with the value of ϕ_o from Eqn 1 and CBDR for each region of constant evader velocity. Eqn 1 sets the pursuer's velocity

perpendicular to the baseline equal to that of the evader, thus ensuring that the baseline vector always points in the same direction, as shown by the constant (‘absolute’) value of α, the constant target angle. For this reason, this strategy is also called constant absolute target direction (CATD). Consequently, the CATD version of motion camouflage can be considered an extension of CBDR to the case of maneuvering prey. CATD results in shorter times to capture than classical pursuit for occasionally erratic prey motion.

The complexity of avian vision must be considered in understanding pursuit strategies used by birds. So far, empirical studies of how birds visually perceive their surroundings during flight (Erichsen et al., 1989; Land, 1999; Martin, 2011) have not explored how the apparent motion of prey on avian predator's visual fields influences pursuit strategies. However, research on the interaction between vision and flight in several avian species has provided evidence that optical flow cues are used by budgerigars navigating their environment (Bhagavatula et al., 2011), zebra finches avoiding obstacles (Eckmeier et al., 2008), hummingbirds feeding (Delafield-Butt et al., 2010; Lee et al., 1991) and Harris hawks and pigeons landing on a perch (Davies and Green, 1990; Lee et al., 1993).

Raptors, in particular, have large, forward-facing eyes with such a limited range of eye motion (2–5 deg) (Jones et al., 2007) that they must move their heads to scan their field of view (O'Rourke et al., 2010; Wallman and Pettigrew, 1985). Their effective visual fields are complicated by the fact that each raptor eye has two foveae (retinal regions with enhanced visual acuity) oriented at different angles with respect to the head axis (Fig. 1D). The shallow, or temporal, fovea has a line of sight oriented at 9–16 deg from the head's forward direction and is used primarily for visualizing objects at close range (<8 m) with either monocular or binocular vision (Tucker, 2000). Perched raptors prefer to view novel or more distant objects monocularly, using the deep, or central, fovea, oriented with its line of sight at an angle >30 deg to the head axis (Frost et al., 1990; Lord, 1956; Tucker, 2000; Wood, 1917).

As a result, a flying falcon cannot view distant prey at an optimal angle for acute vision with its deep fovea and also face its head forward; however, flying with its head turned to favor the deep fovea significantly increases drag. Tucker and colleagues have argued for a pursuit strategy in which falcons can view prey at the optimal visual angle while keeping their heads facing forward by flying along trajectories that resemble logarithmic spirals (Tucker et al., 2000) (Fig. 1E). In this model, the falcon maintains a constant angle, determined by the orientation of its deep foveal field, between the line of sight to the prey and the falcon head axis. The falcon is assumed to fly with its head oriented along its velocity, so its bearing angle is fixed at ϕ_f>30 deg. Using a telescopic tracking device, Tucker and colleagues established that falcon pursuit trajectories are indeed curved, although their results could not quantitatively distinguish between the possible theoretical pursuit models (Tucker et al., 2000). Walking flies have been shown to approach a stationary black–white edge using a logarithmic spiral trajectory, consistent with using a fixed bearing angle (Osorio et al., 1990).

In the optimal visual angle model, the value of ϕ_f should be fixed at a constant value (e.g. ~30 deg or 9–16 deg for the deep or shallow foveal fields, respectively), independent of predator and prey velocities. By contrast, in CATD the value of ϕ_o depends explicitly on v_p/v_e and β, while for classical pursuit, we always have ϕ=0 deg. Thus, one can distinguish between these three strategies by measuring the behavior of ϕ during pursuits: for optimal visual angle and classical pursuit, the value of ϕ is held at the corresponding constant values for all times, while for CATD the value of ϕ should be constant (and equal to ϕ_o) as long as the prey does not maneuver, but assumes a new constant value once the prey changes its velocity.

Even though each strategy considered here predicts the predator maintains a constant bearing angle, it is important to note that constant bearing angle does not necessarily correspond to a spiraling predator trajectory. Because both CATD and CBDR result in both a constant bearing angle and a constant baseline direction, the predator's trajectory is locally straight, not spiral.

Here, the question of which pursuit strategy is used by falcons was addressed using the results of a study of three species of falcons hunting avian prey in the field while wearing miniature videocameras on their heads or backs. This approach overcomes the naturally low pay-off ratio of filming predation in the field through collaboration with several skilled falconers who routinely hunt with falcons. The empirical findings were interpreted using computer simulations of a model predator pursuing flying prey using each of the proposed pursuit strategies.

Computer simulations were performed to model classical pursuit, optimal visual angle and CATD, but not CBDR because the falcon's avian prey maneuvers frequently during chases. The results were displayed as a simulated video image of the prey as it would appear on a camera mounted on the head of the pursuing predator (see details in Materials and methods). Prey positions on the video are displayed as a plot of the prey's horizontal (θ) and vertical (χ) angles with respect to the camera's optical axis, defined as (0 deg, 0 deg) (Fig. 2A). Typical results are shown in Fig. 2B for all times during sample chases for each strategy considered. Constant values of v_e=7.4 m s⁻¹, v_p=10 m s⁻¹ were used, and initial distance between falcon and prey (z) was fixed at 40 m; the direction of prey velocity was initially β≈6.8 deg and 90 time steps between prey maneuvers were chosen to conform with video data discussed below. For classical pursuit, the prey's simulated (θ, χ) was maintained at (0 deg, 0 deg) except at very low values of z (not shown). There, perspective effects shifted the apparent prey location to lower χ values for all models, due to the camera's location above the body and head axes. We did not model CBDR as a distinct case, as its predicted predator motion via Eqn 1 is already incorporated into the algorithm used here for CATD for a maneuvering prey. For optimal visual angle and CATD, the prey's apparent location remained at fixed angles (θ, χ), although the non-zero response time resulted in excursions from the optimal angles when the prey abruptly maneuvered. Thus, for CATD, the simulations yielded a series of apparent prey positions at values of (θ, χ) that depended on prey and predator velocity via Eqn 1, while for optimal visual angle at 45 deg (Tucker et al., 2000), these values were independent of prey maneuvering. For each choice of prey β as it maneuvered, a different average value (θ, χ) was achieved after a small response time lag. Using a realistic response time resulted in a small spread of angles about the optimal values due to the time lag between the prey's maneuvers and the model predator's response. (A more realistic model for how the falcon accelerated during maneuvers would have added to this variation.)

The results of pursuit sequences recorded by bird-mounted cameras were analyzed to measure the prey's apparent motion on the video image, its distance with respect to the falcon and the falcon's roll angle versus time. Here, a chase was defined as a sequence in which the falcon initiated a pursuit and pursued the prey until losing it, capturing it or attempting to capture it. The falcons studied here [peregrine falcon (Falco peregrinus Tunstall) and gyrfalcon (Falco rusticolus L.)/Saker falcon (Falco cherrug Gray), hereafter termed hybrid falcon] used the most common tactics observed in studies of falcon attacks on individual birds (Dekker, 1980; Dekker and Lange, 2001) and flocks (Zoratto et al., 2010): the stoop, a steep, rapid dive from above the prey; and the tail chase, in which the falcon uses powered level flight to pursue prey.

The head-mounted videos of the hybrid falcon recorded 15 chases of carrion crows (Corvus corone) over fields in Belgium, while those for the gyrfalcon recorded five chases of Houbara bustards (Chlamydotis undulata) in the desert in Dubai. A total of 28 back-mounted videos of peregrine falcons hunting crows and other birds in a variety of terrains in the UK, Belgium and the USA were also analyzed. The prey (θ, χ) versus time data fell into three motifs: (1) optical fixation with the prey's image held at approximately constant angles (θ, χ) was present in 78% of the head-mounted and 54% of back-mounted videos (Fig. 3); (2) constant angular rate of change along either θ or χ was observed in 22% of the remaining head-mounted and 29% of the remaining back-mounted videos (Fig. 3B); and (3) erratic motion of the prey on the image was observed in the remaining videos, which occurred at close distances when prey accelerations resulted in large angular excursions. The video data did not indicate that falcons kept the prey's image fixed with respect to distant objects.

The hybrid falcon made saccadic head motions during two chases; head saccades were distinguishable from prey maneuvering or acceleration by the falcon by the very rapid (≤0.1 s) motions of the entire visual field and by its rapid return to the original position in only 0.33±0.12 s (mean ± s.d.). For one chase video that recorded six head saccades, it was possible to extrapolate (θ, χ) for the prey when it was off-camera by using the distant landscape's angular movement (Δθ, Δχ) added to the prey's values of (θ, χ) immediately before and after the saccade (Fig. 4A). Most of the saccadic motion was along θ, with a change in the vertical direction of only Δχ=0.21±1.8 deg (mean ± s.d.). The prey's extrapolated values were θ=36±3 deg (mean ± s.d.) during these events; values extrapolated immediately before and after the falcon turned its head agreed within experimental uncertainties.

The back-mounted video could sometimes image the falcon's head motion directly during foraging for prey and pursuits (Fig. 4B). These videos showed that peregrine falcons usually oriented their heads in the forward direction. However, they did regularly turn their heads from side to side to scan the world during foraging flights, looking downward to scan the ground with either their left or right eyes with equal frequency.

In three chases recorded by a camera on a hybrid falcon, the video recorded a second hybrid falcon intercepting prey birds in the air. These videos showed that the second hybrid falcon approached the crows with a velocity directed to an angle with that of their prey (Fig. 4C). At the pursuit's conclusion, the falcon needs to have its final velocity vector oriented directly toward the prey so it can strike the prey with its feet (Fox, 1995; Goslow, 1971). In our videos, ≤230 ms before impact, the falcon was observed to splay its tail and spread its wings in anticipation of contact; the falcon's feet went from fully retracted to fully extended in ≤67 ms. In addition, we obtained similar values for the times at which raptors spread their wings and extend their feet before impact from our analysis of high speed video of a red-tailed hawk and peregrine falcon attacking falconry lures (Destin, 2012).

The head-mounted video also allowed determination of the prey's position relative to the head axis angle during pursuits. To determine whether falcons use a preferred eye while chasing prey, we performed a statistical analysis of the distribution of θ and χ values during pursuits. We verified that the small (2–5 deg) full range of eye motion measured for other raptor species also applies to falcons by examining close-up videos of gyrfalcon and gyrfalcon/Saker falcon hybrids posted online. As a result, we estimate that θ and χ should agree with the predator's actual bearing angles within ±1–2.5 deg for the head-mounted video. Thus, by determining the fraction of prey images recorded for negative or positive θ, we also could determine whether the falcon favored its left or right eye for viewing prey. We analyzed these data for head-mounted video from the hybrid falcon and the gyrfalcon and from back-mounted video data from the peregrine falcons.

Fig. 5 shows the resulting distributions of prey θ and χ for the hybrid falcon and for the gyrfalcon. These results show the majority of prey positions at negative θ, strongly suggesting that falcons prefer to view prey using their left visual fields. The hybrid falcon data for distant chases (defined here as z>8 m) had a mean θ=−8.8±0.2 deg (n=1277) with 94% of the prey images in the left visual field. For distant chases, the gyrfalcon data had 79% of the prey images in the left visual field, and an estimated mean of θ=−7.6±0.3 deg (n=517, where n=number of video frames analyzed), where the large error bars are due to limitations from the limited camera calibration information available.

By contrast, for pursuits by the hybrid falcon where the prey was ≤8 m away, the θ distribution had a lower mean value of −2.8±0.6 deg (n=231) and 72% of measured θ values were ≤0 deg. For the gyrfalcon, average θ=−1.1±0.8 deg for close-up chases (n=201).

For the hybrid falcon data, the vertical angles were distributed about the horizontal, with the mean of the distribution of χ for chases >8 m away at 2.4±0.1 deg, and for z<8 m at 0.0±0.3 deg; for the gyrfalcon data, the distribution of χ was centered at −1.6±0.3 deg. Given mounting angle uncertainties of ±2 deg along χ, this is consistent with viewing the prey near the center of the falcon's visual field along the vertical.

For the back-mounted geometry, we assumed that the bird's head was primarily oriented along the body axis, such that we could still determine left–right asymmetry from the distribution of prey images along χ and θ. This was consistent with the back-mounted videos in which the bird's head axis was seen to be almost always oriented parallel to the body axis, as well as our survey of videos posted on the internet showing head orientation in flying falcons. The back-mounted video for peregrine falcons showed an equal distribution of prey images on the right and left sides of the image, indicating no observable lateralization of prey θ values during pursuits for the peregrine falcon videos.

The empirical video data showed that, when possible, the falcons flew so as to maintain approximately constant non-zero prey angles along the horizontal and vertical directions. Constant prey angle with θ≠0 deg could be consistent with classical pursuit if the videocamera were oriented with its optical axis at a non-zero angle to the falcon's head axis. In case of such misalignment, for classical pursuit the prey's image would be stationary at a non-zero angle because it would be at the focus of expansion on the optical flow field, which would be offset on the camera's field of view. As mentioned earlier, we ruled out this effect by measuring the optical flow field to rule out such offsets in the camera's mounting angle. Thus, the apparent prey motion captured on video does not agree with classical pursuit.

We now consider the extent to which the finding that falcons maintain the prey at a constant bearing angle for long intervals of time during pursuits is consistent with either CATD or optimal visual angle. For distant chases, θ was peaked at values of −8.8±0.2 deg for the hybrid falcon and −7.6±0.3 deg for the gyrfalcon. Although the shallow fovea's optimal angle has not been measured for these species, values for other raptors range from 9 to 15 deg. The measured θ range disagrees with the range of 30–45 deg found in other raptors for the deep fovea's line of sight. Thus, the range of observed θ is consistent with optimal visual angle strategy using the shallow, but not the deep, fovea.

However, the measured constant θ values were also consistent with those predicted by CATD for plausible values of predator and prey velocity and prey maneuvering. The definitive test for CATD is constancy of the baseline vector's orientation (Fig. 6C), but this measurement was not feasible because of the difficulty of performing 3D imaging of the wide-ranging birds. Instead, computer simulations and animal-borne video were combined to provide an alternative way to distinguish between these two strategies. As previously noted, for optimal visual angle, the prey's position should remain at a fixed value of θ independent of predator and prey velocity and set by the foveal optimal line of sight. By contrast, the values of constant θ predicted by CATD vary when the prey maneuvers, in agreement with what is seen in the video data. For example, compare Fig. 2B and Fig. 3A: while the optimal visual angle predictions always correspond to the same θ value, CATD results in clusters of different constant θ values as the prey maneuvers in the simulations, just as observed in the actual videos.

We might be able to reconcile the optimal visual angle strategy with the video data if the falcon's eyes could undergo saccades of the angular magnitude observed, as this would create a mismatch between the measured θ and the actual angle on the retinal field. However, this discrepancy should be at most ±1–2.5 deg, based on observed raptor eye motions in other species and our own survey of videos showing eye motion in gyrfalcons and gyrfalcon/Saker falcon hybrids. Instead, we observed multiple cases where the change in (θ, χ) values between apparent optical fixes was ≥10 deg. Thus, the observed heterogeneity in the measured constant θ values supports CATD over optimal visual angle, assuming that falcon eye motions this large are unlikely.

Overall, these data provide strong evidence that the falcons studied use visual motion-based steering laws while pursuing flying prey. The video data indicate that the falcon maneuvers so that the prey's constant (θ, χ) values do not contribute to non-zero flow vectors on the optical flow field. We suggest that the actual strategy used may be a synthesis of both CATD and optimal visual angle. Recall that the falcon can control the value of θ with which it pursues the prey by varying its speed (Eqn 1). These birds can fly fast enough to pursue their prey with values of θ not in agreement with their shallow foveal fields. Instead, the falcons studied adjusted their velocities during pursuits so as to utilize CATD while maintaining the prey's θ at values near-optimal for its shallow fovea. This has an additional advantage, as the range of most observed horizontal prey angles was consistent with the measured binocular overlap region of ±10–18 deg measured in other raptor species (Martin and Katzir, 1999; O'Rourke et al., 2010).

When not sustaining constant bearing angles, the falcons often maintained the apparent prey motion at a constant angular rate of change. For small falcon–prey distances, the falcon pursues its prey at smaller average visual and bearing angles before interception and capture. While data for the hybrid falcon and gyrfalcon showed a strong preponderance of prey fixes in the left visual field, data for the peregrine falcon from back-mounted video did not show any evidence of lateralization. Other non-raptor bird species have been found to favor the left or right eye for specific tasks, presumably due to brain lateralization (Rogers, 2012). Further work will attempt to discern the extent to which the observed lateralized prey positions are found in other birds.

These results indicate that the falcons only occasionally view distant prey using their deep fovea during pursuits. By instead viewing the prey using its shallow fovea, the falcon can take advantage of higher acuity vision, (possibly) binocular vision and a shorter interception path while reducing drag by keeping its head oriented forward. The hybrid falcon was observed to perform head saccades consistent with imaging the prey using its deep fovea in one chase. The falcons also occasionally tilted their heads while foraging, presumably to scan the ground for prey. These results agree with findings of an earlier study of head motions used by flying gull-billed terns (Gelochelidon nilotica, another bifoveate species) foraging for and capturing prey (Land, 1999). The resulting eye orientation was consistent with their using the deep fovea for this purpose, without any evident lateral preference. Given these results, it would be worth examining results for helical raptor soaring trajectories for evidence of a preferred handedness (Akos et al., 2008). Finally, the preference for viewing prey with an average χ≈0 deg agrees well with the fact that raptors have a high acuity horizontal retinal streak (Inzunza et al., 1991; Jones et al., 2007).

In the one instance of prey capture recorded on video, we found that the falcons began to spread their wings and brake for impact with the prey at ≤230 ms and that their feet went from fully retracted to fully extended in ≤67 ms, consistent with our analysis of high speed videos of raptor strikes (results not shown) (Destin, 2012) and with published values of 210 ms [the time at which Harris hawks were found to begin preparing to land on a perch (Davies and Green, 1990)] and 60–100 ms (the time for foot extension for peregrine falcons attacking birds on the wing (Goslow, 1971)].

These results have implications for the co-evolution of pursuit-evasion mechanisms. Most prior research on optimal evasive responses by birds has focused on locomotion rather than sensory inputs (Hedenstrom and Rosen, 2001; Howland, 1974; Van den Hout et al., 2010; Weihs and Webb, 1984). A recent review of prey escape strategies has commented on the tendency of prey to double back and move toward the pursuing predator (Domenici et al., 2011), a strategy found to result in lower capture rates for owls (Shifferman and Eilam, 2004). Such prey maneuvers may be attempts to move out of the predator's high acuity visual field. One study allowed pursuer and evader visual systems to evolve computationally in response to the outcomes of simulated pursuits and captures; this work found that the angle for optimal vision for pursuers agreed with the observed orientation of the shallow or the deep fovea (Cliff and Miller, 1996). In the absence of empirical data, computational models of predation on flocks (Ballerini et al., 2008a; Ballerini et al., 2008b; Lee, 2006; Lee et al., 2006; Mecholsky et al., 2010) have assumed classical pursuit. These results should be revisited to model how likely pursuit strategies used by falcons on single birds influence their attacks on flocks.

The emerging evidence indicates that a wide variety of animals utilize visual motion-based cues and CATD/motion camouflage-based strategies for the pursuit and capture of individual prey. It also reinforces the importance of using a sensory ecology approach (Martin, 2012) in understanding problems such as bird collisions with man-made objects (Martin and Shaw, 2010) and antipredator surveillance (Fernández-Juricic, 2012). The complexity and extreme speed of the falcon's attack and its avian prey's evasive response raise important, unresolved questions. Raptor pursuit strategies have evolved under the constraints of avian flight aerodynamics, prey behavior and environment. How optimal are the actual pursuit strategies used by falcons? How do their hunts adapt to different hunting tactics? What import do these results hold for the study of attacks on flocks? How do prey evasive strategies relate to the falcon's hunting methods? Our ongoing experimental and modeling efforts will explore these issues by extending pursuit-evasion models into three dimensions and adding in realistic predator-maneuvering capabilities, as well as exploring different ways to model the fitness of pursuit strategies.

This study used six peregrine falcons (F. peregrinus) and two gyrfalcon (F. rusticolus)/Saker falcon (F. cherrug) hybrids (hybrid falcons) flown and housed by participating falconers who were provided with videocameras, mounts, data storage media and instructions for using the equipment and conducting and documenting the field studies. The experiments explored predation by free-flying falcons in the field in all cases. All prey were wild, free-living birds encountered by the raptors during hunting flights. Most videos analyzed here were recorded in collaboration with four falconers who hunted with six different peregrine falcons and two hybrid falcons, in the USA (Pennsylvania, Arizona and Wyoming) and in the UK, The Netherlands and Belgium. Filming occurred during autumn and winter 2009–2013. All falconers were licensed and had all necessary permits; all personnel involved followed the relevant regulations and laws of each country in question, as well as those of Haverford College's Committee on Animal Care and the ARRIVE guidelines (NC3Rs, 2010). All back-mounted videos were filmed with peregrine falcons. The head-mounted camera was used only with a hybrid falcon because of that species' greater mass. The male hybrid falcons hunted in a pair while one of the birds wore a video camera. Only one of the hybrid falcons was habituated to wearing the head-mounted camera, so the video data presented here are from one individual. In two chases, the hybrid falcon wearing the camera recorded the other hybrid falcon pursuing and capturing a prey bird. In the remaining chases, the falcon wearing the camera recorded the prey bird during a solo pursuit. In a few cases, videos were drawn from publically available archives on the internet where they had been posted by falconers, including five chases with a gyrfalcon wearing a head-mounted camera hunting Houbara bustards (Chlamydotis undulata) in the desert in Dubai (Abu Dhabi Sports Council, 2011), and two pursuits of ring-necked doves (Streptopelia capicola) by a peregrine falcon wearing a back-mounted camera in Arizona (Gonzales, 2010).

The back-mounted cameras and battery packs for the head-mounted cameras were mounted on unanesthetized birds using Marshall Radio Telemetry (North Salt Lake, UT, USA) Trackpack backpacks designed to secure radio telemetry transmitters on falcons and hawks (Fig. 6A). These mounts use ¼ in Teflon ribbons to hold the cameras stably between the falcon's wings, just above and to the rear of the bird's head. In a few cases, we were able to obtain video recorded from videocameras mounted in specially made falconry hoods (Fig. 6B). Falconers experienced in hood-making prepared hoods customized for each falcon and acclimated the birds to wearing them before their actual use in hunting.

We used commercially available miniature (mass 14 g) model 808 camcorders (Toplanter, Huizhou, China). Including mounting hardware, the total mass was 20 g, <3% of the mass of the smallest falcons used, in accordance with international standards (Fair et al., 2010); other sensors were not used to keep within this limit. Units of <5% body mass have been shown to affect falcon flight minimally (Pennycuick et al., 1994). There were no adverse outcomes for any of the falcons in the study due to the use of the videocameras using either mounting method. The videocameras operated at capture rates of 29.97 frames s⁻¹, shutter speeds ≈0.01 s, and 1280×720 pixel resolution. Micro SD memory cards allowed video and audio data to be stored onboard the units during filming for later recovery, allowing recording times of 1–2 h per battery charge.

For videos recorded by cooperating falconers, the camera's optical axis was mounted parallel to the bird's head or body axis and located a distance h=2.4±0.5 cm above its eyes for the head-mounted cameras and 3.0±0.5 cm for the back-mounted cameras (Fig. 6A,B). For videos recorded elsewhere, we had limited information about the videocamera specifications, but were able in most cases to estimate camera focal lengths by measuring approximate distances and sizes of known objects in the videos.

Like most birds, falcons maintain head nystagmus, holding their heads and gaze stable even during rapid turns and banking (Davies and Green, 1990; Warrick et al., 2002; Warrick and Dial, 1998). As a result, the head-mounted video was filmed with the camera image plane oriented along the Earth horizontal and vertical, and along the bird's body axis to within ±2 deg (corresponding to pitch, roll and yaw mounting errors in the bird's body frame). These values for mounting uncertainties were measured from photographs of the hood-mounted cameras on the actual falcons. Falcons and other raptors ordinarily fly with their heads oriented along the forward velocity (something our back-mounted videos and our survey of videos of falcons in flight confirmed); as a result, the back-mounted video cameras should also be centered on the head axis to this same tolerance. Thus, the apparent prey angle on the image should provide a measure of the visual angle (the angle between its line of sight and its head axis) of the prey on the falcon's visual field and, equivalently, the bearing angle, ϕ.

By contrast, back-mounted cameras had the camera image's horizontal axis oriented left to right in the bird's axial plane, and the camera image's vertical axis was oriented along the bird's ventral–dorsal axis (Fig. 6A). As a consequence, objects in the environment rotated on the back-mounted video when the falcon's body rolled from side to side. By separately measuring roll using the orientation of a distant, level horizon, we were able to correct for this by rotating the resulting images to compensate.

We performed calibrations of the bird-mounted videocamera's focal length, distortion parameters and other optical parameters (Bradski and Kaehler, 2012; Cyganek and Siebert, 2009) using OpenCV version 2.3.1 programs written in Python 2.7 (http://www.python.org). The results indicated that the pinhole optics resulted in only small barrel distortions with a standard deviation of 0.8 pixels and a maximum range of deviation of ±1.5 pixels over the entire 1280×720 pixel image. The resulting focal lengths also were measured to ±0.5%, using filming of objects of known length as an additional check. For the head-mounted video of a gyrfalcon from another source, wide-angle lens distortions limited our ability to perform analyses of the angles (θ, χ) (Abu Dhabi Sports Council, 2011) beyond approximate values near the prey's average apparent position.

To analyze the tracks of birds and other objects on video, we used the image analysis program ImageJ (National Institutes of Health, Bethesda, MD, USA). Videos were first converted into image stacks using VirtualDub (http://www.virtualdub.org), checking that no frames were dropped or added. Separate checks were performed by filming a fast clock to make sure that the videocameras did not skip or add frames. The image stacks were then converted to grayscale, the background subtracted when possible, and an image intensity threshold was applied to obtain a binary image of only moving prey birds. Tracking was performed using one of two ImageJ plugins: MTrack2 (for automatic tracking) or MTrackJ (for manual tracking). [Particle image velocimetry methods were attempted, but did not yield useful results because of the falcon's erratic motion and the resulting unstable rotating and moving high-contrast background.] Fitting, statistical analysis, Fourier transforms, correlation functions, and other data analysis was carried out in Origin 8.6 (OriginLab, Northampton, MA, USA). All error bars reported here are s.e. unless noted otherwise. The estimated error in tracked prey angles was ±0.2 deg.

To measure roll angle from the bird camera video, we used the orientation of the distant level horizon when visible. The horizon's orientation was automatically detected using an Opencv 3.2 program written in Python 2.7. Each image frame was individually converted to grayscale and smoothed using a Gaussian filter, then subjected to an adaptive threshold to compensate for non-uniform illumination and five rounds of dilation/erosion filters to remove speckle noise while preserving large features. Finally, a Canny filter and probabilistic Hough Line transform were used to find the orientation of the horizon lines, which were inspected by hand to confirm only distant horizon lines were analyzed (Bradski and Kaehler, 2012).

Computer simulations of birdcamera videos generated by each pursuit model considered were written in Python 2.7. All simulations assumed the chase was confined to a horizontal plane because this sufficed to model the behavior of interest here. Simulation parameters were drawn whenever possible from empirical data for flight speeds and aerodynamic parameters. Maximum flight speeds were unavailable for carrion crows, but values from 7.4 to 15.6 m s⁻¹ have been reported for other species of crows (Broun and Goodwin, 1943; Schnell and Hellack, 1978; Verbeek and Caffrey, 2002). For peregrine falcons, empirical data for maximum speeds in level flapping flight ranged from 10 to 31 m s⁻¹ (Alerstam, 1987; Cochran and Applegate, 1986; Pennycuick, 1997; Pennycuick et al., 1994). Measured flight speeds for diving and stooping falcons range from 30 m s⁻¹ for diving peregrine falcons (Alerstam, 1987) to 58 m s⁻¹ for stooping gyrfalcons (Tucker et al., 1998). Roll angles were computed using the balanced turning approximation (Pennycuik, 2008) and aerodynamic parameter estimates for falcon lift coefficient=0.53 to 1.65 at Reynolds number=10⁵ and wing area=0.132 m² (Tucker, 1998). The simulations assumed no limitations on turning radius or other maneuvers required to implement each strategy, because birds can separately move and shape each wing (and their tails) to achieve low radius turns, abrupt rolls and other maneuvers not permitted by fixed wing aerodynamics (Carruthers et al., 2007; Pennycuik, 2008; Warrick et al., 2002).

Birds, including raptors, can process visual information at rates ≥90 Hz (Fox, 1995; Jones et al., 2007). Consequently, the simulation time step was fixed at Δt=1/(3×29.97 Hz)=11.1 ms≈1/90 Hz to agree with this limit; this value was also close to an integral multiple of the video frame capture rate. Simulated video images were generated every three time steps. The projective geometry for simulating images corresponded to that for the empirical videocamera mounting geometry. The timing and direction of prey accelerations were drawn from approximate values observed on video. In the simulations, ‘capture’ was defined as a falcon–prey distance of ≤0.25 m, and events during the actual capture were not simulated.

The model pursuer moved at a constant speed, so only its bearing and roll angles changed in response to perceived evader motion. To account for predator response time, T_R, to prey maneuvers, the program computed the motion of the pursuer at time t using the prey's position and velocity at t–T_R. The time lag between perceived prey motion and predator response was set at T_R=6Δt, assuming the falcon requires at least three flicker fusion periods to detect prey acceleration and a 38 ms reaction time once the stimulus is received (Pomeroy and Heppner, 1977; Potts, 1984); this value for T_R is also consistent with the 67±24 ms for birds in a flock to respond to the initiation of turning (Potts, 1984).

We wish to thank the falconers who participated in this study: Eddy de Mol and Francois Lorrain, Dennis Decaluwé, Troy Morrs and Stephen Lea, and other falconers who contributed valuable advice: Robert Giroux, Jack Hubley and Patrick Miller. Robert Musters fabricated the hoods used to hold the videocameras, and Bruce Boyes helped to design and adapt the backpack mounts. Emily Cunningham played a key role in the early conception of this project. We would like to acknowledge helpful conversations with Peter J. Love (Haverford College), Charlotte Hemelrijk and Hanno Hildebrandt (University of Groningen). We wish to express our appreciation for the helpful comments and suggestions by two anonymous reviewers.