Introduction

Peregrine falcons Falco peregrinus – hereafter referred to as peregrines – are the world's most widely distributed raptor (Ferguson‐Lees and Christie 2001). They hunt a wide variety of avian prey using a range of alternative attack strategies (Rudebeck 1951, Dekker 1988, Dekker and Taylor 2005, Zoratto et al. 2010). In nature, wild peregrines take prey ranging from small passerines such as goldcrests Regulus regulus and Eurasian blue tits Cyanistes caeruleus (Ratcliffe 2010), up to larger waterfowl such as mallards Anas platyrhynchos (Dekker 2009), spanning well over two orders of magnitude in body mass. The only species that are not attacked are very large birds such as swans and geese (but see Ratcliffe 2010), although the medieval kings of England successfully trained falcons to hunt quarry as large as grey herons Ardea cinerea and common cranes Grus grus (Oggins 2004). Likewise, forest‐dwelling species are rarely taken because the peregrine requires wide open spaces for its preferred hunting modes (Ratcliffe 2010), including the famous stoop in which the falcon soars to a high altitude before diving down at great speed to intercept its prey in mid‐air. The peregrine's choice of prey appears to be opportunistic and dependent on the availability of different prey species (Stevens et al. 2009), which varies by habitat and time of year, but many reports suggest a bias towards certain species, even after accounting for their availability (Ratcliffe 2010). The peregrine is particularly well known for taking pigeons, notably rock doves Columba livia and their feral counterparts – or when these are scarce, for taking other species of a similar size, such as black‐headed gulls Larus ridibundus (Kruuk 1964). The majority of its preferred prey species weigh between 0.05 and 0.5 kg (Ratcliffe 2010). Interestingly, there is a marked difference in the choice of prey between male and female peregrines (Dekker 2009). Males consistently bring smaller prey back to the nest (Parker 1979) and are more often observed to hunt for small passerines (e.g. common starlings Sturnus vulgaris) and small waders (e.g. dunlins Calidris alpina), whereas females hunt more often for larger birds up to the size of ducks (e.g. northern pintails Anas acuta) (Dekker 1980, 1987, 2009).

It has been theorized that the reversed sexual size dimorphism of falcons has evolved such that a pair of birds has a broader selection of prey to choose from (Dekker 2009). The heavier female is supposed to be able to carry larger prey, and the smaller, lighter male is supposed to be more adept at attacking highly maneuverable prey, but it is hard to judge from empirical data whether the relative ease of catching different prey species underlies the observed differences in prey choice between the sexes – not least because the sex of a falcon is hard to identify when observing a high‐speed chase (Dekker 2009). In addition, the empirical relationship between catch success and flight ability is hard to investigate, because the reported success rates of peregrines vary greatly between studies. In one study (Jenkins 2000), catch success was reported to be highest for small doves, and small passerines such as sparrows and queleas; intermediate for small to medium sized birds such as starlings and weavers; and lowest for large pigeons, ducks and waders. In another study (Dekker 2009), the opposite pattern was observed: high success rates for ducks and waders; intermediate for gulls; and lowest for small passerines. Environmental variation aside, one underlying cause of these conflicting findings may be the differing intensity with which the falcons hunted. Many of a falcon's attacks do not appear genuinely intended to kill them (Ratcliffe 2010); perhaps because the falcon is warming up, is playing, is practising or had eaten enough before the attack that its motivation is low. To account for this varying motivation, the terms high‐ versus low‐intensity attacks have been introduced in the literature (Treleaven 1980), where the intensity is judged visually by the observer, but such classification remains subjective. Most problematically, it has been observed that male falcons are more motivated to catch smaller prey than females (Dekker 2009), thereby obscuring any differences in their actual ability to catch them.

The difficulties inherent in studying the factors affecting catch success through field observation motivate us to apply a new and different approach here, using a physics‐based bird flight simulator to study the problem in silico (Fig. 1, Supplementary material Appendix 1). The mathematical details of the flight simulator are summarised here in Supplementary material Appendix 1 and are explained fully in Mills et al. (2018). To allow meaningful inference from the simulation results, the model aims to capture all the key dynamics of the predator–prey scenario. Specifically, it simulates the mechanics and aerodynamics of flapping and gliding flight by both birds, the control mechanisms by which the model birds manipulate their aerodynamic forces, the guidance law by which the model falcons determine how to intercept their prey, and the visual system providing feedback to this guidance law. It has been shown experimentally that the flight trajectories of peregrines follow a guidance law called pure proportional navigation (Brighton et al. 2017). Under this guidance law, the commanded angular rate of change in the falcon's velocity is directly proportional to the angular rate of the line‐of‐sight between the falcon and its prey (Fig. 1d). Our model falcons use the same guidance law, exhibiting a realistically short response delay to maneuvers of the model prey, whose motion they observe with a small degree of visual error. Appropriate tuning of this guidance law is crucial for accurate interception, which we model by optimizing the constant of proportionality in the guidance law (see Mills et al. 2018 for a theoretical exposition and Brighton et al. 2017 for an empirical investigation on proportional navigation in peregrines). Varying this so‐called navigation constant (N) manipulates the trade‐off between higher steering effort and faster convergence to a collision course, as well as influencing the precision of steering in the presence of error. Each simulated species has a different set of morphological parameters (i.e. body mass, moment of inertia, wing area, wingspan, wingbeat frequency, etc.), which in turn determines the mechanical and aerodynamic limits on force production, and hence the ability of the bird to accelerate and maneuver. We simulate attacks of lone falcons intercepting lone prey in mid‐air, and parametrically vary the starting altitude of the falcon and its starting distance to the prey, so as to mimic the variety of attack strategies that real peregrines use – from level chases to stooping. By running these variations in a Monte Carlo simulation, we test whether the optimal attack strategy differs between male and female falcons, and whether the optimal attack strategy depends upon prey species.

Figure 1 Open in figure viewer PowerPoint Visualisation of the bird flight simulator. (a) Low‐speed attack by a falcon. The dotted grey lines connect the positions of the falcon and prey at a given time point with an interval of 100 m s. If the falcon initiates its attack from only 50 m above the prey, then it reaches a speed of approximately 30–40 m s–1 near intercept. Prey in the model fly erratically, exerting high accelerations to the left and right. (b) High‐speed stoop by a falcon on its prey. By initiating its attack from 1500 m above the prey, and by retracting its wings, the falcon is able to reach speeds of over 100 m s–1. (c) Graphical output of the simulator. The real‐time graphical output of the simulator shows how falcon and prey adjust their wingbeat, wing retraction, body orientation and trajectory. The colored lines behind the birds denote their trajectories. (d) A graphical depiction of the pure proportional navigation guidance law of the peregrine. Under this guidance law, the commanded angular rate of change in the falcon's velocity is directly proportional to the angular rate of the line‐of‐sight between the falcon and its prey.

Real prey use a variety of escape strategies, the most ubiquitous of which our simulations attempt to capture. Most ducks drop ‘like a falling stone’ towards the nearest body of water before submerging (Dekker 1980), sometimes reaching a dive speed matching that of the falcon (Dekker 2009). If the duck cannot reach safety before the falcon has caught up, it will maneuver erratically at the last moment (Dekker 2009). Other prey instead aim high in the sky (Lima 1993): many passerines can out‐climb a falcon, and are therefore safe once they reach a slightly higher altitude than the predator. When alerted that a predator is present, many species will start to fly fast and erratically, maneuvering in a way that appears to make it hard for the falcon to catch them. Such behavior is mainly observed in isolated individuals under attack (Kruuk 1964), but is also seen in groups. For instance, the fast, alerted flight of a flock of common starlings manifests itself as dark waves in the murmuration, which are thought to be caused by distinctly‐timed and synchronized zig‐zags on the part of the individuals within the flock (Hemelrijk et al. 2015). Erratic, or ‘jinking’, flight seems to represent an invaluable adaptation for predator avoidance, because non‐alerted, straight‐flying prey are almost always caught (Dekker 1980). This erratic flight mode, with distinctly timed strong acceleration to either side, is fascinating from a theoretical standpoint as it turns out to be the optimal escape strategy when evading missiles that use the same pure proportional navigation guidance law as attacking peregrines (Girard and Kabamba 2015). Yet this jinking escape pattern has never been studied in birds, and previous research has instead focused on studying escape by climbing, diving or turning smoothly (Hedenström and Rosén 2001). We therefore also study the importance of erratic maneuvering by the prey in our physics‐based simulation of aerial attack behaviors.

In summary, our aim in this paper is to provide a comprehensive analysis of how catch success in peregrines is affected by their own flight performance and by that of their prey, and to use this to explore whether intersexual variation in flight performance is expected to lead to intersexual variation in catch success against different prey species. We do this in a simulation environment that accurately captures the physics of the situation, and in which the behavior of predator and prey is optimized to respectively maximize or minimize catch success.