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Figures
- Fig. 1.
Cropped frame from a video of a typical chase involving a gyrfalcon and a ‘Rokarrowan’ Roprey model. Note the proximity of the attacker to its target, and their similar bank angles, which are characteristic of the tail-chasing behaviour that we observed.
- Fig. 2.
Two-dimensional (2D) GPS trajectories for example short dashes and extended chases. (A,B) Example trajectories showing the entirety of a (A) short dash and (B) an extended chase, showing the lines of sight (cyan lines) between the gyrfalcon (blue points) and Roprey (magenta points) at each sample point. Note the small discrepancy in the estimated position of target and attacker at the point of intercept (enlarged sample points), expected because of the positioning error associated with GPS receivers (see Materials and Methods). (C,D) Trajectories for the terminal phase of the same two flights (C, short dash; D extended chase; see Fig. 4G,M for modelling), after shifting the attacker's trajectory to correct for this positioning error. Gridlines are at 10 m spacing.
- Fig. 3.
Box-and-whisker plots comparing model fits and parameter estimates for 2D simulations of 20 flights from n=13 gyrfalcons under proportional pursuit (PP), proportional navigation (PN) and mixed (PN+PP) guidance. (A) Relative error of simulation, showing either the relative error for the longest simulation lasting ≥2 s that met the 1.0% error tolerance threshold for each flight or, if no simulation met this threshold, the minimum relative error achieved on any simulation lasting ≥2 s. Note that whilst the PN+PP simulations fit the flights more closely than either PN or PP, they are almost certainly overfitted (see Results). (B,C) Parameter estimates for the guidance constants N and K for all successfully modelled flights. Note the variable sign of the parameter estimates for K under PN+PP, which confirms that these simulations are overfitted (one outlier for PN+PP not shown). The red line in each box denotes the median value for all flights; the lower and upper bounds of the box denote the 1st and 3rd quartiles; crosses indicate outliers falling >1.5 times the interquartile range beyond the 1st or 3rd quartile; whiskers extend to the farthest datapoints not treated as outliers. The green lines denote the global optimal values of the guidance parameters.
- Fig. 4.
2D attack trajectories for the 18/20 flights from n=13 gyrfalcons that were successfully modelled under PN guidance. Panels show the measured trajectories of the target (magenta points) and attacker (blue points), overlain with the longest simulation fitted to within 1.0% error tolerance (blue lines) in 2D. The corresponding parameter estimate for N is displayed on each plot. Note that among the 9 short dashes (A–I), 7 flights (A–G) are modelled in their entirety from target launch to intercept; the other 9 flights (J–R) each correspond to the terminal phase of an extended chase. Simulations with values of N falling beneath the 1st quartile (N<0.5) coincide with nearly straight sections of flight (D,K,L,N), for which parameter estimation is unreliable. Simulations with values of N falling between the 1st and 3rd quartiles (0.5≤N≤1.4) involve a substantial amount of turning that the model successfully explains (C,E–I,M,O–R). Black arrows display mean wind direction; double-headed arrows correspond to wind speeds >20 km h−1; gridlines are at 10 m spacing. See Fig. S2 for the remaining 2/20 flights that were not successfully modelled under PN.
- Fig. 5.
Three-dimensional (3D) attack trajectories for the subset of 6 gyrfalcon flights involving the greatest altitudinal change among the 12/20 flights that were successfully modelled under PN guidance. (A–F) The measured trajectories of the target (magenta points) and attacker (blue points), overlain with the longest simulation fitted to within 1.2% error tolerance (blue lines) in 3D, corresponding to the following panels in Fig. 4: (A) Fig. 4B; (B) Fig. 4C; (C) Fig. 4J; (D) Fig. 4F; (E) Fig. 4G; and (F) Fig. 4M. The corresponding parameter estimate for N is displayed on each plot. A, B, D and E correspond to short dashes for which almost the entire flight was modelled from target launch to intercept. Gridlines are at 10 m spacing.
- Fig. 6.
Box-and-whisker plots comparing parameter estimates for N in PN guidance models fitted independently to the 13 attack flights from n=4 peregrine falcons and 18 attack flights from n=13 naive gyrfalcons, all in pursuit of manoeuvring targets. The centre line of each box denotes the median for all flights; the lower and upper bounds of the box denote the 1st and 3rd quartiles; crosses indicate outliers falling >1.5 times the interquartile range beyond the 1st or 3rd quartile (one extreme outlier for the peregrine falcons not shown); whiskers extend to the farthest datapoints not treated as outliers.
- Fig. 7.
Effect of the navigation constant N on the dynamics of PN guidance. (A–D) Selection of 4 successfully modelled gyrfalcon flights involving substantial turning, showing the measured trajectory of the target (magenta dots) and attacker (blue dots) overlain with the best-fitting trajectory under PN guidance at the value of N displayed on the panel (blue line), and with the trajectory that would have been followed for the same initial conditions and target motion at the median value of N=2.8 for peregrine falcons (green line). Green circle shows the predicted point of intercept had the gyrfalcon used the median value of N for peregrine falcons; note that this is always sooner than the actual point of intercept (magenta circle). (E–H) Selection of 4 successfully modelled peregrine falcon flights involving substantial turning, showing the measured trajectory of the target (magenta dots) and attacker (blue dots) overlain with the best-fitting trajectory under PN guidance at the value of N displayed on the panel (lilac line), and with the trajectory that would have been followed for the same initial conditions and target motion at the median value of N=1.2 for gyrfalcons (orange line). Lilac circle shows the actual point of intercept; black cross shows the predicted position of the bird had the peregrine falcon used the median value of N for gyrfalcons; note that this is always at some significant distance from the target. Panel letters correspond to the following panels in Fig. 4 and Fig. S3: (A) Fig. 4M; (B) Fig. 4P; (C) Fig. 4Q; (D) Fig. 4R; (E) Fig. S3-5B; (F) Fig. S3-S3D; (G) Fig. S3-S3J; and (H) Fig. S3-5C. Gridlines are at 10 m spacing.
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