Curved paths in raptor flight: Deterministic models

Two deterministic models for flight of Peregrine Falcons and possibly other raptors as they approach their prey are examined mathematically. Both models make two assumptions. The first, applicable to both models, is that the angle of sight between falcon and prey is constant, consistent with observations that the falcon keeps its head straight during flight and keeps on course by use of the deep foveal region in its eye which allows maximum acuity at an angle of sight of about 45°. The second assumption for the first model (conical spiral), is that the initial direction of flight determines the overall path. For the second model (flight constrained to a tilted plane), a parameter that fixes the orientation of the plane is required. A variational calculation also shows that the tilted plane flight path is the shortest total path, and, consequently, the conical spiral is another shortest total path. Numerical calculations indicate that the flight paths for the two models are very similar for the experimental conditions under which observations have been made. However, the angles of flight and bank differ significantly. More observations are needed to investigate the applicability of the two models.