Figure 1.
The local co-ordinate systems of 2 birds with different orientations in space and at different distances to the viewer. is the bird's forward direction;
, its sideways direction; and
, its upward direction. It can change these by rotating around these 3 principal axes (roll, pitch and yaw).
Figure 2.
Social interaction ranges for separation (A), cohesion (B), and alignment (C). Note that the lengths of the different radii in the figure are not to scale with the default values in Table 1.
Figure 3.
Pictorial representation of attraction to the roost. A) horizontal attraction , normal
and intended trajectory for individuals i and j. Trajectories indicate that the turning is sharpest when flying out radially. B) vertical attraction
.
Figure 4.
Aerodynamic forces, A) while flying straight and B) while banking. : lift, weight, thrust, drag respectively.
= effective lift,
= centripetal force,
= centrifugal force.
Table 1.
Default parameter values1.
Figure 5.
Two ways of measuring shape. Left: Measurement of based on PCA analysis. Right: measuring of elongation,
, length in the movement direction and width orthogonally to it. Birds fly from left to right, flock is shown from above.
Figure 6.
Variability of shape over time (60 s). A) trajectories of a flock of 2000 individuals and of 200 individuals above the roost (default parameters), of a flock of 2000 individuals in which individuals are attracted to 50 nearest neighbours (topo50), a flock of 2000 individuals in which individuals are flying without attraction to the roost (free), and a flock of 2000 individuals in which individuals are flying without banking (nobanking). B) Banking behavior over time. C) thickness, I1. D) Longest over shortest dimension, the height, . E) Volume.
Figure 7.
Deviation from average heading and velocity.
Deviation from the average of heading and velocity among individuals in the flock. A) Scale free correlation between correlation length of deviation of velocity from that of the centre of gravity versus length of the flock for default values (flock length is measured by the largest distance (in m) between two individuals in the flock); B) Corresponding snapshots of flocks true to scale. From left to right: N = 200 and Lā20 m, N = 2000 and Lā50 m. C) Polarisation (global and local) versus number of individuals in the flock for default parameters and high number of interaction partners (i.e. 50), N = flock size, lines (continuous and striped) indicate local polarization, points indicate global polarization (fat dots: default parameters, stars: 50 interaction partners).
Figure 8.
A) High and low variability of speed (respectively tau = ā, tau = 1), flock size and degree to which the flock is oblong. B) Oblong in any direction and oblong in movement direction
versus number of individuals in the flock. C) Distribution of angles between the movement direction of the flock and its longest dimension
for flock size of 2000 individuals, D) The turning of a flock (view from above). Flock shape changes relative to the movement direction (from wide to oblong), individuals 1 and 2 follow paths of the same length and their location changes in the flock.
Figure 9.
Shape of fish schools [24]. A) Elongation () and coefficient of variation of speed in fish model and Stardisplay. Circles: fish school, Triangles: StarDisplay. B) Series of snapshots (with fixed time interval) of a school of 600 individuals indicating the initial location of individuals (at right side at front, at front left side, right side at back, left at back) by four grey-colours. C) Snapshot of school during turning in fish model with extremely high variability of speed (tau = 0.4). Individuals in inner corner automatically slow down and in the outer corner they speed up. Darker grey indicates faster movement. The school is oblong. D) Snapshot of school during turning in fish model with extremely low variability of speed (Tau = 0.02). The school is no longer oblong, but approximately as long as wide. For a color version of Fig. 9BCD, see Supplementary material, respectively, Figs. S1, S2, S3.
Table 2.
Hypotheses for empirical testing derived from the model.