Causes of variation of darkness in flocks of starlings, a computational model

Abstract

The coordinated motion of large flocks of starlings is fascinating for both laymen and scientists. During their aerial displays, the darkness of flocks often changes, for instance dark bands propagate through the flock (so-called agitation waves) and small or large parts of the flock darken. The causes of dark bands in agitation waves have recently been shown to depend on changes in orientation of birds relative to the observer rather than changes in density of the flock, but what causes other changes in darkness need to be studied still and this is the aim of the present investigation. Because we cannot empirically relate changes in darkness in flocks to quantities, such as position and orientation of the flock and of its members relative to the observer, we study this in a computational model. We use StarDisplay, a model of collective motion of starlings, because its flocks resemble empirical data in many properties, such as their three-dimensional shape, their manner of turning, the correlation of heading of its group-members, and its internal structure regarding density and stability of neighbors. We show that the change in darkness in the flocks perceived by an observer on the ground mostly depends on the observer’s distance to the flock and on the degree of exposure of the wing surface of flock members to the observer, and that darkness appears to decrease when birds roll during sharp turns. Remarkably, the darkness of the flock perceived by the observer was neither affected by the orientation of the flock relative to the observer nor by the density of the flock. Further studies are needed to investigate changes in darkness for flocks under predation.

Appendix

In the appendix we show additional graphs reporting the darkness as a function of the distance d from a static flock of 10.000 individuals (Fig. 7) and the darkness as a function of the parameter sigma of the Gaussian filter (Fig. 8).

Fig. 7

Darkness as a function of distance d from the static flock of 10.000 individuals. Dashed line: flock showing maximal wing surface: center of mass of the flock at (0,0,d), average heading vector (1,0,0), average up vector (0,0,1), and camera at (0,0,0). Solid line: flock showing minimal wing surface center of mass of the flock at (d,0,150), average heading vector (1,0,0), average up vector (0,0,1), and camera at (0,0,150). Solid vertical lines are guides to the eye that indicate the range of distances reached in the simulations of moving flocks

Fig. 8

Darkness as a function of the parameter sigma of the Gaussian filter. Flock of 10.000 individuals with center of mass at (0,0,250), mean heading vector (1,0,0), and mean up vector (0,0,1), NND = 0.6 (dashed lines) and NND = 1.0 (solid lines). Right panel is a zoom on same data. The value of sigma = 3 chosen for our simulations is a value small enough to get a difference in darkness for different nearest neighbor distances, which confirms that the value of sigma = 3, that was chosen so that the flock was considered as a whole, because white pixels between birds become non-white, is a good choice for our simulations. A smaller value of sigma would produce white pixels between birds such that we could not identify and measure the area of the flock. A larger value of sigma would have the negative effect of flocks with unprecise contours and flocks with different nearest neighbor distances producing the same darkness