Continuum modeling of the equilibrium and stability of animal flocks

Abstract

Groups of animals often tend to arrange themselves in flocks that have characteristic spatial attributes and temporal dynamics. Using a dynamic continuum model for a flock of individuals, we find equilibria of finite spatial extent where the density goes continuously to zero at a well-defined flock edge, and we discuss conditions on the model that allow for such solutions. We also demonstrate conditions under which, as the flock size increases, the interior density in our equilibria tends to an approximately uniform value. Motivated by observations of starling flocks that are relatively thin in a direction transverse to the direction of flight, we investigate the stability of infinite, planar-sheet flock equilibria. We find that long-wavelength perturbations along the sheet are unstable for the class of models that we investigate. This has the conjectured consequence that sheet-like flocks of arbitrarily large transverse extent relative to their thickness do not occur. However, we also show that our model admits approximately sheet-like, ‘pancake-shaped’, three-dimensional ellipsoidal equilibria with definite aspect ratios (transverse length-scale to flock thickness) determined by anisotropic perceptual/response characteristics of the flocking individuals, and we argue that these pancake-like equilibria are stable to the previously mentioned sheet instability.