Spatial instabilities and size limitations of flocks

The movement of flocks with a single leader (and a directed path from it to every agent) can be stabilized over time as has been shown before (for details see [3] and prior references therein, shorter descriptions are given in [1, 4]). But for large flocks perturbations in the movement of the leader may nonetheless grow to a considerable size as they propagate throughout the flock and before they die out over time. We calculate the effect of this “finite size resonance” in two simple cases, and indicate two applications of these ideas. The first is that if perturbations grow as the size of the flock gets larger, then the size of the flock will have a natural limitation. Our examples suggest that for flocks with a symmetric communication graph perturbations tend to grow much slower than in the asymmetric case. The second application concerns a simple traffic-like problem. Suppose the leader accelerates from standstill to a given velocity and a large flock is supposed to follow it. The acceleration of the leader is the ‘perturbation’.