The instability of followers and emergent vorticity in flocking behaviour for an experimental interaction rule

Computational models of collective behavior in birds has allowed us to infer interaction rules directly from experimental data. Using a generic form of these rules we explore the collective behavior and emergent dynamics of a simulated swarm. For a wide range of flock size and interaction extent (the fixed number of neighbors with which an individual will interact) we find that the computational collective is inherently stable --- individuals are attracted to one another and will position themselves a preferred distance from their fixed neighbors within a rigid lattice. Nonetheless, the irregular overall shape of the flock, coupled with the need for individuals on the boundary to move towards their neighbors creates a torque which leads the flock to rotate and then meander. We argue that this "rolling meander" is a very good proxy for real collective behavior in animal species and yet arises from a simple homogeneous and deterministic rule for interaction. Rather than then introduce leaders --- which has already been shown, quite straightforwardly, to drive collective swarms such as this --- we introduce a small number of "followers". Each follower is bound to consider a random fixed individual to be among their neighbors, irrespective of actual metric distance between them. We find that the introduction of a small number of such followers causes a phase transition that quickly leads to instability in the flock structure (as no stable configuration arises) and the previously rigid crystalline interaction among neighbors now becomes fluid: the distance between neighbors decreases, the flock ceases to rotate and meanders less.