Propagating speed waves in flocks: A mathematical model

An efficient collective response to external perturbations is one of the most striking abilities of a biological system. One of the crucial aspect of this phenomenon is given by the information transfer, and resulting propagation of signals, within the group. In this respect the existence of density waves that propagate linearly on a flock of birds is well known. However, most aspects of this phenomenon are still not fully captured by theoretical models. In this work we present a model for the propagation of speed fluctuations inside a flock, which is able to reproduce the observed density waves. We study the full solution of the model in d = 1, and we find a line in the parameter space along which the system relaxes as fast as possible without oscillating, resembling a generalized critical damping condition. By analyzing the parameters plane we show that this "critical damping" line indeed represents an attractor for a steepest descent dynamics of the return time of the system. Finally we propose a method to test the validity of the model through future experiments.