Propagating speed waves in flocks: a mathematical model

TL;DR

This paper introduces a new mathematical model for speed wave propagation in bird flocks, capturing density wave phenomena and identifying conditions for optimal, non-oscillatory relaxation, with implications for understanding collective responses.

Contribution

The work presents a simplified yet effective model for speed fluctuation propagation in flocks, analyzing its solutions and identifying critical damping as an attractor in parameter space.

Findings

Identified the critical damping condition for fastest relaxation without oscillation.

Demonstrated the model's solution in one dimension and its parameter space analysis.

Proposed experimental tests to validate the model.

Abstract

Efficient collective response to external perturbations is one of the most striking abilities of a biological system. Signal propagation through the group is an important condition for the imple- mentation of such a response. Information transfer has been experimentally observed in the turning mechanism of birds flocks. In this context it is well-known also the existence of density waves: birds under predation, attempting to escape, give rise to self-organized density waves that propagates linearly on the flock. Most aspects of this phenomenon are still not fully captured by theoretical models. In this work we present a new model for the propagation of the speed (the modulus of the velocity) fluctuations inside a flock, which is the simplest way to reproduce the observed density waves. We have studied the full solution of the model in d = 1 and we found that there is a line in the parameter space along which the system relaxes in the fastest way with no oscillation after a signal has passed. This is the critical damping condition. By analyzing the parameters plane we show that critical damping 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 through future experiments.