Traveling pulse emerges from individuals coordinating their stop-and-go motion: a case study in sheep

TL;DR

This study models and analyzes how sheep coordinate their stop-and-go behavior to produce a traveling pulse, revealing a robust collective motion emerging from individual decision rules.

Contribution

The paper introduces a microscopic decision-making model for sheep, translates it into macroscopic equations, and analytically derives the shape and speed of the traveling pulse.

Findings

Traveling pulse propagates at constant speed despite individual stop/go states.

Pulse shape is analytically derived as a sech square profile.

The system converges to and recovers from the pulse, showing robustness.

Abstract

Monitoring small groups of sheep in spontaneous evolution in the field, we decipher behavioural rules that sheep follow at the individual scale in order to sustain collective motion. Individuals alternate grazing mode at null speed and moving mode at walking speed, so cohesive motion stems from synchronising when they decide to switch between the two modes. We propose a model for the individual decision making process, based on switching rates between stopped / walking states that depend on behind / ahead locations and states of the others. We parametrize this model from data. Next, we translate this (microscopic) individual-based model into its density-flow (macroscopic) equations counterpart. Numerical solving these equations display a traveling pulse propagating at constant speed even though each individual is at any moment either stopped or walking. Considering the minimal model embedded in these equations, we derive analytically the steady shape of the pulse (sech square). The parameters of the pulse (shape and speed) are expressed as functions of individual parameters. This pulse emerges from the non linear coupling of start/stop individual decisions which compensate exactly for diffusion and promotes a steady ratio of walking / stopped individuals, which in turn determines the traveling speed of the pulse. The system seems to converge to this pulse from any initial condition, and to recover the pulse after perturbation. This gives a high robustness to this coordination mechanism.