Signal propagation and linear response in the delay Vicsek model

TL;DR

This study investigates how time delays in orientation interactions affect the collective behavior of swarms modeled by the Vicsek model, revealing impacts on stability, information spread, and response linearity.

Contribution

It introduces the effect of discrete time delays into the Vicsek model and analyzes their influence on swarm stability, information propagation, and linear response behavior.

Findings

Delayed interactions reduce leader-following ability but increase stability.

Longer delays and higher speeds promote ballistic information spread.

Linear response breaks down when orientation conservation is broken.

Abstract

Retardation between sensation and action is an inherent biological trait. Here we study its effect in the Vicsek model, which is a paradigmatic swarm model. We find that: (i) a discrete time delay in the orientational interactions diminishes the ability of strongly aligned swarms to follow a leader and, in return, increases their stability against random orientation fluctuations; (ii) both longer delays and higher speeds favor ballistic over diffusive spreading of information (orientation) through the swarm; (iii) for short delays, the mean change in the total orientation (the order parameter) scales linearly in a small orientational bias of the leaders and inversely in the delay time, while its variance first increases and then saturates with increasing delays; (iv) the linear response breaks down when orientation conservation is broken.