On Cucker-Smale model with noise and delay

TL;DR

This paper extends the Cucker-Smale model by including stochasticity and delay, deriving conditions for flocking and revealing that intermediate delays can promote collective behavior, supported by analytical and numerical results.

Contribution

It introduces a generalized Cucker-Smale model with noise and delay, providing new analytical conditions for flocking and insights into delayed stochastic systems.

Findings

Derived sufficient conditions for flocking in the generalized model

Established new results on delayed geometric Brownian motion

Numerical simulations suggest intermediate delays can enhance flocking

Abstract

A generalization of the Cucker-Smale model for collective animal behaviour is investigated. The model is formulated as a system of delayed stochastic differential equations. It incorporates two additional processes which are present in animal decision making, but are often neglected in modelling: (i) stochasticity (imperfections) of individual behaviour; and (ii) delayed responses of individuals to signals in their environment. Sufficient conditions for flocking for the generalized Cucker-Smale model are derived by using a suitable Lyapunov functional. As a byproduct, a new result regarding the asymptotic behaviour of delayed geometric Brownian motion is obtained. In the second part of the paper results of systematic numerical simulations are presented. They not only illustrate the analytical results, but hint at a somehow surprising behaviour of the system - namely, that an introduction of intermediate time delay may facilitate flocking.