ACUCKER?SMALEMODELWITHNOISEANDDELAY

Abstract

A generalization of the Cucker?Smale model for collective animal behavior 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 modeling: (i) stochasticity (imperfections) of individual behavior 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 by-product, a new result regarding the asymptotic behavior 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 behavior of the system?namely, that the introduction of an intermediate time delay may facilitate flocking. Key words. Cucker?Smale system, flocking, asymptotic behavior, noise, delay, geometric Brownian motion.