Effect of time-delayed interactions on milling: A minimal model

Abstract

Models of collective motion show a rich variety of patterns. One of these is milling, in which the individuals of a group are circling around a common center. Milling has been generated in a Vicsek-like model of collective motion, i.e., a minimal model where individuals coordinate their headings only via alignment with close neighbors, without being attracted to each other and without avoiding collisions. However, in this Vicsek-like model information propagates instantaneously among neighbors, whereas in nature transfer and processing of information need time. How this delay affects patterns of collective motion, particularly milling, is unknown. Here we investigate the effect of time-delayed interactions on the emergence of milling in a Vicsek-like model. We show that delays may either destroy milling or induce it, depending on the parameters of the system. The range of speeds and fields of view of individuals at which milling occurs is shifted to smaller values if there are time delays in the model. Our findings may help to understand what causes milling in nature.