On following behaviour as a mechanism for collective movement.

During collective movement, animals display a wide variety of mechanisms to maintain cohesion. In some species, individuals rely mainly on following their direct predecessor, thereby forming spectacular processions of individuals in single file. Despite being the simplest case of following behaviour, it is largely absent from the theoretical literature on collective migrations. The objective of this study is to quantify the efficiency of following the predecessor, in terms of ensuring cohesion. The situation we consider is a sequence of individuals facing a bifurcation. The choice between left and right is influenced by the choice of the predecessor. First, we model this situation with a two-state Markov chain with a symmetric transition matrix. Cohesion is quantified as the expected number of individuals on either side, and the expected number of consecutive followers. Although cohesion increases with the probability of following, it remains surprisingly low unless the probability is almost equal to one. Furthermore, cohesion decreases with group size regardless of the probability of following. Then, we generalise our model to situations in which individuals have a preference for one of the two choices (asymmetric transition matrix). For some parameter sets, the tendency to follow each other leads a large fraction of the individuals to the non preferred side. Moreover, this fraction increases with the total population size. Finally, we include the possibility to follow N individuals. This provides the link between our model and other collective migration models. If enough individuals are perceived, the results shift from symmetrical (low cohesion) to asymmetrical (high cohesion) distribution of the individuals. All in all, our results suggest that following the direct predecessor must be complemented with other cohesive behaviours (involving the perception of more individuals or a navigation system) to guarantee its efficiency. We discuss our findings in the context of the different following behaviours covered in the literature.