Generic Instability at the Crossing of Pedestrian Flows

Abstract

Diagonal stripe formation is a well-known phenomenon in the pedestrian traffic community. Here we define a minimal model of intersecting traffic flows. It consists in an M × M space-discretized intersection on which two types of particles propagate towards east (\(\mathcal{E}\)) and north (\(\mathcal{N}\)), studied in the low density regime. It will also be shown that the behaviour of this model can be reproduced by a system of mean field equations. Using periodic boundary conditions the diagonal striped pattern is explained by an instability of the mean-field equations, supporting both the correspondence between equations and particle model and the generality of this pattern formation. With open boundary conditions, translational symmetry is broken. One then observes an asymmetry between the organization of the two types of particles, leading to tilted diagonals whose angle of inclination slightly differs from 45∘ both for the particle system and the equations. Even though the chevron effect does not appear in the linear stability analysis of the mean-field equations it can be understood in terms of effective interactions between particles, which enable us to isolate a macroscopic nonlinear propagation mode which accounts for it. The possibility to observe this last chevron effect on real pedestrians is then quickly discussed.