Lane formation by side-stepping

TL;DR

This paper models pedestrian lane formation using a system of nonlinear PDEs derived from a lattice approach, analyzing stationary solutions, proving global existence, and illustrating behavior through simulations.

Contribution

It introduces a PDE model for pedestrian lane formation driven by aversion and cohesion, with rigorous existence proofs and numerical illustrations.

Findings

Existence of stationary lane solutions

Global bounded weak solutions established

Numerical simulations demonstrate system behavior

Abstract

In this paper we study a system of nonlinear partial differential equations, which describes the evolution of two pedestrian groups moving in opposite direction. The pedestrian dynamics are driven by aversion and cohesion, i.e. the tendency to follow individuals from the own group and step aside in the case of contraflow. We start with a 2D lattice based approach, in which the transition rates reflect the described dynamics, and derive the corresponding PDE system by formally passing to the limit in the spatial and temporal discretization. We discuss the existence of special stationary solutions, which correspond to the formation of directional lanes and prove existence of global in time bounded weak solutions. The proof is based on an approximation argument and entropy inequalities. Furthermore we illustrate the behavior of the system with numerical simulations.