A Mathematical Model of the Formation of Lanes in Crowds of Pedestrians Moving in Opposite Directions

Guillermo H Goldsztein

doi:10.1155/2015/872807

Abstract

In crowded environments, pedestrians moving in opposite directions segregate into lanes of individuals moving in the same direction. It is believed that this formation of lanes that facilitates the flow results from the individuals acting on their behalf, responding to local stimuli, without the intention of benefiting the crowd as a whole. We give evidence that this is true by developing and analyzing a simple mathematical model. Our results suggest that the simple behavior of moving out of the way to avoid imminent collisions leads to the formation of lanes of individuals moving in the same direction.

Highlights

Self-organization refers to the pattern formation or emergence of order in systems composed of several units or individuals, where these units or individuals respond only to local stimuli, independently of the global state of the system
Examples include the occurrence of alternating flows through bottlenecks, as two groups of a large number of individuals cross the bottleneck in opposite directions [4], the formation of trails [5], and the different pattern formations when the crowd is a collection of several small groups [6]
We have introduced a very simple model to study the formation of lanes in crowds of individuals moving in opposite directions

Summary

Introduction

Self-organization refers to the pattern formation or emergence of order in systems composed of several units or individuals, where these units or individuals respond only to local stimuli, independently of the global state of the system. Examples include the occurrence of alternating flows through bottlenecks, as two groups of a large number of individuals cross the bottleneck in opposite directions [4], the formation of trails [5], and the different pattern formations when the crowd is a collection of several small groups (friends) [6]. These coordinated behaviors are not planned by the crowd as a whole or by any member or group of members within the crowd.

The Mathematical Model

The System Self-Organizes with Probability 1

Direction of the Pedestrians in Each Lane after Self-Organization

Number of Collisions Required for Self-Organization

Time Required for Self-Organization

Summary