Discrete Dynamics in Nature and Society

Volume 2015, Article ID 872807, 7 pages

http://dx.doi.org/10.1155/2015/872807

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

School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, USA

Received 5 May 2015; Accepted 9 June 2015

Academic Editor: Tetsuji Tokihiro

Copyright © 2015 Guillermo H. Goldsztein. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. 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.

Primary research goals in the field of self-organization are to understand the local interactions among individuals and with the environment and how these local interactions lead to the global self-organization. A major objective is to discover common features of different self-organizing systems to establish general principles that apply to classes of systems. Some books and review articles on self-organization in biological systems include [1–3].

Self-organization in human crowds is commonly observed in real life scenarios and is well documented in the literature. 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. Instead, they emerge as a consequence of the individuals acting in response to local stimuli and motivated by their own goals.

Examples of why the understanding of the dynamics of crowds is of interest include the following: (1) It is a source of interesting examples of self-organization. (2) It can provide guidelines to increase the safety in very transited areas like some bridges [7] or in events attended by massive crowds, such as events in stadiums. (3) It can provide guidelines in the design of buildings, such as movie theaters and shopping malls, where optimal crowd flow is desired because of economical and safety reasons. Accordingly, the study of dynamics of crowds, both theoretically (early works include [8, 9]; see also [10] for a review) and experimentally [11–13], is a very active area of research.

The existing mathematical models of crowd dynamics can be categorized according to the length scale modeled as microscopic, mesoscopic, and macroscopic.

Microscopic models solve for the position of each individual. This class includes cellular automata models [14–16], lattice gas automata models [17, 18], algorithms [19], and large systems of odes, where the mass times acceleration of each individual is set equal to the sum of* generalized *or* social forces* the individual feels [20–24]. These* social forces* are not real forces between individuals but are introduced to model the responses of the individuals to stimuli they feel, which include the environment and the presence of the other individuals. In most models, the social forces are of four types: (1) repulsive force acting in a very short range, to avoid collisions with other individuals or walls or obstacles, (2) attractive force that acts in a longer range and attempts to model the desire of individuals to move in groups, (3) a force that would result in the desired speed and direction of motion of the individual in the absence of the other forces, and (4) a random force to account for differences in individuals. In some models, the speed of each individual remains constant, and only its direction changes in response to the social forces [25]. Some of these models are known as self-propelled particles models [26] and others as individual based models [25, 27].

Mesoscopic, kinetic, or Boltzmann type models are integropartial differential equations that describe the evolution of probability densities of the position and velocities of the pedestrians [28, 29]. Macroscopic or continuum models are partial differential equations (conservation equations), where the dependent variables are the density and* local average* velocity of individuals [30–34]. Some works connect microscopic to macroscopic models [35–37]. Also, network models have been introduced [38] and optimal control theory has been used [39].

Consider a crowded long corridor or street. Assume some of the pedestrians move in one direction and the others in the opposite direction. A self-organizing phenomenon familiar to all of us is that the individuals segregate into lanes, where each lane contains only individuals moving in one direction [40–42]. This formation of lanes facilitates the flow in both directions [43]. In this paper, we introduce and analyze a new model motivated by this phenomenon. This new model is very different to the ones that can be found in the literature. The results we obtain suggest that the simple behavior of moving out of the way to avoid collisions leads to the self-organization of the system.

In Section 2, we introduce our model, in which we assume that individuals move around a two-lane circular track with the same angular speed, half of them in one direction and the other half in the opposite direction. Each time two individuals moving in the opposite direction and in the same lane meet; one of them, randomly chosen with each having the same probability of of being chosen, moves to the other lane so they both can continue their motion around the track. In Section 3, we prove that, with probability 1, the system will* self-organize*; that is, all the individuals moving in the counterclockwise direction end up in the same lane, and all other individuals, moving in the clockwise direction, end up in the other lane. More details on the dynamics and final state of our system are given in the subsequent sections. In Section 4, we provide the probability that, after the system is self-organized, the pedestrians in lane 1 move in the clockwise direction. In Section 5, we find the expected number of collisions required for the system to self-organize, and in Section 6 we compute numerically the expected time to self-organization. We finish the paper with a small discussion in Section 7.

#### 2. The Mathematical Model

Assume that individuals or pedestrians move around a two-lane circular track, half of them in one direction and the other half in the opposite direction. Each time two individuals moving in opposite directions and in the same lane meet; one of them, randomly chosen, with each having the same probability of of being chosen, moves to the other lane so they both can continue their motion around the track. More precisely, our model is defined as follows.

*Rules.* The following set of rules define the dynamics of our model.(1)individuals move around a circular track with two lanes. The inner lane is referred to as lane 1 and the outer lane as lane 2. Each individual is labeled with an integer between 1 and . We assume is even.(2) individuals move in the counterclockwise direction and the other in the clockwise direction.(3)All individuals move with the same constant angular speed .(4)The position of the individual at time is described by an angle and by , the number of the lane the individual is in at time . Note that the angle is not uniquely determined by the position of the individual . Changing by adding to it an integer multiple of does not change the position of the individual .(5)We take all the angles to satisfy for all . The angle is assumed to increase counterclockwise. Thus, if the individual moves in the counterclockwise direction, we have , where is only the integer (that depends on ) that makes satisfy the constraint . Analogously, if the individual moves in the clockwise direction, then for an integer value function . With this rule, is uniquely determined.(6)We assume that initially all the angles are different; that is, for all .(7)Two individuals, and , collide at time if they reach the same location at that time; that is, and (they have the same angle and are in the same lane).(8)If two individuals, and , collide at time , exactly one of them changes lanes at that time. The probability that the individual changes lanes is and, thus, is also the probability that changes lanes. The lane change is instantaneous; that is, it does not slow down the angular speed of the individual changing lanes.(9)An individual can only change lanes when it collides with another individual.

*Observation 1. *Note that the assumptions that all the initial angles are different and that the angular speed of all individuals is the same imply that two individuals moving in the same direction will never have the same angle. Thus, if two individuals have the same angle at some time , they must be moving in opposite directions. These facts imply that when two individuals collide, there is no other individual next to them in the other lane, and, thus, our rule that one of them changes lanes can be followed with no problems and no exceptions.

Our model is very simple and is based on local responses only. The selection of the individual that changes lanes is independent of the location of all the other individuals. In other words, the individuals do not make* smart* decisions attempting to minimize the number of lane changes they will have to make.

Figure 1 shows the results of a numerical simulation with individuals. The individuals move at one revolution per unit time. Figure 1 shows the positions of the individuals at three different times: , , and . The number of collisions that occurred by those times is also indicated in Figure 1. The initial angle of the individuals, , was randomly chosen with uniform probability distribution around the track. The lane where each pedestrian started, , was also randomly selected, with each lane having as the probability of being chosen.