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doi:10.1016/j.ph

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Physica A 375 (2007) 668–678

www.elsevier.com/locate/physa

A behavior-based model for pedestrian counter flow

W.G. Weng�, S.F. Shen, H.Y. Yuan, W.C. Fan

Center for Public Safety Research, Department of Engineering Physics, Tsinghua University, Beijing 100084, PR China

Received 28 March 2006; received in revised form 14 August 2006

Available online 10 November 2006

Abstract

A behavior-based lattice-gas model for pedestrian dynamics is presented. This model adopts the behaviorism for mobile

robot, and the walk task of pedestrian can be divided into three basic behaviors, i.e., ‘‘move’’, ‘‘avoid’’, and ‘‘swirl’’ basic

behaviors. The walk direction is determined from the walk weight, which is the sum of the product of each vector of basic

behavior multiplied by the weight in the corresponding direction. This model can simulate pedestrian movement with

different walk velocities through the update at different time-step intervals. The periodic boundary for pedestrian counter

flow with six simulation conditions in the channel is considered, and the dynamical characteristics are discussed.

Simulation results show this presented behavior-based model can simulate some characteristics of pedestrian counter flow,

e.g., lane formation and jammed configuration, etc. In addition, the different simulation conditions result in the different

numbers of phases and their different critical total densities. In general, the mean flow /JS is always high if the

corresponding mean velocity /VS is high, and their phases also turn at the same critical total density.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Pedestrian dynamics; Counter flow; Behavior-based

1. Introduction

Recently, traffic and pedestrian flows have attracted considerable attention from physicists [1–6]. Thereason for pedestrian study is that pedestrian movement is an important component in the analysis and designof transportation facilities, pedestrian walkways, traffic intersections, markets, and other public buildings. Forthe design of walking infrastructure, a working knowledge of the characteristics of pedestrian flows is requiredin order to design the infrastructure as well as to assess its efficiency and safety. In particular, a goodunderstanding of the emergent patterns is required to predict how the flow will behave under differentcircumstances. It is also important to avoid the jammed state of pedestrian in the channel of the walkwayssuch as the subways, etc.

Pedestrian flow is a kind of many-body system of strongly interacting persons. The pedestrian flowdynamics is closely connected with the driven many particle systems. Many observed self-organizationphenomena in pedestrian flows have been successfully reproduced with physical methods [7,8]: the lattice-gasmodel of biased-random walkers [9–11], the molecular dynamic model of active walkers [6,12,13], and the

e front matter r 2006 Elsevier B.V. All rights reserved.

ysa.2006.09.028

ing author.

ess: wgweng@tsinghua.edu.cn (W.G. Weng).

ARTICLE IN PRESSW.G. Weng et al. / Physica A 375 (2007) 668–678 669

mean-field rate-equation model [10]. Henderson [14] has conjectured that pedestrian crowds behave similarlyto gases or fluids. Helbing and Mulnar [6] has shown that human trail formation is interpreted as self-organization effect due to nonlinear interactions among persons. The escape panic [12,13,15,16], counterchannel flow [9], and bottleneck flow [11,17,18] have been studied numerically. Muramatsu and Nagatani[19,20] have found that the jamming transition occurs in the pedestrian counter flow within a channel when thedensity is higher than the threshold [9]. Tajima et al. have shown that the clogging transition occurs in theunidirectional channel flow with a bottleneck if the density is higher than the threshold [10]. The cloggingtransition is similar to that of the simple asymmetric exclusion model with a barrier. Schadschneider andothers [21,22] also presented a 2-dimensional cellular automaton model introducing a so-called floor field tocollective effects and self-organization encountered in pedestrian dynamics. Maniccam has studied the effectsof back step and update rule on the traffic congestion properties of mobile objects [23], and a 2-dimensionaltraffic system using simple congestion-avoiding traffic rules [24].

The purpose of this paper is to present a behavior-based model for pedestrian flow. The idea of behavior-based model is from Brooks’ behaviorism for mobile robot. Based on behavior-based view, Brooks [25]developed subsumption architecture and Arkin [26] presented motor schemas. In behaviorism, individualprimitive behaviors express separate goals or constrains for a task, and a high-level behavior (task) maysubsume many low-level behaviors, which run concurrently. As an example, important behaviors for anavigational task would include ‘‘move’’, ‘‘avoid’’, and ‘‘swirl’’ behaviors. Pedestrian behaves in the channellike mobile robot’s navigation. So, this paper adopts the idea of behaviorism for the simulation of pedestriancounter flow. Weng et al. [27] have presented ideas of behavior-based model and given a simulation ofevacuation from a hall. And here pedestrian counter flow using the behavior-based model will be studied tocheck the effects of weights of some basic behaviors.

This paper proposes a behavior-based lattice-gas model for pedestrian dynamics. This model can simulatepedestrian movement with different walk velocities through update at different time-step intervals. Theperiodic boundary condition for pedestrian counter flow is considered, and the dynamical characteristics arediscussed. In the following section, model is presented in detail. Section 3 gives simulation results anddiscussions, followed by conclusions.

2. Behavior-based model

We concentrate on the simplest case, which seems to be sufficient for most purposes. We will simulate thepedestrian flow by use of a lattice-gas model. Each pedestrian is represented by a walker on a lattice withL�W sites reflecting the channel. We choose the lattice spacing as 0.4m, since the typical space occupied by apedestrian is about 0.4� 0.4m2. For simulating the pedestrian movement with different walk velocities, weintroduce an idea for the model, i.e., pedestrians’ update at different time-step intervals. For example, for thepedestrians with walk velocities of 1.0 and 1.5m/s, the former ones update at every 3 time steps, and the latterones are at every 2 time steps, if the lattice spacing corresponds to approximately 0.4� 0.4m2 and one timestep is approximately 2/15 s.

In the lattice-gas model, a rule defines the state of a site in dependence of the neighbor of the site. In thismodel, the neighbor setup shown in Fig. 1 is used. Thus, the state of the core site at the next update time stepdepends on the states of the sites in the neighbor including the site above, below, right, and left, also the coresite itself of this update time step. Each pedestrian is only allowed to move to a neighbor site in the directionsof east, west, south, and north at a given time step. In each update time step, for each pedestrian a desiredmove is chosen according to the walk weights of four directions. The walk weights are determined based onthe behavior described below.

In this model, pedestrian counter flow is considered, and there are four types of walkers including the rightwalker with the walk velocities of 1.0 and 1.5m/s moving from the left to the right boundary, and the leftwalkers with walk velocities of 1.0 and 1.5m/s moving from the right to the left boundary. And so the walktask can be divided into three basic behaviors, i.e., ‘‘move’’ (moves to the given direction), ‘‘avoid’’ (avoidsother pedestrians), and ‘‘swirl’’ (swirls other pedestrians) basic behaviors. Fig. 2 gives the schematicillustration of three basic behaviors. For simplicity, the magnitudes of all three basic behaviors are set as 1, i.e.,j~Smoj ¼ j~Savj ¼ j~Sswj ¼ 1.

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Fig. 1. Neighbor setup in a 2-dimensional lattice.

V2<V1

V2<V1

(a) (b) (c) (d)

(h)(e) (f) (g)

Smo Smo Sav

Sav

SavSav

Ssw

Ssw Ssw

Ssw

Fig. 2. Schematic illustration of ‘‘move’’, ‘‘avoid’’, and ‘‘swirl’’ basic behaviors. (a) and (b) are the directions of ‘‘move’’ basic behavior for

the right and the left walkers; (c) and (d) show the directions of ‘‘avoid’’ basic behavior for the right and the left walkers; (e) and (f) indicate

the directions of ‘‘swirl’’ basic behavior for the right and the left walkers, when they encounter pedestrians with the opposite walk

velocities; (g) and (h) give the directions of ‘‘swirl’’ basic behavior for the right and the left walkers when they exceed pedestrians with the

lower walk velocities and the same walk directions, respectively.

W.G. Weng et al. / Physica A 375 (2007) 668–678670

The direction of ‘‘move’’ basic behavior is along a line from pedestrian to the goal, moving to the goal.Figs. 2(a) and (b) are the directions of ‘‘move’’ basic behavior for the right and the left walkers, i.e., go to rightand left, respectively. The direction of ‘‘avoid’’ basic behavior is along a line from pedestrian to the center ofother pedestrian, moving away from the other pedestrian. Figs. 2(c) and (d) show the directions of ‘‘avoid’’basic behavior for the right and the left walkers, respectively. The directions of ‘‘swirl’’ basic behavior dependon the neighbor pedestrians in the walk direction, considering the traffic rule and custom. The pedestrian isobligated to walk on the right-hand side of the walkway in China, if he encounters another pedestrian with theopposite walk velocity, and accustomed to exceeding another pedestrian with lower walk velocity and thesame walk direction from the left-hand side. Figs. 2(e) and (f) indicate the directions of ‘‘swirl’’ basic behaviorfor the right and the left walkers when they encounter pedestrians with the opposite walk velocities,respectively. And Figs. 2(g) and (h) give the directions of ‘‘swirl’’ basic behavior for the right and the leftwalkers, when they want to exceed pedestrians with the lower walk velocities, and the same walk directions,

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Left boundary

Right boundary

V=1.5 m /s

V=1.0 m /s V=1.5 m /s

V=1.0 m /s

L

W

Fig. 3. Schematic illustration of the pedestrian counter flow in the channel, which is composed of top and bottom walls. The right walkers

going to the right with walk velocities of 1.0 and 1.5m/s are indicated by the white and black full circles, respectively; and the left ones with

walk velocities of 1.0 and 1.5m/s are indicated by the red and green ones.

W.G. Weng et al. / Physica A 375 (2007) 668–678 671

respectively. If pedestrians encounter those with the same walk velocities and directions, they will queue, notswirl or exceed the encountering pedestrians.

These basic behaviors are independent, and they can run concurrently. The importance of basic behaviorsrelative to each other is indicated by a weight for each one, i.e., Wmo, Wav, and Wsw are for ‘‘move’’, ‘‘avoid’’,and ‘‘swirl’’ basic behaviors, respectively. This paper will set the weight a constant for simplifying the updaterules for the lattice-gas model. Each vector of basic behavior is multiplied by the corresponding weight. Andthe walk weights of four directions are the sum of these products in the corresponding directions.

The update rules of the presented model have the following structure:

(1)

The walk weights of a pedestrian are computed based on behaviorism described above. (2) If the direct neighbor in a direction is occupied by another pedestrian or obstacle, its walk weight in the

corresponding direction is set to be zero.

(3) The pedestrian will walk in the direction whose walk weight is the biggest among four directions. (4) If there are many similar walk weights, the pedestrian will choose stochastic one. If the walk weights of

four directions are all zero, the pedestrian will not walk.

In this model, sequential update is chosen. In each update time step, the pedestrians are numbered randomlyfrom 1 to N, where N is the total number of pedestrians in the system, and then each pedestrian is updatedonce in the sequential order from 1 to N.

Fig. 3 shows the schematic illustration of the pedestrian counter flow in the channel with L�W sites, whichis composed of the top and bottom walls. The right walkers going to the right with walk velocities of 1.0 and1.5m/s are indicated by the white and black full circles, respectively; and the left ones with walk velocities of1.0 and 1.5m/s are indicated by the red and green ones. Here we consider the periodic boundary condition.The pedestrians of four types are initially distributed randomly. If the right walker arrives at the rightboundary, he moves to the left boundary; if the left walker arrives at the left boundary, he moves to the rightboundary. Thus, the total number of walkers of each type is conserved as PsrLW, PfrLW, PslLW, and PflLW,where Psr, Pfr, Psl, and Pfl are the initial densities for white (slow velocity, right motion), black (fast velocity,right motion), red (slow velocity, left motion), and green ones (fast velocity, left motion). The total density isP ¼ Psr þ Pfr þ Psl þ Pfl; and only Psr ¼ Pfr ¼ Psl ¼ Pfl is considered.

3. Simulation results and discussions

Using the lattice-gas model described above, we carried out the simulation for pedestrian counter flow withthe periodic boundary condition. The channel size is set to be L ¼ 100 sites in the x direction and W ¼ 40 sitesin the y direction. We check the mean velocity /VS and the mean flow /JS of pedestrian counter flow. Themean velocity /VS of pedestrians moving at one update time step is defined as the value of the number of

ARTICLE IN PRESSW.G. Weng et al. / Physica A 375 (2007) 668–678672

walkers moving forward divided by the total number of walkers existing in the system N( ¼ PLW). The meanflow /JS of pedestrians is defined as the sum of the number of the right walkers moving through the rightboundary and that of the left ones moving through the left boundary at one update time step. For eachsimulation, 10,000 time steps are carried out, and the values of /VS and /JS are computed according to thelast 2000 time steps averaged over 5 different random initial conditions, although in all of our simulations, ittakes no more than 6000 time steps to reach the steady states.

In order to study the different effects with the different weights Wmo, Wav, and Wsw for ‘‘move’’, ‘‘avoid’’,and ‘‘swirl’’ basic behaviors, we conducted simulations with all of six possible conditions, Wsw4Wav (Wav 6¼0,here Wsw ¼ 0.6 and Wav ¼ 0.4), Wsw4Wav (Wav ¼ 0, here Wsw ¼ 0.1), WswoWav (Wsw 6¼0, here Wsw ¼ 0.4and Wav ¼ 0.6), WswoWav (Wsw ¼ 0, here Wav ¼ 0.1), Wsw ¼Wav (Wsw 6¼0, here Wsw ¼ 0.5 and Wav ¼ 0.5),and Wsw ¼Wav (Wsw ¼Wav ¼ 0). The weight of ‘‘move’’ basic behavior Wmo is the most among Wmo, Wav,and Wsw (here all Wmo ¼ 1.0), which means the pedestrians know their desired walk directions. Fig. 4 gives theplots of (a) the mean velocity /VS and (b) the mean flow /JS against the total density in the pedestriancounter flow on the lattice with L ¼ 100 sites and W ¼ 40 sites. The full squares, open squares, full circles,open circles, full triangle, and open triangle indicate the weight variables of Wsw4Wav (Wav 6¼0), Wsw4Wav

(Wav ¼ 0), WswoWav (Wsw 6¼0), WswoWav (Wsw ¼ 0), Wsw ¼Wav (Wsw 6¼0), and Wsw ¼Wav (Wsw ¼ 0),respectively. From this figure, it is indicated that all /VS always change from 1 to 0, which means allsimulations have the phase transition at some critical total density, except that with the simulation conditionof Wsw ¼Wav (Wsw ¼ 0). /VS is only 0.626 at the total density P ¼ 0.1 for Wsw ¼Wav (Wsw ¼ 0), but it willapproach to 1 with the decrease of the total density. In general, the mean flow /JS is always high if thecorresponding mean velocity /VS is high, and their phases also turn at the same critical total density. Theabove simulations show that the different simulation conditions result in different numbers of phases and theirdifferent critical total densities.

For the simulation condition of Wsw4Wav (Wav 6¼0), the system has three phases, i.e., the freely movingphase, the lane formation phase, and the stopped phase. In the freely moving phase, pedestrians can walkfreely, /VS ¼ 1. In the lane formation phase, pedestrians walk along the certain lane with the same pedestriancharacteristics, e.g., the walk direction and velocity. If pedestrian is in the stopped phase, his /VS approachesto 0 with the increase of the total density. The phases transit at the critical total densities of 0.02570.005 and0.31670.004, respectively. If the condition is Wsw4Wav (Wav ¼ 0), the system has four phases, i.e., the freelymoving phase, the lane formation phase, the partially stopped phase, and the perfectly stopped phase. Herethe partially stopped phase means only few pedestrians can walk forward and the rest cannot walk. In theperfectly stopped phase, none of the pedestrians can walk forward, i.e., /VS ¼ 0 and /JS ¼ 0. For thecondition of WswoWav (Wsw 6¼0), the system has three phases as that of Wsw4Wav (Wav 6¼0); only the criticaltotal densities are different, i.e., 0.05070.005 and 0.33670.002, respectively. For the condition of WswoWav

(Wsw ¼ 0), the system has only two phases, the moving phase and the stopped phase. If pedestrian is in the

0.0 0.2 0.4 0.6 0.8 1.0

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Wsw > Wav (Wav ≠ 0)Wsw > Wav (Wav ≠ 0) Wsw < Wav (Wsw ≠ 0) Wsw < Wav (Wsw ≠ 0) Wsw = Wav (Wsw ≠ 0) Wsw = Wav (Wsw ≠ 0)

Wsw > Wav (Wav ≠ 0)Wsw > Wav (Wav ≠ 0) Wsw < Wav (Wsw ≠ 0) Wsw < Wav (Wsw ≠ 0) Wsw = Wav (Wsw ≠ 0) Wsw = Wav (Wsw ≠ 0)

0

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w <

J>

(a) (b)

Fig. 4. (a) Plot of the mean velocity /VS and (b) plot of the mean flow /JSagainst the total density in the pedestrian counter flow on the

lattice with L ¼ 100 sites and W ¼ 40 sites. The full squares, open squares, full circles, open circles, full triangle, and open triangle indicate

the weight variables of Wsw4Wav (Wav6¼0), Wsw4Wav (Wav ¼ 0), WswoWav (Wsw 6¼0), WswoWav (Wsw ¼ 0), Wsw ¼Wav (Wsw6¼0), and

Wsw ¼Wav (Wsw ¼ 0), respectively.

ARTICLE IN PRESSW.G. Weng et al. / Physica A 375 (2007) 668–678 673

moving phase, his /VS approach to 0 with the decrease of the total density. The two phases transit at thecritical total density of 0.29770.002. For the condition of Wsw ¼Wav (Wsw 6¼0), the system also has threephases as that of Wsw4Wav (Wav 6¼0). And the critical total densities are 0.08070.010 and 0.29770.003,respectively. If the condition is Wsw ¼Wav (Wsw ¼ 0), the system has two stopped phases, the partiallystopped phase, and the perfectly stopped phase. And the critical total density is 0.06070.010.

The next observations are from the pedestrian patterns obtained at time step ¼ 10,000 and the timeevolution of the velocity V(t) for the six simulation conditions, shown in Figs. 5–16. Fig. 5 gives the pedestrianpattern obtained at time step ¼ 10,000 in the pedestrian counter flow with the weight variables of Wsw4Wav

(Wav 6¼0) on the lattice with L ¼ 100 sites and W ¼ 40 sites. (a) The lane formation phase at the total densityP ¼ 0.30; (b) the stopped phase at P ¼ 0.40. In Fig. 5(a), obviously there are lanes, in which the pedestriansare of the same type, i.e., the same walk direction and velocity. Here Fig. 5(a) shows the empirically confirmeddevelopment of dynamically varying lanes consisting of pedestrians who intend to walk in the same walkdirection and with the same walk velocity. Periodic boundary condition in the transversal direction wouldform and stabilize these lanes since they would no longer be destroyed at the ends of the walkway by enteringpedestrians. The segregation effect of lane formation is not only a result of the initial pedestrian distribution,but also a consequence of the pedestrians’ interactions. Pedestrians walking in a mixed crowd or walkingagainst the stream will have frequent and strong interactions. In each interaction, the encountering pedestriansmove a little aside according to traffic rule and custom in order to pass each other. This sideward movementtends to separate oppositely walking pedestrians and those with different walk velocities. Moreover, once thepedestrians move in uniform lanes, they will have very rare and weak interactions. Hence, the tendency tobreak up existing lanes is negligible [6]. Fig. 5(b) is the jammed configuration, and less and less pedestrianswalk forward with the increase of the total density in the stopped phase. Fig. 6 shows the V(t) plot thatcorresponds to the evolution of lane formation and jammed configuration. At P ¼ 0.3, in the beginning, V(t)decreases, but with time evolution, it will stabilize at a certain value due to the self-organized characteristic [6].The V(t) decreases with time evolution corresponding to the formation and growth of jam clusters that causedecrease in walk of pedestrians.

Fig. 7 shows the pedestrian pattern obtained at time step ¼ 10,000 in the pedestrian counter flow with theweight variables of Wsw4Wav (Wav ¼ 0) on the lattice with L ¼ 100 sites and W ¼ 40 sites: (a) The laneformation phase at the total density P ¼ 0.30; (b) the stopped phase at P ¼ 0.52. Wav ¼ 0 indicatespedestrians have no back-step, which means pedestrians only walk forward, leftward, or rightward. So, theperfectly stopped phase forms if the total density exceeds its critical value, i.e., 0.33670.002. The pedestrianpattern of the lane formation phase is different from that in Fig. 5(a); there are three group lanes composed offour lanes of pedestrians of different walk directions and velocities. From Fig. 7(a), it is indicated that the laneof pedestrians with fast walk velocity is narrower than that of pedestrians with slow walk velocity. Thisaccords to practical observation; pedestrians always exceed those with slow walk velocity through a narrowwalkway. The perfectly stopped phase in Fig. 7(b) is also different from that in Fig. 5(b) due to Wav ¼ 0; thecrowd is sparse. No ‘‘avoid’’ basic behavior results in difficulty in crowd together. Fig. 8 is the time evolutionof the velocity V(t) corresponding to Fig. 7. When the total density P is 0.52, V(t) quickly reaches 0.Figs. 9 and 10 are for the simulation condition of WswoWav (Wsw 6¼0). The pedestrian pattern of lane

Fig. 5. Pedestrian pattern obtained at time step ¼ 10,000 in the pedestrian counter flow with the weight variables of Wsw4Wav (Wav6¼0)

on the lattice with L ¼ 100 sites and W ¼ 40 sites. (a) Lane formation phase at the total density P ¼ 0.30; (b) stopped phase at P ¼ 0.40.

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0 2000 4000 6000 8000 100000.0

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V (

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P = 0.30 P = 0.40

Fig. 6. Time evolution of the velocity V(t) in the pedestrian counter flow with the weight variables of Wsw4Wav (Wav6¼0) on the lattice

with L ¼ 100 sites and W ¼ 40 sites at the total density P ¼ 0.30 and 0.40.

Fig. 7. Pedestrian pattern obtained at time step ¼ 10,000 in the pedestrian counter flow with the weight variables of Wsw4Wav (Wav ¼ 0)

on the lattice with L ¼ 100 sites and W ¼ 40 sites. (a) Lane formation phase at the total density P ¼ 0.30; (b) perfectly stopped phase at

P ¼ 0.52.

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Fig. 8. Time evolution of the velocity V(t) in the pedestrian counter flow with the weight variables of Wsw4Wav (Wav ¼ 0) on the lattice

with L ¼ 100 sites and W ¼ 40 sites at the total density P ¼ 0.30 and 0.52.

W.G. Weng et al. / Physica A 375 (2007) 668–678674

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Fig. 9. Pedestrian pattern obtained at time step ¼ 10,000 in the pedestrian counter flow with the weight variables of WswoWav (Wsw6¼0)

on the lattice with L ¼ 100 sites and W ¼ 40 sites. (a) Lane formation phase at the total density P ¼ 0.30; (b) stopped phase at P ¼ 0.40.

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Fig. 10. Time evolution of the velocity V(t) in the pedestrian counter flow with the weight variables of WswoWav (Wsw6¼0) on the lattice

with L ¼ 100 sites and W ¼ 40 sites at the total density P ¼ 0.30 and 0.40.

Fig. 11. Pedestrian pattern obtained at time step ¼ 10,000 in the pedestrian counter flow with the weight variables of WswoWav

(Wsw ¼ 0) on the lattice with L ¼ 100 sites and W ¼ 40 sites. (a) Moving phase at the total density P ¼ 0.20; (b) stopped phase at

P ¼ 0.30.

W.G. Weng et al. / Physica A 375 (2007) 668–678 675

formation phase is different from that in Fig. 5, though they are at the same total density. In Fig. 9,segregation effect is only from the walk direction, not like that from both the walk direction and velocity inFig. 5. The pedestrians with the same direction mix each other, though those with fast velocity have smallwalkway. For the simulation condition of WswoWav (Wsw ¼ 0) in Figs. 11 and 12, the system has no laneformation phase since pedestrians have no ‘‘swirl’’ behavior. For the same weight of Wsw and Wav inFigs. 13–16, the pattern of Wsw 6¼0 is somewhat same as that in Fig. 5. Figs. 15 and 16 take on differentcharacteristics; the system has partially stopped phase and perfectly stopped phase. Since pedestrian has no

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Fig. 12. Time evolution of the velocity V(t) in the pedestrian counter flow with the weight variables of WswoWav (Wav ¼ 0) on the lattice

with L ¼ 100 sites and W ¼ 40 sites at the total density P ¼ 0.20 and 0.30.

Fig. 13. Pedestrian pattern obtained at time step ¼ 10,000 in the pedestrian counter flow with the weight variables of Wsw ¼Wav (Wsw 6¼0)

on the lattice with L ¼ 100 sites and W ¼ 40 sites. (a) Lane formation phase at the total density P ¼ 0.20; (b) stopped phase at P ¼ 0.30.

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Fig. 14. Time evolution of the velocity V(t) in the pedestrian counter flow with the weight variables of Wsw ¼Wav (Wav6¼0) on the lattice

with L ¼ 100 sites, and W ¼ 40 sites at the total density P ¼ 0.20 and 0.30.

W.G. Weng et al. / Physica A 375 (2007) 668–678676

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Fig. 15. Pedestrian pattern obtained at time step ¼ 10,000 in the pedestrian counter flow with the weight variables of Wsw ¼Wav

(Wsw ¼ 0) on the lattice with L ¼ 100 sites and W ¼ 40 sites. (a) Partially stopped phase at the total density P ¼ 0.06; (b) perfectly stopped

phase at P ¼ 0.20.

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Fig. 16. Time evolution of the velocity V(t) in the pedestrian counter flow with the weight variables of Wsw ¼Wav (Wav ¼ 0) on the lattice

with L ¼ 100 sites and W ¼ 40 sites at the total density P ¼ 0.06 and 0.20.

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‘‘swirl’’ and ‘‘avoid’’ behaviors, he will stop if he encounters other pedestrian with the opposite walk direction.For the small total density (P ¼ 0.06), there are some lanes for pedestrians with the same walk direction,which means these pedestrians would not encounter other pedestrian with the opposite walk direction.

From Figs. 5–16, it is indicated that there are always the perfectly stopped phases in all weight variablessetting, though the configurations are different mainly due to the effect of Wav. The situation of Wav ¼ 0 inFigs. 7(b) and 15(b) has sparser crowd than that of Wav 6¼0 in Figs. 5(b), 9(b), 11(b) and 13(b). For one or morenonzero weight variables setting, there are three phases, i.e., the freely moving phase, the lane formationphase, and the perfectly stopped phase. For the situations of Wsw 6¼0 in Figs. 5(a), 7(a), Figs. 9(a), and 13(a),the lane formation phase always occurs. In the situations of Wsw ¼ 0, Wav 6¼0 will let pedestrians move freely,otherwise Wav ¼ 0 lets few pedestrians move freely only when these pedestrians have no encountering ones intheir corresponding ways. Only in the situations of Wav ¼ 0, the partially stopped phase occurs due toreversing behavior. From the above discussions, it is indicated that with the periodic boundary condition, wecan simulate the collective effects observed empirically such as lane formation and jammed configuration, etc.using this presented model.

4. Conclusions

This paper proposed a behavior-based lattice-gas model for pedestrian dynamics. From the idea of Brooks’behaviorism for mobile robot, we divide the walk task of pedestrian into three basic behaviors, i.e., ‘‘move’’,

ARTICLE IN PRESSW.G. Weng et al. / Physica A 375 (2007) 668–678678

‘‘avoid’’, and ‘‘swirl’’ basic behaviors. Each vector of basic behavior is multiplied by the corresponding weight.And the walk weights of four directions are the sum of these products in the corresponding directions. Thepedestrian will walk in the direction whose walk weight is the biggest among four directions. This model alsoconsiders the traffic rule and custom through the ‘‘swirl’’ basic behavior. This model can simulate pedestrianmovement with different walk velocities through the update at different time-step intervals. The periodicboundary for pedestrian counter flow with six simulation conditions in the channel is considered, and thedynamical characteristics are discussed. One of the most important conclusions is drawn that this presentedbehavior-based model can simulate some characteristics of pedestrian counter flow, e.g., lane formation andjammed configuration, etc. And the different simulation conditions have different numbers of phases and theirdifferent critical total densities.

Acknowledgments

The authors acknowledge the supports provided by the China NKBRSF project (no. 2001CB409600).

References

[1] T. Nagatani, Rep. Prog. Phys. 65 (2002) 1331.

[2] D. Helbing, Rev. Mod. Phys. 73 (2001) 1067.

[3] D. Chowdhury, L. Santen, A. Schadschneider, Phys. Rep. 329 (2000) 199.

[4] B.S. Kerner, Phys. World 12 (1999) 25.

[5] D.E. Wolf, M. Schrekenberg, A. Bachem (Eds.), Traffic and Granular Flow, World Scientific, Singapore, 1996.

[6] D. Helbing, P. Mulnar, Phys. Rev. E 51 (1995) 4282.

[7] M. Isobe, T. Adachi, T. Nagatani, Physica A 336 (2004) 638.

[8] R. Nagatani, M. Fukamachi, T. Nagatani, Physica A 358 (2005) 516.

[9] M. Muramatsu, T. Irie, T. Nagatani, Physica A 267 (1999) 487.

[10] T. Nagatani, Physica A 300 (2001) 558.

[11] Y. Tajima, T. Nagatani, Physica A 292 (2001) 545.

[12] D. Helbing, I. Farkas, T. Vicsek, Nature 407 (2000) 487.

[13] D. Helbing, I. Farkas, T. Vicsek, Phys. Rev. Lett. 84 (2000) 1240.

[14] L.F. Henderson, Nature 229 (1971) 381.

[15] A. Kirchner, A. Schadschneider, Physica A 312 (2002) 260.

[16] D. Helbing, M. Isobe, T. Nagatani, K. Takimoto, Phys. Rev. E 67 (2003) 067101.

[17] Y. Tajima, K. Takimoto, T. Nagatani, Physica A 294 (2001) 257.

[18] T. Itoh, T. Nagatani, Physica A 313 (2002) 695.

[19] M. Muramatsu, T. Nagatani, Physica A 275 (2000) 281.

[20] M. Muramatsu, T. Nagatani, Physica A 286 (2000) 377.

[21] A. Kirchner, A. Schadschneider, Physica A 312 (2002) 260.

[22] A. Schadschneider, cond-mat/0112117, 2001.

[23] S. Maniccam, Physica A 346 (2005) 631.

[24] S. Maniccam, Physica A 363 (2006) 512.

[25] R. Brooks, IEEE J. Robotics Auto. 1 (1986) 14.

[26] R.C. Arkin, Int. J. Robotics Res. 8 (1989) 92.

[27] W.G. Weng, Y. Hasemi, W.C. Fan, Int. J. Mod. Phys. C 17 (2006) 853.