ORIGINAL RESEARCH Andreas Schadschneider, 1 Ansgar Kirchner, 1 Katsuhiro Nishinari 2 1 Institute for Theoretical Physics, Universität zu Köln, Köln, Germany; 2 Deartment of Alied Mathematics and Informatics, Ryukoku University, Shiga, Jaan Abstract: This aer resents a model for the simulation of edestrian dynamics insired by the behaviour of ants in ant trails. Ants communicate by roducing a heromone that can be smelled by other ants. In this model, edestrians roduce a virtual heromone that influences the motion of others. In this way all interactions are strictly local, and so even large crowds can be simulated very efficiently. Nevertheless, the model is able to reroduce the collective effects observed emirically, eg the formation of lanes in counterflow. As an alication, we reroduce a surrising result found in exeriments of evacuation from an aircraft. Keywords: collective henomena, traffic flow, comuter simulations, evacuation rocesses, cellular automata, chemotaxis Introduction During the last decade, considerable research has been devoted to the toic of highway traffic using methods from hysics (Chowdhury et al 2; Helbing 21). However, edestrian dynamics (Schreckenberg and Sharma 21) has not been studied as extensively as vehicular traffic. One reason is robably its generically two-dimensional nature. Therefore, only few models exist that can reroduce the emirically observed behaviour accurately. An imortant examle is the social force model (Helbing and Molnar 1995; Helbing 21), where edestrians are treated as articles 1 subject to long-ranged forces induced by the social behaviour of the individuals. This leads to (couled) equations of motion similar to Newtonian mechanics. Such a descrition has many similarities with the modelling of sand and other granular materials. Due to the comlexity, any edestrian model for realistic alications has to be treated on a comuter. Efficient comuter simulations of the social force model are difficult for large crowds consisting of hundreds or thousands of individuals. Due to the long-ranged nature of forces, many interaction terms have to be calculated. An additional difficulty arises in comlex geometries. It has to be checked whether an interaction between two individuals is ossible, or whether it is blocked by walls. This requires comlicated algorithms, which slow down the seed of the simulations. Efficient comuter simulations require simle models that nevertheless rovide an accurate descrition of reality. One simle class of models are so-called cellular automata that have been studied in statistical hysics for a long time (Wolfram 1986, 22). In a cellular automaton, sace, time and state variables are discrete, which makes them ideally suited for high-erformance comuter simulations. However, this still leaves the roblem of the long-ranged interactions. In rincile, a model is needed where the interactions are only local (restricted to the nearest neighbourhood) so that the resence of thousands of other eole and walls does not influence erformance of the simulations. It is exactly at this oint, where we can learn from nature. The roblem of transforming the effects of a long-ranged satial interaction into local ones has already been solved by many insects. In fact, we will develo a model for edestrian dynamics that is insired by the formation of ant trails (Wilson 1971; Hölldobler and Wilson 199; Camazine et al 21; Mikhailov and Calenbuhr 22). Here, the henomenon of chemotaxis is the clue (see Ben- Jacob (1997) for a review). The model, which is introduced in the section Modelling edestrian dynamics, uses a form of virtual chemotaxis to mediate the interactions between the edestrians. To achieve comlex behaviour in a simle fashion, one often resorts to a stochastic descrition. A realistic situation can seldom be described comletely by a deterministic aroach. Minor events can lead to a very different behaviour due to the comlexity of the interactions involved. For the roblem of edestrian motion this becomes evident in the case of a anic, for examle, where the behaviour of eole seems almost unredictable. But also for normal situations, a stochastic comonent in the dynamics can lead to a more accurate descrition of comlex henomena, since Corresondence: Andreas Schadschneider, Institute for Theoretical Physics, Universität zu Köln, Köln 5937, Germany; tel +49 221 47 4312; fax +49 221 47 5159; email as@th.uni-koeln.de Alied Bionics and Biomechanics 23:1(1) 11 19 23 Oen Mind Journals Limited. All rights reserved. 11
Schadschneider et al a b c Figure 1 Clogging near a bottleneck (door). Shown are tyical stages of an evacuation simulation: initial state (t = ) (a); middle stages (b); end stage with only a few articles left (c). it takes into account that we usually do not have full knowledge about the state of the system and its dynamics. Collective henomena One of the reasons why the investigation of edestrian dynamics is attractive for hysicists is that many interesting collective effects and self-organisation henomena can be observed. Here, we give only a brief overview; refer to Helbing (21) and Schreckenberg and Sharma (21) for a more comrehensive discussion and list of references. Jamming At large densities, various kinds of jamming henomena occur; for examle, when many eole try to leave a large room at the same time, and the flow is limited by a door or narrowing (see Figure 1). Therefore, this kind of jamming henomenon does not deend strongly on the microscoic dynamics of the articles. This clogging effect is tyical for a bottleneck situation. It is imortant for ractical alications, esecially evacuation simulations. Other tyes of jamming occur in the case of counterflow where two grous of edestrians mutually block each other. This haens tyically at high densities and when it is not Figure 2 Lane formation in counterflow in a narrow corridor. ossible to turn around and move back, eg when the flow of eole is large. Lane formation In counterflow, ie two grous of eole moving in oosite directions, a kind of sontaneous symmetry breaking occurs (see Figure 2). The motion of the edestrians can selforganise in such a way that (dynamically varying) lanes are formed where eole move in just one direction (Helbing and Molnar 1995). In this way, strong interactions with oncoming edestrians are reduced, and a higher walking seed is ossible. Oscillations In counterflow at bottlenecks, eg doors, one can observe oscillatory changes in the direction of motion. Once a edestrian is able to ass the bottleneck, it becomes easier for others to follow her/him in the same direction until somebody is able to ass the bottleneck (eg through a fluctuation) in the oosite direction (see Figure 3). Patterns at intersections At intersections, various collective atterns of motion can be formed. Tyical examles are short-lived roundabouts, which make the motion more efficient. Even if these are connected with small detours, the formation of these atterns can be favourable since they allow for a smoother motion. Panic situations In anic situations, many counter-intuitive henomena can occur. In the faster-is-slower effect, a higher desired velocity leads to a slower movement of a large crowd 12 Alied Bionics and Biomechanics 23:1(1)
Figure 4). The cells are labelled by the index i (i=1, 2,..., L); L being the length of the lattice. With each ant cell, we associate another cell for the heromones (Figure 4). For simlicity, only the two states heromone resent and no heromone resent are noted. The dynamics of the model consists of two stages (see Figure 4). In stage I, ants are allowed to move, whereas stage II corresonds to the dynamics of the heromone. Stage I: motion of ants An ant in cell i that has an emty cell in front of it moves forward with: Q if a heronome is resent at i+ 1 robability = (1) q if no heromone is resent at i+ 1 where, to be consistent with real ant trails, we assume q<q. The value of the hoing robability reresents the average velocity v of non-interacting ants. Without heromones, a single ant moves with v = q, whereas v = Q in the resence of heromones. Thus, stage I, in a simle way, reresents the fact that ants move faster towards their destination if they are guided by heromones. Figure 3 Oscillations of the flow direction at a door with counterflow. (Helbing et al 2). Such effects are extremely imortant for evacuations in emergency situations. Modelling ant trails Before introducing our model of edestrian dynamics, we will discuss a model for ant trails (Chowdhury et al 22; Nishinari et al 23), which shows the basic rinciles, esecially the influence of chemotaxis. Ants communicate with each other by droing a chemical (generically called heromone ) on the substrate as they crawl forward (Wilson 1971; Hölldobler and Wilson 199; Camazine et al 21; Mikhailov and Calenbuhr 22). Although we cannot smell it, the trail heromone sticks to the substrate long enough for the other following ants to ick u its smell and follow the trail. Ant trails may serve different uroses (trunk trails, migratory routes) and may also be used in a different way by different secies. Therefore, one-way trails are observed as well as trails with counterflow of ants. Here, we consider one-dimensional trails with ants moving only in one direction. The trail is now divided into cells that can accommodate at most one ant at a time (see Stage II: evaoration of heromones At each cell i occuied by an ant after stage I, a heromone will be created. In this way, a kind of heromone trace is roduced by moving ants. Any free heromone at a site i not occuied by an ant will evaorate with the robability f a b c Figure 4 Schematic reresentation of tyical configurations; it also illustrates the udate rocedure. (a) Configuration at time t, ie before stage I of the udate. The non-vanishing hoing robabilities of the ants are also shown exlicitly. (b) Configuration after one ossible realisation of stage I. Two ants have moved comared with the to art of the figure. Also indicated are the heromones that may evaorate in stage II of the udate scheme. (c) Configuration after one ossible realisation of stage II. Two heromones have evaorated and one heromone has been created due to the motion of an ant. Alied Bionics and Biomechanics 23:1(1) 13
Schadschneider et al er unit time; therefore, the trace created by a moving ant will disaear after some time if it is not renewed by other ants. Desite its simlicity, this model contains the basics that describe the motion of ants in a trail (Chowdhury et al 22; Nishinari et al 23). The forward motion is reresented by hoing, whereas the mutual hindrance of ants comes from the exclusion rincile that allows at most one ant er cell. The influence of the heromone is encoded in the hoing robabilities q and Q, and finally it has also been taken into account that heromones have a finite lifetime through the evaoration robability f. Of course, many subtle details have been neglected. However, these details turn out to be of less imortance for an understanding of the basic rinciles underlying the rocess of chemotaxis. Modelling edestrian dynamics In the revious section, the basic rinciles for modelling ant trails was resented. Based on similar rinciles, a model for edestrian dynamics was develoed (Burstedde et al 21; Kirchner and Schadschneider 22). Guided by the henomenon of chemotaxis the interactions between edestrians are local and thus allow for comutational efficiency. General rinciles The edestrians create a virtual trace, which then influences the motion of other edestrians. This allows for a very efficient imlementation on a comuter since now all interactions are local. The transition robabilities for all edestrians only deend on the occuation numbers and strength of the virtual trace in the neighbourhood, ie the long-ranged satial interaction has been translated into a local interaction with memory. The number of interaction terms in other long-ranged models, eg the social-force model, is of the order N 2 (where N is the number of articles), whereas it is only of the order N in our model. The idea of a virtual trace can be generalised to a socalled floor field. This floor field includes the virtual trace created by the edestrians as well as a static comonent, which does not change with time. The latter allows modelling, eg referred areas, walls and other obstacles. The edestrians then react to both tyes of floor fields. To kee the model simle, the articles needed to be rovided with as little intelligence as ossible, and there needed to be the formation of comlex structures and collective effects by means of self-organisation. In contrast to other aroaches, it is not necessary to make detailed assumtions about human behaviour. Nevertheless, the model is able to reroduce many of the basic henomena. The key feature to substitute individual intelligence is the floor field. Aart from the occuation number, each cell carries an additional quantity (field) similar to the heromone field in the ant trail model discussed. This field has its own dynamics given by diffusion and decay. Interactions between edestrians are reulsive for short distances. One likes to kee a minimal distance from others to avoid buming into them. In the simlest version of the model, this is taken into account through hard-core reulsion, which revents multile occuation of the cells. For longer distances, the interaction is often attractive; for examle, when walking in a crowded area it is usually advantageous to follow directly behind the redecessor. Large crowds may also be attractive due to curiosity. These basic rinciles are already sufficient to reroduce the effects described in the section Collective henomena, which so far only has been achieved by the social force model (Helbing and Molnar 1995). This shows the ability of cellular automata to create comlex behaviour out of simle rules, and the great alicability of this aroach to all kinds of traffic flow roblems (Chowdhury et al 2). Definition of the model and its dynamics The area available for edestrians is divided into small cells of aroximately 4 cm 4 cm. This is the tyical sace occuied by a edestrian in a dense crowd. As in the ant trail model, each cell can either be emty or occuied by exactly one article (edestrian). For secial situations it might be desirable to use a finer discretisation, such that each edestrian occuies four cells, for examle, instead of one. Each edestrian can move to one of the unoccuied next-neighbour cells (i, j) (or stay at the resent cell) at each discrete time ste t t +1, according to certain transition robabilities ij (Figure 5) as exlained below.,-1-1,,,1 1, Figure 5 Possible target cells for a erson at the next time ste. 14 Alied Bionics and Biomechanics 23:1(1)
For the case of evacuation rocesses, the static floor field S describes the shortest distance to an exit door. The field strength S ij is set inversely roortional to the distance from the door. The dynamic floor field D is a virtual trace left by the edestrians similar to the heromone in chemotaxis (see Ben-Jacob (1997) for a review). It has its own dynamics, namely diffusion and decay, which leads to broadening, dilution and finally vanishing of the trace. At t = for all sites (i, j) of the lattice, the dynamic field is zero, ie D ij =. Whenever a article jums from site (i, j) to one of the neighbouring cells, D at the origin cell is increased by one. The udate is done in arallel for all articles (synchronous dynamics). This introduces a time scale into the dynamics, which can roughly be identified with the reaction time t reac. From the emirically observed average velocity of a edestrian of about 1.3 m/s, one can estimate that one time ste in our model corresonds to aroximately.3 s in real time. This is of the order of the reaction time t reac and thus consistent with our microscoic rules. It also agrees nicely with the time needed to reach the normal walking seed, which is about.5 s. The udate rules of our cellular automaton are divided into five stes that have the following structure: 1. The dynamic floor field D is modified according to its diffusion and decay rules, controlled by the arameters α and δ. In each time ste of the simulation, each single article of the whole dynamic field D decays with robability δ and diffuses with robability α to one of its neighbouring cells. 2. For each edestrian, the transition robabilities ij for a move to an unoccuied next-neighbour cell (i, j) are determined by the two floor fields and one s inertia (Figure 5). The values of the fields D (dynamic) and S (static) are weighted with two sensitivity arameters k D and k S : ij = Nex( kddij) ex( kssij)( 1 nij ) ξ ij (2) with a normalisation N that guarantees ij = 1. nij is i, j the occuation number of the target cell (i, j), ie n ij =1 for an occuied cell and n ij = otherwise. Similarly, ξ ij is a hindrance factor, which is if a cell cannot be reached, eg due to the resence of a wall or other obstacle, and 1 otherwise. 3. Each edestrian randomly chooses a target cell based on the transition robabilities ij determined by ste 2. 4. Whenever two or more edestrians attemt to move to the same target cell, the movement of all involved t t+ 1 Figure 6 Refused movement due to the friction arameter µ. µ articles is denied with robability µ [,1] (see Figure 6), ie all edestrians remain at their site (Kirchner, Nishinari et al 23). This means that with robability 1 µ, one of the individuals moves to the desired cell. The one allowed to move is decided using a robabilistic method (Burstedde et al 21; Kirchner, Nishinari et al 23). In the following, µ is called friction arameter, since it has similar effects as friction between, for examle, sand grains. 5. The edestrians who are allowed to move erform their motion to the target cell chosen in ste 3. D, at the origin cell (i, j) of each moving article, is increased by one: D ij D ij + 1, ie D can take any non-negative integer value. This corresonds to an ant droing a heromone. In contrast to the ant trail model, an arbitrary number of heromones can exist at each site so that it is ossible to model a varying strength of the trace. The above rules are alied to all edestrians at the same time (arallel udate). The rules and their interretation will be exlained in more detail below. In rincile we can give each article a referred walking direction. From this direction, a 3 3 matrix of references (Burstedde et al 21) is constructed, which contains the robabilities for a move of the article in a certain direction when interactions are neglected. The robabilities can be related to the average velocity and its longitudinal and transversal standard deviations (see Burstedde (21) for details). So the matrix of references contains information about the referred walking direction and seed. In rincile, it can differ from cell to cell deending on the geometry and aim of the edestrians. Interretation of the rules To reroduce the collective henomena discussed in a revious section, it is necessary to take into account the motion of all other edestrians in a certain neighbourhood. This imlies the introduction of non-local or longer-ranged interactions. In some continuous models this is done using Alied Bionics and Biomechanics 23:1(1) 15
Schadschneider et al the idea of a social force (Helbing and Molnar 1995; Helbing 21). The aroach described above is different. Since we want to kee the model as simle as ossible we avoid using a long-range interaction exlicitly. Instead, the concet of a floor field was introduced that takes into account interactions between edestrians and the geometry of the system (building) in a unified and simle way without loosing the advantages of local transition rules. The floor field modifies the transition robabilities in such a way that a motion into the direction of larger fields is referred. The dynamic floor field D is just the virtual trace left by the edestrians. It is modified by the resence of edestrians and has its own dynamics, ie diffusion and decay. Usually the dynamic floor field is used to model a ( longranged ) attractive interaction between the articles. Each edestrian leaves a trace, ie the floor field of occuied cells is increased. Diffusion and decay lead to a broadening, dilution and, finally, the vanishing of the trace. The static floor field S does not evolve with time and is not changed by the resence of edestrians. Such a field can be used to secify regions of sace that are more attractive, eg emergency exits or sho windows. Since the total transition robability is roortional to the dynamic floor field, it becomes more attractive to follow in the footstes of other edestrians. This effect cometes with the referred walking direction and the effects of the geometry encoded in S ij. The relative influence of the contributions is controlled by the couling arameters k S and k D. These deend on the situation to be studied. Consider, for examle, a situation where eole want to leave a large room. Normal circumstances, where everybody is able to see the exit, can be modelled by solely using a static floor field, which decreases radially with the distance from the door. Since transitions in the direction of larger fields are more likely, this will automatically guarantee that everybody is walking in the direction of the door. If, however, the exit cannot be seen by everybody, eg in a smoke-filled room or in the case of failing lights, eole will try to follow others, hoing that they know the location of the exit. In this case, the couling to the dynamic floor field is much stronger, and the static field has a considerable influence only in the vicinity of the door. The influence of the friction arameter µ [,1] (see Kirchner, Nishinari et al (23) for details) in ste 4 of the udate rules, which describes clogging effects between the edestrians, will now be discussed. Due to the use of arallel dynamics, it might haen that in ste 3 two (or more) edestrians choose the same target cell. Such a situation is called a conflict. Since the hard-core exclusion allows at most one article er cell, at most one erson can move. If friction is included, the movement of all involved articles is denied with the robability µ, ie all edestrians remain at their site (see Figure 6). This means that with robability 1 µ, one of the individuals moves to the desired cell. The article that actually moves is then determined by a robabilistic method, eg by choosing one of them with equal robabilities. The effects of this friction arameter are similar to those of contact friction in granular materials, which exlains the name. It can be interreted as a moment of hesitation when eole meet and try to avoid each other. It is a local effect that can have enormous influence on macroscoic quantities like flow and evacuation time (Kirchner, Nishinari et al 23). Note that the kind of friction introduced here only influences interacting articles, not the average velocity of a freely moving edestrian. Results The model defined in the revious section is easy to imlement on a comuter. Due to its simlicity, very efficient Monte Carlo simulations, even of large crowds, are ossible. In the following, some interesting results are highlighted. Collective effects The model is able to reroduce various fundamental henomena described in the section Collective henomena, such as lane formation in a corridor, herding and oscillations at a bottleneck (Burstedde et al 21; Kirchner and Schadschneider 22). This is an indisensable roerty for any reliable model of edestrian dynamics, esecially for discussing safety issues. As an examle, a simulation of counterflow in a corridor is shown in Figure 7. One can clearly see that in the lower art of the figure lanes have formed, whereas in the uer art the motion is still disordered. In simulations of an evacuation rocess from a large room with only one exit, three different regimes can be observed (Kirchner and Schadschneider 22) deending on the values of the couling constants k S and k D to the static and dynamic field, resectively. In these simulations, the edestrians did not have a referred direction of motion. The only information about the location of the exit was 16 Alied Bionics and Biomechanics 23:1(1)
Figure 7 Snashot from a simulation of a corridor with two grous of edestrians moving in oosite directions. The left art shows the arameter control. The central window is the corridor and the light and dark squares are right- and left-moving edestrians, resectively. The right art shows the floor fields for the two secies. obtained through the static floor field, which increases in the direction of the door. For strong couling k S and very small couling k D we find an ordered regime. This corresonds to a situation where all edestrians have a good knowledge about the location of the exit and try to go there using the shortest ath. At large densities, this leads to clogging at the door, since many edestrians arrive there trying to leave the room. On the other hand, the disordered regime is characterised by strong couling k D and weak couling k S. This describes situations where the individuals do not know the exact location of the exit, eg in a smoke-filled room or in the case of failing lights. In such situations, eole try to follow others, hoing that they know better where to go. This herding behaviour is tyical for anic situations. Between these two regimes an otimal regime exists where the combination of interaction with the static and the dynamic floor fields minimises the evacuation time (Kirchner and Schadschneider 22). In such evacuation simulations, counter-intuitive effects can be observed. Usually, one would exect that increasing the couling k S to the static field would lead to a decrease of the evacuation time. For large k S eole try to use the shortest way to the exit, whereas for small k S they erform a random motion, ie the effective velocity should increase with k S. However, this is not always true. Figure 8 shows the influence of the friction arameter on the evacuation time T. As exected, T is monotonically Evacuation time (s) 15 1 5 µ=. µ=.6 µ=.9 2 4 6 8 1 k s Figure 8 Deendence of evacuation times on the friction arameter µ as a function of k S for density ρ =.3. Alied Bionics and Biomechanics 23:1(1) 17
Schadschneider et al increasing with µ. For µ = the evacuation time is monotonically decreasing with increasing k S, as exected. For large µ, however, T(k S ) shows a minimum at an intermediate couling strength k S 1. This behaviour is related to clogging at the door; here, the influence on the evacuation time. If k S is large, clogging is strong since all edestrians arrive at the door quickly. For small and intermediate coulings, clogging is less strong because eole arrive at very different times due to the almost random-walk like behaviour. Therefore, fewer conflicts occur. Note that this observation is similar to the faster-isslower effect. An alication: cometitive versus cooerative behaviour An interesting exerimental result (Muir et al 1996) shows that the motivation level has a significant influence on the egress time from a narrow body aircraft. The exeriment was carried out with grous of 5 7 ersons, where in one case (cometitive), a bonus was aid for the first 3 ersons reaching the exit, and in the non-cometitive (cooerative) case, no bonus was aid. The time of the 3th erson to reach the exit was measured (t com and t non-com, resectively) for variable exit widths w. The surrising result is t com > t non-com for w < w c, whereas t com < t non-com for w > w c. The critical width was determined exerimentally to be about 7 cm (see Figure 9). This shows that cometition is beneficial if the exit width exceeds a certain minimal value. For small exit widths, however, cometition is harmful. Within the framework of the model described above, cometition is described as an increased assertiveness (large k S ) and at the same time strong hindrance in conflict situations, ie large µ. Cooeration is then reresented by small k S and vanishing µ. This allows quantitatively distinguishing cometition from cooeration and comaring the exerimental results to simulations. The exerimental results have been reroduced qualitatively using a simlified scenario (Kirchner, Klüfel et al 23). Instead of a real aerolane, an evacuation from a room without additional internal structure was simulated. Figure 9 shows tyical average evacuation times for the noncometitive and the cometitive regime with an initial article density of ρ =.3 (116 articles). The door width is variable and ranges from 1 to 11 lattice sites. Clearly, the simulations are able to reroduce the observed crossing of the two curves at a small door width qualitatively. Without friction (µ = ), increasing k S alone always decreases t egress. The effect is therefore only obtained by increasing both k S and µ. Thus, there are two factors that determine the egress of ersons and the overall evacuation time in our scenario: walking seed (controlled by the arameter k S ) and friction (controlled by µ). These arameters deend in a different way on the door width: the influence of the friction dominates for very narrow doors, which leads the crossing shown in Figure 9. Summary and conclusions We have introduced a stochastic cellular automaton to simulate edestrian behaviour. 2 The general idea of our model is insired by the henomenon of chemotaxis, which is used by certain insects for communication. However, a 35 b 4 3 cometitive non-cometitive 3 k S =1., µ=.6 k S =1., µ=. Evacuation time (s) 25 2 T 2 15 1 1 5 1 15 2 Aerture width (cm) 1 3 5 7 9 11 w(cells) Figure 9 (a) Exerimental results for evacuation times for an aircraft with variable door width for cometitive and non-cometitive behaviour of the assengers. (b) Results from comuter simulations of the cellular automaton model. 18 Alied Bionics and Biomechanics 23:1(1)
edestrians leave a virtual trace rather than a chemical one. This virtual trace has its own dynamics (diffusion and decay) that, for examle, restricts the interaction range (in time). It is imlemented as a dynamic floor field, which allows the use of local interactions only. Furthermore, we do not need to rovide the edestrian in the model with any intelligence. Together with the static floor field, it offers the ossibility to take different effects into account in a unified way, eg the social forces between the edestrians or the geometry of the building. The floor fields translate satial long-ranged interactions into local interactions with memory. The latter can be imlemented much more efficiently on a comuter. Another advantage is an easier treatment of comlex geometries, eg the formation of lanes in counterflow. Furthermore, counterintuitive results were found in evacuation simulations like the faster-is-slower effect. The examle resented here shows once more that nature has found efficient solutions to some difficult roblems. In the case of edestrian dynamics, the recise nature of interactions between eole is difficult to determine. Therefore, it is not entirely clear whether our aroach is just a technical trick or not. The virtual trace created by moving edestrians might corresond to some reresentation of the ath of others formed in the mind of each individual. Nevertheless, our results so far give a strong indication that there are more similarities between the motion of insects and intelligent individuals than exected. Acknowledgements We thank D Chowdhury and H Klüfel for helful discussions and their collaboration on some of the results resented here. Notes 1 In this aer, we use edestrian and article interchangeably. 2 Further information and Java alets for the scenarios studied here can be found at htt://www.th.uni-koeln.de/~as/ References Ben-Jacob E. 1997. From snowflake formation to growth of bacterial colonies. II: cooerative formation of comlex colonial atterns. Contem Phys, 38:25 41. Burstedde C, Klauck K, Schadschneider A et al. 21. Simulation of edestrian dynamics using a two-dimensional cellular automaton. Physica A, 295:57 25. Camazine S, Deneubourg JL, Franks NR et al. 21. Self-organization in biological systems. Princeton, NJ: Princeton Univ Pr. Chowdhury D, Guttal V, Nishinari K et al. 22. A cellular-automata model of flow in ant trails: non-monotonic variation of seed with density. J Phys, A35:L573 7. Chowdhury D, Santen L, Schadschneider A. 2. Statistical hysics of vehicular traffic and some related systems. Phys Re, 329:199 329. Helbing D. 21. Traffic and related self-driven many-article systems. Rev Mod Phys, 73:167 141. Helbing D, Farkas I, Vicsek T. 2. Simulating dynamical features of escae anic. Nature, 47:487 9. Helbing D, Molnar P. 1995. Social force model for edestrian dynamics. Phys Rev E Stat Phys Plasmas Fluids Relat Interdisci Toics, 51:4282 6. Hölldobler B, Wilson EO. 199. The ants. Cambridge, MA: Belkna. Kirchner A, Klüfel H, Nishinari K et al. 23. Simulation of cometitive egress behavior: comarison with aircraft evacuation data. Physica A, 324:689 97. Kirchner A, Nishinari K, Schadschneider A. 23. Friction effects and clogging in a cellular automaton model for edestrian dynamics. Phys Rev E Stat Nonlin Soft Matter Phys, 67(5 Pt 2):56122. Kirchner A, Schadschneider A. 22. Simulation of evacuation rocesses using a bionics-insired cellular automaton model for edestrian dynamics. Physica A, 312:26 76. Mikhailov AS, Calenbuhr V. 22. From cells to societies. Models of comlex coherent action. Berlin: Sringer-Verlag. Muir HC, Bottomley DM, Marrison C. 1996. Effects of motivation and cabin configuration on emergency aircraft evacuation behavior and rates of egress. Int J Aviat Psychol, 6:57 77. Nishinari K, Chowdhury D, Schadschneider A. 23. Cluster formation and anomalous fundamental diagram in an ant-trail model. Phys Rev E Stat Nonlin Soft Matter Phys, 67(3 Pt 2):3612. Schreckenberg M, Sharma SD, eds. 21. Pedestrian and evacuation dynamics. Berlin: Sringer-Verlag. Wilson EO. 1971. The insect societies. Cambridge, MA: Belkna. Wolfram S. 1986. Theory and alications of cellular automata. Singaore: World Scientific. Wolfram S. 22. A new kind of science. Illinois: Wolfram Media. Alied Bionics and Biomechanics 23:1(1) 19