Active walker model for the formation of human and animal trail systems

Active walker models have recently proved their great value for describing the formation of clusters, periodic patterns, and spiral waves as well as the development of rivers, dielectric breakdown patterns, and many other structures. It is shown that they also allow to simulate the formation of trail systems by pedestrians and ants, yielding a better understanding of human and animal behavior. A comparison with empirical material shows a good agreement between model and reality.

Our trail formation model includes an equation of motion, an equation for environmental changes, and an orientation relation. It contains some model functions, which are specified according to the characteristics of the considered animals or pedestrians. Not only the kind of environmental changes differs:

Whereas pedestrians leave footprints on the ground, ants produce chemical markings for their orientation. Nevertheless, it is more important that pedestrians steer towards a certain destination, while ants usually find their food sources by chance, i.e. they reach their destination in a stochastic way. As a consequence, the typical structure of the evolving trail systems depends on the respective species. Some ant species produce a dendritic trail system, whereas pedestrians generate a minimal detour system.

The trail formation model can be used as a tool for the optimization of pedestrian facilities: It allows urban planners to design convenient way systems which actually meet the route choice habits of pedestrians.

The emergence of complex behavior in a system consisting of simple, interacting elements [1–3] is among the most fascinating phenomena of our world. Examples can be found in almost every field of today’s scientific interest, ranging from coherent pattern formation in physical and chemical systems [4–6], to the motion of animal swarms in biology [7,8], and the behavior of social groups [9–11].

In the life and social sciences, one is usually convinced that the evolution of social systems is determined by numerous factors, such as cultural, sociological, economic, political, ecological etc. However, in recent years, the development of the interdisciplinary field “science of complexity” has lead to the insight that complex dynamic processes may also result from simple interactions, and even social structure formation could be well described within a mathematical approach [10–14]. Moreover, at a certain level of abstraction, one can find many common features between complex structures in very different fields.

A recent field of particular interest is the microsimulation of self-organization phenomena occuring in traffic systems. This includes the formation of jammed states in freeway or city traffic [15–28] as well as the various collective patterns of motion developing in pedestrian crowds [28–32] like oscillatory changes of the walking direction at narrow passages or roundabout traffic at crossings.

In this paper, we draw the attention to the specific collective phenomenon of trail formation [33,34], which is widely spread in the world of animals and humans. Regarding their shape, duration and extension, trail systems of different animal species and humans differ, of course. However, more striking is the question, whether there is a common underlying dynamics which allows for a generalized description of the formation and evolution of trail systems.

As our experience tells us, trails are adapted to the requirements of their users. In the course of time, frequently used trails become more developed, making them more attractive, whereas rarely used trails vanish again. Trails with large detours become optimized by creating shortcuts. New destinations or entry points are connected to an existing trail system. These dynamical processes occur basically without any common planning or direct communication among the users. Instead, the adaptation process can be understood as a self-organization phenomenon, resulting from the non-linear feedback between the users and the trails [35].

In order to simulate this process, we propose here a particle-based, multi-agent approach to structure formation, which belongs to the class of active walker models. Like random walkers, active walkers are subject to fluctuations and influences of their environment. However, they are additionally able to locally change their environment, e.g. by altering an environmental potential, which in turn influences their further movement and their behavior. In particular, changes produced by some walkers can influence other walkers. Hence, the non-linear feedback can be interpreted as an indirect interaction between the active walkers via environmental changes, which may lead to the self-organization of spatial structures.

Active walker models have proved their versatility in a variety of applications, such as formation of complex structures [36–42], pattern formation in physico-chemical systems [43–46], aggregation in biological [47,48] or urban [49] systems, and generation of directed motion [50,51]. The approach provides a quite stable and fast numerical algorithm for simulating processes involving large density gradients, and it is applicable also in cases where only small particle numbers govern the structure formation. In particular, the active walker model is applicable to processes of pattern formation which are intrinsically determined by the history of their creation, such as the formation of trail systems, discussed in this paper.

In Section II, the active walker model for trail formation is formulated in terms of a Langevin equation for the movement of the walkers, an equation for environmental changes, and a relation describing the orientation of the walkers with respect to existing trails. As one application of the model, Section III describes the formation of trunk trails in ant colonies, which are commonly used to exploit food sources. As a second application, in Section IV the evolution of pedestrian trail systems is modelled. Both Sections III and IV present a comparison of computational results with real trail systems, indicating a good agreement between model and empirical facts. In Section IV.A, the equations for pedestrian trail systems are scaled to dimensionless equations, in order to demonstrate that the evolving trail systems are (apart from the boundary conditions) only determined by two parameters. In Section IV.B, a macroscopic formulation of human trail formation is derived from the microscopic equations, allowing analytical investigations and an efficient calculation of the stationary solution by a self-consistent field method. Our conclusions and an outlook, which suggests an application of the model to the optimization of trail systems, are presented in Section V.

In order to introduce our model, we first describe the process of trail formation within a general stochastic framework. Hence, in this section the active walkers are not specified as pedestrians or animals. Rather, they are considered as arbitrary moving agents, who continuously change their environment by leaving markings while moving. These markings can, for example, be imagined as damaged vegetation on the ground (as in the case of hoofed animals or pedestrians) or as chemical markings (as in the case of ants).

The spatio-temporal distribution of the existing markings will be described by a ground potential $G\_k(\backslash mathbf\{r\},\; t)$. Trails are characterized by particularly large values of $G\_k(\backslash mathbf\{r\},\; t)$. The subscript $k$ allows to distinguish different kinds of markings. Due to weathering or chemical decay, the markings have a certain life time $T\_k(\backslash mathbf\{r\})$ which characterizes their local durability. Therefore, existing trails tend to fade, and the ground potential would exponentially adapt to the natural ground conditions $G\_k^0(\backslash mathbf\{r\})$, if the production of markings would be stopped. However, the creation of new markings by agent $\backslash alpha$ is described by the term $Q\_\backslash alpha(\backslash mathbf\{r\}\_\backslash alpha,\; t)\; \backslash delta(\backslash mathbf\{r\}\; -\; \backslash mathbf\{r\}\_\backslash alpha)$, where Dirac's delta function $\backslash delta(\backslash mathbf\{r\}\; -\; \backslash mathbf\{r\}\_\backslash alpha)$ gives only a contribution at the actual position $\backslash mathbf\{r\}\_\backslash alpha(t)$ of the walker. The quantity $Q\_\backslash alpha(\backslash mathbf\{r\}\_\backslash alpha,\; t)$ represents the strength of new markings and will be specified later. In summary, we obtain the following equation for the spatio-temporal evolution of the ground potential:

The motion of the active walker $\backslash alpha$ on a two-dimensional surface will be described by the following Langevin equation:

Eq. (2) considers both deterministic and stochastic influences on the motion of the active walker. $\backslash mathbf\{v\}\_\{\backslash alpha\}$ denotes the actual velocity of walker $\backslash alpha$. $\backslash gamma\_\{\backslash alpha\}$ represents some kind of friction coefficient. It is given by the relaxation time $\backslash tau\_\{\backslash alpha\}$ of velocity adaptation, specified later: $\backslash gamma\_\{\backslash alpha\}\; =\; 1/\backslash tau\_\{\backslash alpha\}$. The last term describes random variations of the motion in accordance with the fluctuation-dissipation theorem. $\backslash varepsilon\_\{\backslash alpha\}$ is the intensity of the stochastic force $\backslash boldsymbol\{\backslash xi\}\_\{\backslash alpha\}(t)$, which was assumed to be Gaussian white noise:

The $\backslash alpha$-dependence of $\backslash varepsilon\_\{\backslash alpha\}$ takes into account that different walkers could behave more or less erratic, dependent on their current situation.

Finally, the term $\backslash mathbf\{f\}\_\{\backslash alpha\}$ represents deterministic influences on the motion, such as intentions to move into a certain direction with a certain desired velocity, or to keep distance from neighboring walkers. According to the social force concept [28,30], $\backslash mathbf\{f\}\_\{\backslash alpha\}$ is specified as follows:

Here, $v\_\{\backslash alpha\}^0$ describes the desired velocity and $\backslash mathbf\{e\}\_\{\backslash alpha\}$ the desired direction of the walker. The term $\backslash mathbf\{f\}\_\{\backslash alpha\backslash beta\}$ delineates the effect of pair interactions between walkers $\backslash alpha$ and $\backslash beta$ on the motion of walker $\backslash alpha$ [28,30,33]. Since we will focus on cases of rare direct interactions, $\backslash mathbf\{f\}\_\{\backslash alpha\backslash beta\}$ can be approximately neglected here. Thus, eq. (2b) becomes

where the first term reflects an adaptation of the actual walking direction $\backslash mathbf\{v\}\_\backslash alpha/\backslash |\backslash mathbf\{v\}\_\backslash alpha\backslash |$ to the desired walking direction $\backslash mathbf\{e\}\_\backslash alpha$ and an acceleration towards the desired velocity $v\_\backslash alpha^0$ with a certain relaxation time $\backslash tau\_\backslash alpha$. Assuming that the time $\backslash tau\_\backslash alpha$ is rather short compared to the time scale of trail formation (which is characterized by the durability $T\_k$), equation (5) can be adiabatically eliminated. This leads to the following equation of motion:

To complete our trail formation model, we must finally specify the orientation relation

which determines the desired walking direction in dependence of the ground potentials $G\_k(\backslash mathbf\{r\},\; t)$. Since the concrete orientation relation for pedestrians differs from that for ants, it will be introduced later on, in the respective sections. However, it is clear that the presence of a trail will have an attractive effect, i.e. it will induce an orientation towards it. According to Eq. (6), this will cause a tendency to approach and to use the trail.

Therefore, the mechanism of trail formation is based on some kind of agglomeration process, which is delocalized due to the directedness of the walkers' motion. Starting with a plain, spatially homogeneous ground, the walkers will move arbitrarily. However, by continuously leaving markings, they produce trails which have an attractive effect on nearby walkers. Thus, the agents begin to use already existing trails after some time. By this, a kind of selection process between trails occurs (cf. [43]): Frequently used trails are reinforced, which makes them even more attractive, whereas rarely used trails may vanish again. The trails begin to bundle, especially where different trails meet or intersect. Therefore, even walkers with different entry points and destinations use and maintain common parts of the trail system.

As a first example, we want to model the formation of trunk trails, which is a widely observed phenomenon in ant colonies, such as in the Myrmicinae, Dolichoderinae and Formicinae species, commonly foraging for food from a central nest [51–53]. The trails are used to connect the food sources with the nest to allow for a collective exploitation of the food. In the case of ants, the markings are chemical signposts, so-called pheromones, which also provide the basic orientation for foraging and homing of the animals. However, note that not all ants species form trails. There is a variety of very complex foraging patterns in ants, such as swarm riding of army ants (e.g. in the species of Eciton and Dorylus) [54]. Therefore, we restrict here to cases, in which trunk trail formation of group riding ants is reported.

Before we present our model, we would like to mention some differences between active walkers and ants. The latter are rather complex biological creatures which are capable of using additional information (e.g. landmark use) or egocentric navigation [55] for their food searching and homing. Moreover, they can store information in an individual memory and communicate with nest mates in a very complex manner [56].

We will neglect these abilities, in order to show that they are not necessary for trail formation. The active walkers in our model merely count on the local information provided by the chemical trail, in order to guide themselves. They do not have additional navigation or information processing capabilities, and are not subject to long-range attracting forces to the food sources or to the nest. Hence, the formation of trunk trails in the following model is clearly a self-organizing process, based on the local interactions of the walkers [51].

Trunk trails used for foraging are typically dendritic in form. Each one starts from the nest vicinity as a single thick pathway that splits first into branches and then into twigs to convey large numbers of ants rapidly into the foraging areas (see Fig. 1).

In order to distinguish those trails which lead to a food source, the ants, after discovering a food source, use another pheromone to mark their trails, which stimulates the recruitment of additional ants to follow that trail. In our active walker model, we count on that fact by using two different chemical markings: Chemical 0 is used by the active walkers as long as they have not reached a food source, i.e. on their way from the nest to the food or during search periods. Chemical 1 is only used by active walkers after they have reached a food source, i.e. on their way back from food sources to the nest. An internal parameter $k\_\backslash alpha\; =\; \backslash \{0,\; 1\backslash \}$ indicates which of these markings is produced by the active walker $\backslash alpha$. Hence, the production term for the ground potential in Eq. (1) is defined as follows:

The first term is relevant for $k\_\backslash alpha\; =\; 0$, i.e. when searching a food source, whereas the second term contributes only for $k\_\backslash alpha\; =\; 1$, i.e. after having found some food. Since the capacity of producing chemical markings is limited, we have assumed that the quantity of chemical produced by a walker after leaving the nest or the food source decreases exponentially in time, where $\backslash beta\_0$ and $\backslash beta\_1$ are the respective decay parameters. $q\_0$ and $q\_1$ denote the initial production, and $t\_0^\backslash alpha$, $t\_1^\backslash alpha$ are the times, when the walker $\backslash alpha$ has started from the nest or the food source, respectively.

Due to the two chemical markings, we have two different ground potentials $G\_0(\backslash mathbf\{r\},\; t)$ and $G\_1(\backslash mathbf\{r\},\; t)$ here, which provide orientation for the walkers. In the following, we need to specify how they influence the motion of the agents, especially their desired directions $\backslash mathbf\{e\}\_\backslash alpha(\backslash mathbf\{r\}\_\backslash alpha,\; \backslash mathbf\{v\}\_\backslash alpha,\; t)$. At this point, we take into account that the walkers $\backslash alpha$ will not directly be affected in their behavior by the ground potentials $G\_k(\backslash mathbf\{r\},\; t)$ themselves, which reflect the pure existence of markings of type $k$ at place $\backslash mathbf\{r\}$. They will rather be influenced by the perception of their environment from their actual positions $\backslash mathbf\{r\}\_\backslash alpha$, which will be described by the trail potentials

For the detection of chemical markings, insects like ants use specific receptors which are located at their so-called antennae. Their perception is mainly determined by the angle $2\backslash varphi$ of perception, which is given by the angle between the antennae (cf. Fig. 2). Therefore, we make the assumption

where the angle $\backslash omega\_\backslash alpha$ is given by the current walking direction

According to (10), our active walkers integrate over the ground potential between the antennae of length $\backslash Delta\; r$. Note, however, that the restriction to the angle of perception is not an indispensable assumption for the generation of trails [57]. Thus, it could be neglected in a minimal model. Nevertheless, it has been introduced to mimic the biological constitution of the ants and to keep close to biology.

The perception of already existing trails will have an attractive effect $\backslash mathbf\{f\}\_\{tr\}(\backslash mathbf\{r\}\_\backslash alpha,\; \backslash mathbf\{v\}\_\backslash alpha,\; t)$ to the active walkers. This has been defined by the gradients of the trail potentials,

The above formula takes into account that walkers which move out from the nest to reach a food source ($k\_\backslash alpha\; =\; 0$) orientate by chemical 1, whereas walkers which move back from the food ($k\_\backslash alpha\; =\; 1$) orientate by chemical 0. This implies that initially, in the absense of chemical 1, the walkers move as random walkers which discover a food source only by chance.

We complete our model of trunk trail formation by specifying the orientation relation of the walkers. Assuming $\backslash mathbf\{e\}\_\backslash alpha(\backslash mathbf\{r\},\; \backslash mathbf\{v\},\; t)\; =\; \backslash mathbf\{f\}\_\{tr\}^\backslash alpha(\backslash mathbf\{r\},\; \backslash mathbf\{v\},\; t)\; /\; \backslash |\backslash mathbf\{f\}\_\{tr\}^\backslash alpha(\backslash mathbf\{r\},\; \backslash mathbf\{v\},\; t)\backslash |$, the desired walking direction $\backslash mathbf\{e\}\_\backslash alpha(\backslash mathbf\{r\},\; \backslash mathbf\{v\},\; t)$ points into the direction of the steepest increase of the relevant trail potential $V\_\{tr\}^k(\backslash mathbf\{r\},\; \backslash mathbf\{v\},\; t)$. However, this formula does not take into account the ants' persistence to keep the previous direction of motion [58]. The latter reduces the probability of changing to the opposite walking direction by fluctuations, which would cause the ants to move backwards before reaching their goal. Therefore, we modify the above formula to

where $\backslash mathcal\{N\}\_\backslash alpha(\backslash mathbf\{r\},\; \backslash mathbf\{v\},\; t)\; =\; \backslash |\backslash mathbf\{f\}\_\{tr\}^\backslash alpha(\backslash mathbf\{r\},\; \backslash mathbf\{v\},\; t)\; +\; \backslash mathbf\{e\}\_\backslash alpha^*(t\; -\; \backslash Delta\; t)\backslash |$ is a normalization factor. That means, on a ground without markings, the walking direction tends to agree with the one at the previous time $t\; -\; \backslash Delta\; t$, but it can change by fluctuations.

Finally, it is known from ant species that they are able to leave a place where they do not find food and increase their mobility to reach out for other areas. Since active walkers do not reflect their situation, they stick on their local markings even if they did not find any food source. In order to increase the mobility of the active walkers in those cases, we assume that every walker has an individual noise intensity $\backslash varepsilon\_\backslash alpha(t)$, which is related to the walker's spatial diffusion coefficient and should increase continuously, as long as the walker does not find a food source:

$t\_0^\backslash alpha$ is again the starting time of walker $\backslash alpha$ from the nest, $\backslash varepsilon\_0$ is the initial noise level and $r\_\backslash varepsilon$ its growth rate. If the noise intensity $\backslash varepsilon\_\backslash alpha$ has reached a critical upper value, the walker $\backslash alpha$ behaves more or less as a random walker which does not pay attention to the trail potential. But if the walker found some food, its individual noise intensity is set back to the initial value $\backslash varepsilon\_0$.

Figure 1b shows the result of computer simulations of trunk trail formation. The related dendritic trail system of Pheidole milicida, a harvesting ant of the southwestern U.S. deserts, is displayed in Figure 1a. In our simulation, a nest is assumed in the middle of a triangular lattice of size $100\; \backslash times\; 100$ with periodic boundary conditions. Initially, there are no chemical markings on the lattice. At time $t\; =\; 0$, a number $N\_0$ of walkers start from the nest with a random direction, leaving markings of chemical 0. If a walker discovers a food source by chance, it begins to produce chemical 1. Should such a walker find its way back to the nest, it activates an additional number of walkers, the recruits, to move out. The maximum number of walkers in the simulation is limited to $N\_\{max\}$, which denotes the population size.

For the food sources, an extended food distribution at the top and bottom line of the lattice is assumed [59]. These sources could be exhausted by the visiting walkers, but the accidental discovery of new ones in the neighborhood results in a branching of the main trails in the vicinity of the food sources and eventually leads to the dendritic structures. The trail system observed in Figure 1b remains unchanged in its major parts as has been reported also in the biological observations of trunk trail formation by ants [52]. Nevertheless, some minor trails in the vicinity of the food sources slightly shift in the course of time due to fluctuations.

Trail formation by pedestrians has been investigated only very recently [60]. It can be interpreted as a complex interplay between pedestrian motion, human orientation, and environmental changes: On the one hand, pedestrians tend to take the shortest way to their destination. On the other hand, they avoid to walk on bumpy ground, since this is uncomfortable. Therefore, they prefer to use existing trails, but they build a new shortcut, if the relative detour would be too large. In the latter case they generate a new trail, since footprints clear some vegetation. Examples of the resulting trail systems can be found in green areas, like public parks (cf. Fig. 3).

Empirical studies have shown that pedestrian motion can be surprisingly well described by the social force model sketched in Section II [11,29]. In particular, it has been demonstrated that this model allows a realistic simulation of various observed self-organization phenomena in pedestrian crowds [28–33]. This includes the emergence of collective patterns of motion, e.g. lanes of uniform walking direction [30,33] or roundabout traffic at intersections [31–33].

In this section, however, we want to model the evolution of human trail patterns. We will assume that the pedestrians behave ‘reasonably’ and, as before, we will restrict our model to the most important factors. It is obvious that pedestrians are able to show a much more complicated behavior than described here.

Since the equation (6) of motion can be also applied to pedestrians, we have to specify how moving pedestrians change their environment by leaving footprints, now. This time, we do not have to distinguish different kinds of markings. Thus, we will need only one ground potential $G(\backslash mathbf\{r\},\; t)$, and the subscript $k$ can be omitted. The value of $G$ is a measure of the comfort of walking. (Therefore, it can considerably depend on the weather conditions, which is not discussed here any further.)

For the strength $Q\_\backslash alpha(\backslash mathbf\{r\},\; t)$ of the markings produced by footprints at place $\backslash mathbf\{r\}$ we assume

where $I(\backslash mathbf\{r\})$ is the location-dependent intensity of clearing vegetation. The saturation term $[1\; -\; G(\backslash mathbf\{r\},\; t)/G\_\{\backslash max\}(\backslash mathbf\{r\})]$ results from the fact that the clarity of a trail is limited to a maximum value $G\_\{\backslash max\}(\backslash mathbf\{r\})$.

On a plain, homogeneous ground without any trails, the desired direction $\backslash mathbf\{e\}\_\backslash alpha$ of a pedestrian $\backslash alpha$ at place $\backslash mathbf\{r\}$ is given by the direction $\backslash mathbf\{e\}\_\backslash alpha^*$ of the next destination $\backslash mathbf\{d\}\_\backslash alpha$, i.e.

with the destination potential

However, the perception of already existing trails will have an attractive effect $\backslash mathbf\{f\}\_\{tr\}(\backslash mathbf\{r\},\; t)$ on the walker, which will again be defined by the gradient of the trail potential $V\_\{tr\}(\backslash mathbf\{r\},\; t)$, specified later on:

Since the potentials $U$ and $V\_\{tr\}$ influence the pedestrian at the same time, it seems reasonable to introduce an orientation relation similar to (13), by taking the sum of both potentials:

Here, $\backslash mathcal\{N\}(\backslash mathbf\{r\},\; t)\; =\; \backslash |\backslash nabla[U\_\backslash alpha(\backslash mathbf\{r\})\; +\; V\_\{tr\}(\backslash mathbf\{r\},\; t)]\backslash |$ serves as normalization factor. By relation (19) we reach that the vector $\backslash mathbf\{e\}\_\backslash alpha(\backslash mathbf\{r\}\_\backslash alpha,\; t)$ points into a direction which is a compromise between the shortness of the direct way to the destination and the comfort of using an existing trail.

Finally, we need to specify the trail potential $V\_\{tr\}$ for pedestrians. Obviously a trail must be recognized by the walkers and near enough in order to be used. Whereas the ground potential $G(\backslash mathbf\{r\},\; t)$ describes the existence of a trail segment at position $\backslash mathbf\{r\}$, the trail potential $V\_\{tr\}(\backslash mathbf\{r\}\_\backslash alpha,\; t)$ reflects the attractiveness of a trail from the actual position $\backslash mathbf\{r\}\_\backslash alpha(t)$ of the walker. Since this will decrease with the distance $\backslash |\backslash mathbf\{r\}\; -\; \backslash mathbf\{r\}\_\backslash alpha\backslash |$, we have applied the relation

where $\backslash sigma(\backslash mathbf\{r\}\_\backslash alpha)$ characterizes the sight, i.e. the range of visibility. In analogy to (10), this formula could be easily generalized to include conceivable effects of a pedestrian's angle of sight. However, we will not do this here, since we would have to calculate different trail potentials $V\_\{tr\}^\backslash alpha$ for all walkers $\backslash alpha$, then. This would make the model much more complicated.

The simulation results of the above described trail formation model are in good agreement with empirical observations, as can be seen by comparison with photographs. Our multi agent simulations begin with plain, homogeneous ground. All pedestrians have their own destinations and entry points (like shops, houses, underground stations, or parking lots), from which they start at a randomly chosen time. In Figure 4 the entry points and destinations are distributed over the small ends of the ground, while in Figure 5 (Fig. 7) pedestrians move between all possible pairs of three (four) fixed places.

At the beginning, pedestrians take the direct ways to their respective destinations. However, after some time pedestrians begin to use already existing trails, since this is more comfortable than to clear new ways. The frequency of usage decides which trails are reinforced and which ones vanish in the course of time. If the attractiveness of the forming trails is large, the final trail system is a minimal way system (which is the shortest way system that connects all entry points and destinations). However, because of the pedestrians' dislike of taking detours the evolution of the trail system normally stops before this state is reached. In other words, a so-called minimal detour system develops if the model parameters are chosen realistically (cf. Fig. 5). The resulting trails can considerably differ from the direct ways which the pedestrians would use if these were equally comfortable.

We showed that the active walker concept is suitable for modeling and understanding trail formation by pedestrians and animals. Our model turned out to be in good agreement with observations. It included an equation of motion of the walkers, an equation describing environmental changes by the markings which they leave and their decay, a relation reflecting the attractiveness of already existing trails, and an equation delineating their influence on orientation. Whereas frequently used trails are reinforced, rarely chosen trails are vanishing in the course of time. This causes a tendency of trail bundling, which can be interpreted as an agglomeration phenomenon. However, the evolving patterns are not localized since the active walkers intend to reach certain destinations, starting from their respective entry points.

The structure of the resulting trail system can considerably vary with the species. This depends decisively on the main effect which counteracts the trail attraction. Whereas our model ants find their destinations (the food sources) by chance, pedestrians can directly orient towards their destinations, so that fluctuations are no necessary model component in this case. Thus, for certain ant species a dendritic trail system is found, the detailed form of which depends on random events, i.e. the concrete history of its evolution. Pedestrians, however, produce a minimal detour system, i.e. an optimal compromise between a direct way system and a minimal way system.

As a consequence, we could derive a macroscopic model for the trail formation by pedestrians, but not for ants. It implied a self-consistent field method for a very efficient calculation of the finally evolving trail system. This is determined by the location of the entry points and destinations (e.g. houses, shops, or parking lots) and the rates of choosing the possible connections between them. Apart from this it depends on two parameters only, which was demonstrated by scaling to dimensionless equations. These are related to the trail attractiveness and the average velocity of motion.

The authors would like to thank K. Humpert and B. Hölldobler for providing some of their interesting (photo)graphical material.