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Previous studies have shown that various observed self-organization phenomena in pedestrian crowds can be simulated very realistically. These phenomena include the emergence of lanes of uniform walking direction and oscillatory changes of the passing direction at bottlenecks7,10. Another interesting collective effect of pedestrian motion, which we have investigated very recently, is the formation of trail systems in green areas. In many cases, the pedestrians' desire to take the shortest way and the specific properties of the terrain are insufficient for an explanation of the trail characteristics. It is essential to include the effect of human orientation. To simulate the typical features of trail systems, we have extended the earlier model of pedestrian motion to an active-walker model by introducing equations for environmental changes and their effect on the chosen walking direction.

First, we represent the ground structure at place r and time t by a function G(r,t) which reflects the comfort of walking. Trails are characterized by particularly large values of G. On the one hand, at their positions r = rα(t), all pedestrians α leave footprints on the ground (for example, by trampling down some vegetation). Their intensity is assumed to be I(r)[1 − G(r,t)/Gmax(r)], as the clarity of a trail is limited to a maximum value Gmax(r). This causes a saturation effect [1 − G(r,t)/Gmax(r)] of the ground's alteration by new footprints. On the other hand, the ground structure changes owing to the regeneration of vegetation. This will lead to a restoration of the natural ground conditions G0(r) with a certain weathering rate 1/T(r) which is related to the durability T(r) of trails. Thus, the equation of environmental changes reads:

where δ(r − rα) denotes Dirac's delta function (which yields only a contribution for r = rα).

The attractiveness of a trail segment at place r from the perspective of place rα decreases with its distance |r − rα(t)| and depends on the visibility σ(rα). Taking this into account by using a factor exp(−|r − rα|/σ(rα)) and taking the spatial average by integration of the weighted ground structure over the green area, we obtain:

The trail potential Vtr(rα,t) reflects the attractiveness of walking at place rα. It describes indirect long-range interactions via environmental changes, which are essential for the characteristics of the evolving patterns18.

On a plain, homogenous ground, the walking direction eα of pedestrian α is determined by the direction of the next destination dα, that is eα(rα) = (dα − rα)/|dα − rα|. Without a destination, a pedestrian is expected to move into the direction of the largest increase of ground attraction, which is given by the (normalized) gradient ∇rαVtr(rα,t) of the trail potential. However, because the choice of the walking direction eα is influenced by the destination and existing trails at the same time, the orientation relation

was taken as the arithmetic average of both effects. Considering cases of rare interactions, the approximate equation of motion of a pedestrian α with desired velocity vα0is:

A comparison of simulation results with photographs shows that the above described model is in good agreement with empirical observations. In particular, the evolution of the unexpected ‘island’ in the middle of the trail system in Fig. 1 can be correctly described (Fig. 2). The goodness of fit of the model is quite surprising, as it contains only two independent parameters κ= IT/σ and λ= V0T/σ, where V0denotes the average of the desired velocities vα0. This can be shown by scaling the model to dimensionless equations. The parameter λ was kept constant.

Figure 1: Between the straight, paved ways on the university campus in Stuttgart-Vaihingen a trail system has evolved (centre of the picture).
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Two types of nodes are observed: intersections of two trails running in a straight line and junctions of two trails which smoothly merge into one trail.

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Figure 2: The structure of the emerging trail system (yellow to blue) depends essentially on the attractiveness parameter κ.
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If κ is small, a direct way system develops (left); if κ is large, a minimal way system is formed (right); otherwise, a compromise between both extremes will result (middle) which looks similar to the trail system in the centre of Fig. 1.

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Our simulations are based on a discretization of the considered area in small quadratic elements of equal size, which converts the integral in equation (2)into a sum. Temporal and spatial derivatives are approximated by difference quotients. The examples we present here begin with plain, homogenous ground. All pedestrians have their own destinations and entry points, from which they start at a randomly chosen point in time. In Fig. 2 (Fig. 3) pedestrians move between all possible pairs of three (four) fixed places; in Fig. 4, the entry points and destinations are distributed over the small ends of the ground.

Figure 3: The places and walking directions of pedestrians are represented here by arrows.
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The trail potential Vtr(r, t) is represented by a colour scale (red indicates small values, blue indicates large values). Starting with a plain ground, the structure of the trail system changes considerably during the simulation. Initially, pedestrians use the direct ways (left). Because frequently used trails become more comfortable, a bundling of trails sets in which reduces the overall length of the trail system (right). The resulting way system (whose asymmetry is caused by differences in the frequency of trail usage) could serve as a planning guideline. It provides a suitable compromise between minimal construction costs and maximal comfort. Moreover, it balances the relative detours of all walkers.

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Figure 4: When pedestrians leave footprints on the ground, trails will develop, and only parts of the ground are used for walking (in contrast to pa.
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ved areas). The similarity between the simulation result (left) and the trail system on the university campus of Brasilia (right, reproduction by permission of K. Humpert) is obvious.

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At the beginning, pedestrians take the direct ways to their respective destinations. But after some time they begin to use already existing trails, as this is more comfortable than clearing new ways. By this, a kind of selection process16,20,21 between trails sets in: frequently used trails are more attractive than others. For this reason they are chosen very often, and the resulting reinforcement makes them even more attractive. However, the weathering effect destroys rarely used trails and limits the maximum length of the way system which can be supported by a certain rate of trail usage. As a consequence, the trails begin to bundle, especially where different trails meet or intersect. This explains why pedestrians with different destinations use and produce common parts of the trail system (Figs 2 and 3).

A direct way system (which provides the shortest connections, but covers a lot of space) only develops if all ways are almost equally comfortable. If the advantage κ of using existing 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). For realistic values of κ, the evolution of the trail system stops before this state is reached (Fig. 2). Thus, κ is related to the average relative detour of the walkers. We conjecture (but cannot yet prove) that the resulting way system is the shortest one which is compatible with a certain accepted relative detour. In this sense, it yields an optimal compromise between convenience and shortness.

Therefore, we suggest the above model could be used by urban planners and landscape gardeners, who face the task of constructing the most comfortable and convenient way systems at minimal construction costs. For planning purposes, one needs to know the entry points and destinations within the considered area and the rates of usage of their connections. If necessary, these can be estimated by trip chaining models22, which are also needed in cases of complex lines of access and sight. The effects of the physical terrain and already existing ways can be taken into account by the function G0(r). By varying the model parameter κ, the overall length of the resulting trail system can be influenced (Figs 2 and 3). In the same way, one can check its structural stability. We are at present evaluating typical parameter values of κ and λ by comparison of simulation results with real pedestrian flows which are reconstructed from video films by image processing. These values could be used for designing convenient way systems in residential areas, parks and recreation areas by means of computer simulations (Fig. 3).

It would be interesting to relate our model to work on space syntax23. Repulsive interactions between pedestrians, which are relevant in cases of frequent pedestrian interactions, can be taken into account by generalizing equation (4)in accordance with the social force model of pedestrian motion7,10; in general, they lead to broader trails.