(A) Illustration of a pedestrian p₁ facing three other subjects and trying to reach the destination point O₁ marked in red. The blue dashed line corresponds to the line of sight. (B) Illustration of the same situation, as seen by pedestrian p₁. (C) Abstraction of the scene by a black and white visual field. Here, darker areas represent a shorter collision distance. (D) Graphical representation of the function f(α) reflecting the distance to collision in direction α. The left-hand side of the vision field is limited by a wall. Pedestrian p₄ is hidden by pedestrian p₂ and, therefore, not visible. Pedestrian p₃ is moving away, so a collision would occur in position p′₃, but only if p₁ moved toward the right-hand side.

Results of computer simulations for the heuristic pedestrian model (solid lines) compared with experimental results (shaded lines) during simple avoidance maneuvers in a corridor of 7.88 m length and 1.75 m width (data from ref. 10). (A) Average trajectory of a pedestrian passing a static individual standing in the middle of the corridor (n = 148 replications). (B) Average trajectory of a pedestrian passing another individual moving in the opposite direction (n = 123 replications). Dashed shaded lines indicate the SD of the average trajectory. Pedestrians are moving from left to right. The computer simulations were conducted in a way that reflected the experimental conditions. The model parameters are τ = 0.5 s, ϕ = 75°, d_max = 10 m, k = 5 × 10³, and = 1.3 m/s.

Characterization of turbulent flows in front of a bottleneck for an occupancy value of 0.98. (For the analysis of a turning corridor as in the Love Parade disaster in Duisburg in 2010, see Fig. S4). (A) The local body compression reveals two critical areas of strong compression in front of the bottleneck (shown in red). (B) Analyzing the “crowd pressure” (defined as local density times the local velocity variance) (Materials and Methods) reveals areas with a high risk of falling (in red), indicating the likelihood of a crowd disaster (4). (C) Distribution of displacements (i.e., location changes between two subsequent stops, defined by speeds with ). The double logarithmic representation reveals a power law with slope k = −1.95 ± 0.09, in good agreement with empirical findings (see figure 3e in ref. 4, where the slope is k = −2.01 ± 0.15). The local speed, local pressure, and local compression coefficients are defined in Materials and Methods. The above results are based on simulations of 360 pedestrians during 240 s in a corridor of length l = 10 m and width w = 6 m, with a bottleneck of width 4 m, assuming periodic boundary conditions.

Evaluation of different kinds of collective dynamics resulting from unidirectional flows in a street of length l = 8 m and width w = 3 m. The total number of pedestrians varied from 6 to 96, assuming periodic boundary conditions. (A) Velocity–density relation, determined by averaging over the speeds of all pedestrians for 90 s of simulation. The occupancy corresponds to the fraction of area covered by pedestrian bodies. Our simulation results (black curve) are well consistent with empirical data (dots), which were collected in real-life environments (21). The Inset indicates the average body compression where the brackets indicate an average over all pedestrians i and over time t (Materials and Methods). (B) Correlation coefficient between the average local speeds V(x,t) and , measuring the occurrence of stop-and-go waves (see Materials and Methods for the analytical definition of the local speed). Here, the value of X is set to 2 m. The increase at intermediate densities indicates that speed variations at positions x and x − X are correlated for an assumed time delay T of 3 s. Significant P values for the correlation coefficient are found for occupancies between 0.4 and 0.65, indicating the boundaries of the stop-and-go regime (Fig. S3). (C) Typical space–time diagrams at four density levels, representing different kinds of collective motion. The color coding indicates the local speed values along the street (where pedestrians move from left to right). At occupancy level 1, the diagram displays a smooth, laminar flow with occasional variations in speed. For occupancy levels 2 and 3, stop-and-go waves appear, as they have been empirically observed at high densities (figure 2a in ref. 4). At occupancy level 4, the average traffic flow is almost zero, but turbulent fluctuations in the flow occur (Fig. 4). The underlying model parameters are τ = 0.5 s, ϕ = 45°, d_max = 8 m, and k = 5 × 10³. The desired speed was chosen according to a normal distribution with mean value 1.3 m/s and SD = 0.2.