Modelling vehicle and pedestrian collective dynamics: Challenges and advances

TL;DR

This work surveys the modelling of vehicle and pedestrian collective dynamics in urban environments, highlighting the limitations of classical reactive, force-based approaches and the need for long-term anticipation and multiscale coupling. It covers car-following and pedestrian models (including OV, FVD, social-force, velocity-obstacle, and hybrid methods) and analyzes four collective behaviours—stop-and-go waves, lane formation, long-horizon navigation, and load balancing in evacuation—using both empirical observations and stability/mean-field analyses. Key insights include stability conditions for OV/FVD models, the role of anticipatory control and mean-field game frameworks in capturing strategic behaviours, and the potential of port-Hamiltonian formulations as generic order parameters. The findings underscore the importance of integrating tactical planning, socio-psychological factors, and robust multiscale modelling to improve traffic safety, efficiency, and crowd management in complex urban systems.

Abstract

In our urbanised societies, the management and regulation of traffic and pedestrian flows is of considerable interest for public safety, economic development, and the conservation of the environment. However, modelling and controlling the collective dynamics of vehicles and pedestrians raises several challenges. Not only are the individual entities self-propelled and hard to describe, but their complex nonlinear physical and social interactions makes the multi-agent problem of crowd and traffic flow even more involved. In this chapter, we purport to review the suitability and limitations of classical modelling approaches through four examples of collective behaviour: stop-and-go waves in traffic flow, lane formation, long-term avoidance behaviour, and load balancing in pedestrian dynamics. While stop-and-go dynamics and lane formation can both be addressed by basic reactive models (at least to some extent), the latter two require anticipation and/or coordination at the level of the group. The results highlight the limitations of classical force-based models, but also the need for long-term anticipation mechanisms and multiscale modelling approaches. In response, we review new developments and modelling concepts.

Figures (6)

• Figure 1: Main variables for a car-following situation in one dimension. The dynamics of the vehicle $n$ are totally asymmetric. They solely depend on the distance and speed of the predecessor $n+1$.
• Figure 2: Main variables of pedestrian dynamics in two dimensions. $d_{n,m}$ is the distance from pedestrian $n$ to pedestrian $m$, while $\vec{e}_{n,p}$ is the unit vector joining pedestrian $p$ to pedestrian $n$.
• Figure 3: Trajectories in the experiment of Sugiyama et al. with 22 vehicles on a circuit of length 230 meters starting from an uniform configuration sugiyama2008traffic. After a while, a stop-and-go wave appears, which causes a decrease in the average speed and an increase in the speed standard deviation.
• Figure 4: Trajectories of pedestrians in a counterflow experiment cao2017fundamental. The blue trajectories are pedestrians moving from right to left, while the grey trajectories are pedestrians moving from left to right. We can see the formation of lanes moving in opposite directions.
• Figure 5: Experiment nicolas2019mechanical in which a dense crowd is crossed by an cylindrical intruder (Left). Pedestrians anticipate the passage of the cylinder as seen from the void in front of the cylinder, and temporarily accept the discomfort of the dense regions on both sides of the cylinder (Right, from butano_a_u2024a), because they know it will not last.