ON THE MODELLING CROWD DYNAMICS FROM SCALING TO HYPERBOLIC MACROSCOPIC MODELS

BELLOMO, NICOLA; DOGBÉ, CHRISTIAN · 2008 · OpenAlex-citations

DOI: 10.1142/s0218202508003054

archive: archived pipeline: cataloged verified

Get this paper ↗ (DOI — opens at the source; we link to it, we don't host it)

Summary

This paper establishes a mathematical framework for modeling crowd dynamics in bounded domains, addressing the complexity of systems where individuals interact nonlinearly. Motivated by the need for improved safety in transportation and architectural design, particularly regarding panic scenarios, the authors aim to develop a general theory that captures pedestrian strategies and environmental interactions. Unlike vehicular traffic models, which are typically one-dimensional and assume uniform strategies, crowd dynamics occur in two or three dimensions with behaviors that shift significantly based on context, such as the onset of panic. The methodology employs a multiscale approach, identifying microscopic, kinetic, and macroscopic scales. The authors focus on deriving macroscopic hydrodynamic models using continuum mechanics. The state of the system is defined by locally averaged density and mean velocity. The core mathematical framework consists of a mass conservation equation and a linear momentum equilibrium equation. The critical modeling challenge lies in defining the acceleration term, which represents the internal driving force or motivation of pedestrians rather than external physical forces. The authors categorize second-order models into three classes based on how pedestrians select their direction of motion and respond to density. Class I models assume pedestrians move along straight lines toward a target, with acceleration determined by relaxation toward a density-dependent equilibrium velocity and the influence of density gradients. Class II models incorporate a visibility zone, allowing pedestrians to adjust their direction to avoid high-density areas within their visual field, thereby modeling path selection more realistically. Class III models utilize a conservation law for total pressure, adapting a one-dimensional traffic flow model to two dimensions to predict crowd behavior responses over time and space. The paper provides specific phenomenological expressions for equilibrium velocities and pressure terms, drawing analogies to existing vehicular traffic models like those of Kladek, Zhang, and Payne. The significance of this work lies in its systematic derivation of hyperbolic macroscopic models that account for the cognitive and strategic aspects of pedestrian movement. By analyzing the qualitative properties of these models, particularly their hyperbolicity, the authors provide a foundation for simulating crowd dynamics under normal conditions. The paper also highlights the conceptual differences between crowd and swarm modeling and outlines future research directions, including the modeling of boundary conditions and the transition from normal to panic states. This theoretical framework offers a rigorous basis for understanding complex crowd behaviors, aiding in the design of safer public spaces and the management of pedestrian flows.

Provenance

The full processing record for this entry. Every stage of this paper's journey through the pipeline is logged — what ran, with which tool and model, how many attempts it took, and when it last completed.

Summary generated by qwen3.6-27b-prismaquant on 2026-06-26; verification: verified.

Topics

Ranked by relevance to this paper. Hover a topic for its definition.

• mental model of traffic