This paper presents a review and critical analysis of the mathematical literature concerning the modeling of vehicular traffic and crowd phenomena. The survey of models deals with the representation scales and the mathematical frameworks that are used for the modeling approach. The paper also considers the challenging objective of modeling complex systems consisting of large systems of individuals interacting in a nonlinear manner, where one of the modeling difficulties is the fact that these systems are difficult to model at a global level when based only on the description of the dynamics of individual elements. The review is concluded with a critical analysis focused on research perspectives that consider the development of a unified modeling strategy.

This paper, that deals with the modelling of crowd dynamics, is the first one of a project finalized to develop a mathematical theory refereing to the modelling of the complex systems constituted by several interacting individuals in bounded and unbounded domains. The first part of the paper is devoted to scaling and related representation problems, then the macroscopic scale is selected and a variety of models are proposed according to different approximations of the pedestrian strategies and interactions. The second part of the paper deals with a qualitative analysis of the models with the aim of analyzing their properties. Finally, a critical analysis is proposed in view of further development of the modelling approach. Additional reasonings are devoted to understanding the conceptual differences between crowd and swarm modelling.

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In this paper, we study, on a very simple kinetic model, the flow structure induced by a discontinuity of the boundary data. The model considered is a stationary one-speed transport equation posed in a half-plane; for simplicity, the boundary data consist of the number density of incoming particles. The propagation of singularities is studied with the velocity averaging method.11

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This paper deals with the modelling of complex systems composed of a large number of elements grouped into different functional subsystems. The modelling framework is that of the thermostatted kinetic theory, which consists of a set of nonlinear integro-differential equations. Another source of nonlinearity is the presence of a mathematical thermostat that ensures the control of the global energy of the system. Specifically, this paper is devoted to the derivation of evolution equations for the macroscopic variables (density and momentum) from the underlying description at the microscopic scale delivered by the thermostatted kinetic models. With this as the aim, hyperbolic-type and parabolic-type scalings of the thermostatted kinetics for the active particles model are performed and the resulting macroscopic equations are obtained. Finally, asymptotic methods are applied to the relaxation model.

A macroscopic limit for a binary gas mixture in terms of the Boltzmann system with three small parameters: the Knudsen number, the Mach number and the diameter of particles, is considered in the whole physical space. When the small positive parameter ∊ goes to zero, it is shown that the Boltzmann system results in the compressible Euler equations decoupled with Navier–Stokes equations. In this first part of our paper, the results are of a conditional (formal) nature: both existence of a solution and existence of appropriate limits are assumed.

This paper is concerned with the asymptotic analysis of space-velocity dependent thermostatted kinetic frameworks which include conservative, nonconservative and stochastic operators. The mathematical frameworks are integro-partial differential equations that can be proposed for the modeling of most phenomena occurring in biological and chemical systems. Specifically the paper focuses on the derivation of macroscopic equations obtained by performing a low-field and a high-field scaling into the thermostatted kinetic framework and considering the related convergence when the scaling parameter goes to zero. In the low-field limit, the macroscopic equations show diffusion with respect to both the space variable and a scalar variable that is introduced for the modeling of the strategy of the particle system. In the high-field limit, the macroscopic equations show hyperbolic behavior. The asymptotic analysis is also generalized to systems decomposed in various functional subsystems.