This paper presents a review and critical analysis on the modeling of the dynamics of vehicular traffic, human crowds and swarms seen as living and, hence, complex systems. It contains a survey of the kinetic models developed in the last 10 years on the aforementioned topics so that overlapping with previous reviews can be avoided. Although the main focus of this paper lies on the mesoscopic models for collective dynamics, we provide a brief overview on the corresponding micro and macroscopic models, and discuss intermediate role of mesoscopic model between them. Moreover, we provide a number of selected challenging research perspectives for readers’ attention.

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This paper deals with the derivation of macroscopic tissue models from the underlying description delivered by a class of equations that models binary mixtures of multicellular systems by methods of the kinetic theory for active particles. Cellular interactions generate both modification of the biological functions and proliferative and destructive events. The asymptotic analysis deals with suitable parabolic and hyperbolic limits, and is specifically focused on the modeling of the chemotaxis phenomena.

In this paper the stability of the Vlasov–Poisson–Fokker–Planck with respect to the variation of its constant parameters, the scaled thermal velocity and the scaled thermal mean free path, is analyzed. For the case in which the scaled thermal velocity is the inverse of the scaled thermal mean free path and the latter tends to zero, a parabolic limit equation is obtained for the mass density. Depending on the space dimension and on the hypothesis for the initial data, the convergence result in L1 is weak and global in time or strong and local in time.

This paper deals with the modeling and simulation of swarms viewed as a living, hence complex, system. The approach is based on methods of kinetic theory and statistical mechanics, where interactions at the microscopic scale are nonlinearly additive and modeled by stochastic games.

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This paper presents a revisiting, with developments, of the so-called kinetic theory for active particles, with the main focus on the modeling of nonlinearly additive interactions. The approach is based on a suitable generalization of methods of kinetic theory, where interactions are depicted by stochastic games. The basic idea consists in looking for a general mathematical structure suitable to capture the main features of living, hence complex, systems. Hopefully, this structure is a candidate towards the challenging objective of designing a mathematical theory of living systems. These topics are treated in the first part of the paper, while the second one applies it to specific case studies, namely to the modeling of crowd dynamics and of the immune competition.

The analysis of dissipative transport equations within the framework of open quantum systems with Fokker-Planck-type scattering is carried out from the perspective of a Wigner function approach. In particular, the well-posedness of the self-consistent whole-space problem in 3D is analyzed: existence of solutions, uniqueness and asymptotic behavior in time, where we adopt the viewpoint of mild solutions in this paper. Also, the admissibility of a density matrix formulation in Lindblad form with Fokker-Planck dissipation mechanisms is discussed. We remark that our solution concept allows to carry out the analysis directly on the level of the kinetic equation instead of on the level of the density operator.