In this paper, we are concerned with a new stochastic system of nonlinear partial differential equations modeling the Lotka–Volterra interactions of predators and preys in the presence of prey-taxis, spatial diffusion, and noises. The spatial and temporal variations of the predator’s velocity are determined by the prey gradient. In the first part, we derive a macroscopic model from stochastic kinetic equations by using the micro–macro decomposition method. In the second part, we sketch the proof of the existence of weak martingale solutions by using a Faedo–Galerkin method. In the last part, we develop a one- and two-dimensional finite volume approximation for the stochastic kinetic and macroscopic models, respectively. Our one-dimensional space numerical scheme is uniformly stable along the transition from kinetic to macroscopic regimes. We close with various numerical tests illustrating the convergence of our numerical method and some features of our stochastic macro-scale system.

This paper is concerned with the modeling and mathematical analysis of vehicular traffic phenomena. We adopt a kinetic theory point of view, under which the microscopic state of each vehicle is described by: (i) position, (ii) velocity and also (iii) activity, an additional varible that we use to describe the quality of the driver-vehicle micro-system. We use methods coming from game theory to describe interactions at the microscopic scale, thus constructing new models within the framework of the Kinetic Theory of Active Particles; the resulting models incorporate some of the symmetries that are commonly found in the mathematical models of the kinetic theory of gases. Short-range interactions and mean field interactions are introduced and modeled to depict velocity changes related to passing phenomena. Our main goal is twofold: (i) to use continuum-velocity variables and (ii) to introduce a non-local acceleration term modeling mean field interactions, related to, for example, the presence of tollgates or traffic highlights.

This paper presents a survey and critical analysis of the mathematical literature on modeling of dynamic populations living in a fluid medium. The present review paper is divided into two main parts: The first part deals with the multiscale derivation of deterministic and stochastic cross-diffusion systems governed by the incompressible Navier–Stokes equations. The derivation is obtained from the underlying description at the microscopic scale in kinetic theory models according to the micro–macro decomposition method. In the second part of this review, we are delighted to present a new variety of mathematical models describing different applications, namely, the pursuit–evasion dynamics, cancer invasion, and virus dynamics. Finally, critical analysis and future research perspectives are discussed.