Multiscale derivation of deterministic and stochastic cross-diffusion models in a fluid: A review.

Abstract

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.