Narrow Results

This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller–Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions. The presentation is organized in three parts. The first part focuses on a survey of some sample models, namely the original model and some of its developments, such as flux limited models, or models derived according to similar concepts. The second part is devoted to the qualitative analysis of analytic problems, such as the existence of solutions, blow-up and asymptotic behavior. The third part deals with the derivation of macroscopic models from the underlying description, delivered by means of kinetic theory methods. This approach leads to the derivation of classical models as well as that of new models, which might deserve attention as far as the related analytic problems are concerned. Finally, an overview of the entire contents leads to suggestions for future research activities.

This paper proposes a review focused on exotic chemotaxis and cross-diffusion models in complex environments. The term exotic is used to denote the dynamics of models interacting with a time-evolving external system and, specifically, models derived with the aim of describing the dynamics of living systems. The presentation first, considers the derivation of phenomenological models of chemotaxis and cross-diffusion models with particular attention on nonlinear characteristics. Then, a variety of exotic models is presented with some hints toward the derivation of new models, by accounting for a critical analysis looking ahead to perspectives. The second part of the paper is devoted to a survey of analytical problems concerning the application of models to the study of real world dynamics. Finally, the focus shifts to research perspectives within the framework of a multiscale vision, where different paths are examined to move from the dynamics at the microscopic scale to collective behaviors at the macroscopic scale.