Mathematical Research for Models Which is Related to Chemotaxis System

Abstract

This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the Chemotaxis system 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 six parts. The first part focuses on background of the models which is related to Chemotaxis system and its development. The second–five part are devoted to the qualitative analysis of the (quasilinear) Keller–Segel model, the (quasilinear) chemotaxis–haptotaxis model, the (quasilinear) chemotaxis system with consumption of chemoattractant, and the (quasilinear) Keller–Segel–Navier–Stokes system. Finally, an overview of the entire contents leads to suggestions for future research activities.