A review on the qualitative behavior of solutions in some chemotaxis-haptotaxis models of cancer invasion

Chemotaxis is an oriented movement of cells and organisms in response to chemical signals, and plays an important role in the life of many cells and microorganisms, such as the transport of embryonic cells to developing tissues and immune cells to infection sites. Since the pioneering works of Keller and Segel, there has been a great deal of literature on the qualitative analysis of chemotaxis systems. As an important extension of the Keller-Segel system, a variety of chemotaxis-haptotaxis models have been proposed in order to gain a comprehensive understanding of the invasion-metastasis cascade. From a mathematical point of view, the rigorous analysis thereof is a nontrivial issue due to the fact that partial differential equations (PDEs) for the quantities on the macroscale are strongly coupled with ordinary differential equations (ODEs) modeling the subcellular events. It is the goal of this paper to describe recent results of some chemotaxis-haptotaxis models, inter alia macro cancer invasion models proposed by Chaplain et al., and multiscale cancer invasion models by Stinner et al., and also to introduce some open problems.