Can primitive chemotaxis generate spatial structures?

Parabolic models for the collective behavior in populations of single-cell species are considered. A predominant emphasis will be on cases in which individuals are particularly primitive in the sense that beyond a partially oriented movement toward increasing concentrations of a nutrient, further activity can essentially be neglected. Recent developments in the analysis of such nutrient taxis systems are to be described, with a special focus set on mathematical challenges related to the fundamental question how far models of this type are capable of adequately reflecting aspects of colorful dynamics known from experimental observations. In particular, some results on nontrivial large time stabilization toward either trivial nor nonconstant equilibria, as well as on persistent localization of spatial positivity sets are to be discussed.