Mathematical models of taxis

Taxis is an oriented movement of biological cells or individuals toward gradient of some stimulus distributed in the environment. This mechanism, ubiquitous in nature, occurs at all levels of complexity of biological systems, starting from a sub-cellular level through tissue levels up to the interspecies interactions on ecological or social levels. Most of models are systems of degenerate or non-degenerate quasi-linear parabolic equations. During the last few decades many efforts of mathematicians have been focused on the understanding of the singularity formation in finite time in such systems which is related to the process of aggregation and pattern formation. We will focus on recent models of indirect taxis which appear in ecology to describe chemical signalling e.g taxis of predators toward gradient of some chemical (a scent) released by prey.