Several transport-diffusion systems arise as simple models in chemotaxis (motion of bacterias or amebia interacting through a chemical signal) and in angiogenesis (development of capillary blood vessels from an exhogeneous chemoattractive signal by solid tumors). These systems describe the evolution of a density (of cells or blood vessels) coupled with the evolution equation for a chemical substance, through a nonlinear transport term depending on the gradient of the chemoattracting substance. Such systems are successful in recovering various qualitative behavior (chemotactic collapse, ring dynamics). Endothelial (i.e. cells forming blood vessels) have a tendency to form different patterns, initiating the vessels shape. Then hyperbolic models seem better adapted to describe this kind of network formation.

We will present these models, their main mathematical properties (quantitative and qualitative), numerical simulations and, for bacteria E. Coli, we will give a microscopic picture based on a kinetic modelling of the interaction (nonlinear scattering equation). We show that such models can have global solutions that converge in finite time to the Keller-Segel model, as a scaling parameter vanishes. This point of view has also the advantage of unifying all the models.