A Class of Kinetic Models for Chemotaxis with Threshold to Prevent Overcrowding

We introduce three new examples of kinetic models for chemotaxis, where a kinetic equation for the phase-space density is coupled to a parabolic or elliptic equation for the chemo-attractant, in two or three dimensions. We prove that these models have global-in-time existence and rigorously converge, in the drift-diffusion limit to the Keller–Segel model. Furthermore, the cell density is uniformly-in-time bounded. This implies, in particular, that the limit model also has global existence of solutions.