  The evolution of a chemotactic system involving a population of cells
attracted to self-produced chemicals is described by the Keller-Segel system.
In spacial dimension 2, this system demonstrates a balance between the
spreading effect of diffusion and the concentration due to self-attraction. As
a result, there exists a critical "mass" (i.e. total cell's population) above
which the solution of this system collapses in a finite time, while below this
critical mass there is global existence in time. The existence of this critical
mass is related to a functional inequality known as the Moser-Trudinger
inequality.
  An extension of the Keller-Segel model to several cells populations was
considered before in the literature. Here we review some of these results and,
in particular, consider the case of conflict between two populations, that is,
when population one attracts population two, while, at the same time,
population two repels population one. This assumption leads to a new functional
inequality which generalizes the Moser-Trudinger inequality.

Analysis of PDEs