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The evolution of a chemotactic system involving a population of cellsattracted to self-produced chemicals is described by the Keller-Segel system.In spacial dimension 2, this system demonstrates a balance between thespreading effect of diffusion and the concentration due to self-attraction. Asa result, there exists a critical "mass" (i.e. total cell's population) abovewhich the solution of this system collapses in a finite time, while below thiscritical mass there is global existence in time. The existence of this criticalmass is related to a functional inequality known as the Moser-Trudingerinequality.  An extension of the Keller-Segel model to several cells populations wasconsidered 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 functionalinequality which generalizes the Moser-Trudinger inequality.

Chemotactic systems in the presence of conflicts: a new functional inequality