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We introduce a multi-species chemotaxis type system admitting an arbitrarilylarge number of population species, all of which are attracted vs. repelled bya single chemical substance. The production vs. destruction rates of thechemotactic substance by the species is described by a probability measure. Forsuch a model we investigate the variational structures, in particular we provethe existence of Lyapunov functionals, we establish duality properties as wellas a logarithmic Hardy-Littlewood-Sobolev type inequality for the associatedfree energy. The latter inequality provides the optimal critical value for theconserved total population mass.

A multi-species chemotaxis system: Lyapunov functionals, duality, critical mass