  In this article we deal with a class of strongly coupled parabolic systems
that encompasses two different effects: degenerate diffusion and chemotaxis.
Such classes of equations arise in the mesoscale level modeling of biomass
spreading mechanisms via chemotaxis. We show the existence of an exponential
attractor and, hence, of a finite-dimensional global attractor under certain
'balance conditions' on the order of the degeneracy and the growth of the
chemotactic function.
