  In this article we investigate a parabolic-parabolic-elliptic two-species
chemotaxis system with weak competition and show global asymptotic stability of
the coexistence steady state under a smallness condition on the chemotactic
strengths, which seems more natural than the condition previously known.
  For the proof we rely on the method of eventual comparison, which thereby is
shown to be a useful tool even in the presence of chemotactic terms.
