On the weakly competitive case in a two-species chemotaxis model
  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. 
Tobias Black, Johannes Lankeit, Masaaki Mizukami
math.AP