  In this paper, we consider two species chemotaxis systems with Lotka-Volterra
competition reaction terms. Under appropriate conditions on the parameters in
such a system, we establish the existence of traveling wave solutions of the
system connecting two spatially homogeneous equilibrium solutions with wave
speed greater than some critical number c*. We also show the non-existence of
such traveling waves with speed less than some critical number c*_0, which is
independent of the chemotaxis. Moreover, under suitable hypotheses on the
coefficients of the reaction terms, we obtain explicit range for the chemotaxis
sensitivity coefficients ensuring c*= c*_0, which implies that the minimum wave
speed exists and is not affected by the chemoattractant.
