  By employing the N-barrier method developed in the paper, we establish a new
N-barrier maximum principle for diffusive Lotka-Volterra systems of two
competing species. As an application of this maximum principle, we show under
certain conditions, the existence and nonexistence of traveling waves solutions
for systems of three competing species. In addition, new
$(1,0,0)$-$(u^{\ast},v^{\ast},0)$ waves are given in terms of the tanh function
provided that the parameters satisfy certain conditions.
