  We consider a two-species competition-diffusion system involving a small
parameter $\varepsilon>0$ and discuss the validity of formal asymptotic
expansions of solutions near the sharp interface limit $\varepsilon\approx0$.
We assume that the corresponding ODE system has two stable equilibria. As in
the scalar Allen--Cahn equation, it is known that the motion of the sharp
interfaces of such systems is governed by the mean curvature flow with a
driving force. The formal expansion also suggests that the profile of the
transition layers converges to that of a traveling wave solution as
$\varepsilon\rightarrow0$. In this paper, we rigorously verify this latter
ansatz for a large class of initial data.
  The proof relies on a rescaling argument, the super--subsolution method and a
Liouville type theorem for eternal solutions of parabolic systems. Roughly
speaking, the Liouville type theorem states that any eternal solution that lies
between two traveling waves is itself a traveling wave. The same Liouville type
theorem was established for the scalar Allen--Cahn equation by Berestycki and
Hamel. In view of their importance, we prove the Liouville type theorems in a
rather general framework, not only for two-species competition-diffusion
systems but also for $m$-species cooperation-diffusion systems possibly with
time periodic or spatially periodic coefficients.
