  This paper is concerned with non-cooperative parabolic reaction--diffusion
systems which share structural similarities with the scalar Fisher--KPP
equation. In a previous paper, we established that these systems admit
traveling wave solutions. It is then natural to investigate their asymptotic
distributions. This is the main object of this paper. Non-cooperative KPP
systems can model various phenomena where the following three mechanisms occur:
local diffusion in space, linear cooperation and superlinear competition.
