  This paper is concerned with non-cooperative parabolic reaction--diffusion
systems which share structural similarities with the scalar Fisher--KPP
equation. These similarities make it possible to prove, among other results, an
extinction and persistence dichotomy and, when persistence occurs, the
existence of a positive steady state, the existence of traveling waves with a
half-line of possible speeds and a positive minimal speed and the equality
between this minimal speed and the spreading speed for the Cauchy problem.
Non-cooperative KPP systems can model various phenomena where the following
three mechanisms occur: local diffusion in space, linear cooperation and
su-perlinear competition.
