The paper concerns the system \[ (*)\quad -\Delta u=u[a-u-cv],\quad - \Delta v=v[d-eu-v] \] on a smooth bounded domain \(\Omega \subseteq {\mathbb{R}}^ N\) subject to zero boundary data. A componentwise positive solution to (*) represents a steady-state to a corresponding diffusive Volterra-Lotka model for two competing species in which both species survive. Consequently, questions of existence, uniqueness, dependence on the parameters a,c,d,e, and stability (when viewed as a steady-state to a parabolic problem) of such solutions are of considerable interest.