  We consider a family of positive solutions to the system of $k$ components \[
  -\Delta u_{i,\beta} = f(x, u_{i,\beta}) - \beta u_{i,\beta} \sum_{j \neq i}
a_{ij} u_{j,\beta}^2 \qquad \text{in $\Omega$}, \] where $\Omega \subset
\mathbb{R}^N$ with $N \ge 2$. It is known that uniform bounds in $L^\infty$ of
$\{\mathbf{u}_{\beta}\}$ imply convergence of the densities to a segregated
configuration, as the competition parameter $\beta$ diverges to $+\infty$. In
this paper %we study more closely the asymptotic property of the solutions of
the system in this singular limit: we establish sharp quantitative point-wise
estimates for the densities around the interface between different components,
and we characterize the asymptotic profile of $\mathbf{u}_\beta$ in terms of
entire solutions to the limit system \[
  \Delta U_i = U_i \sum_{j\neq i} a_{ij} U_j^2. \] Moreover, we develop a
uniform-in-$\beta$ regularity theory for the interfaces.
