  In this article we study the positive solutions of the parabolic semilinear
system of competitive type \[ \left\{\begin{array} [c]{c}% u_{t}-\Delta
u+v^{p}=0, v_{t}-\Delta v+u^{q}=0, \end{array} \right. \] in
$\Omega\times\left(0,T\right) $, where $\Omega$ is a domain of
$\mathbb{R}^{N},$ and $p,q>0,$ $pq\neq1.$ Despite of the lack of comparison
principles, we prove local upper estimates in the superlinear case $pq>1$ of
the form \[ u(x,t)\leqq Ct^{-(p+1)/(pq-1)},\qquad v(x,t)\leqq
Ct^{-(q+1)/(pq-1)}% \] in $\omega\times\left(0,T_{1}\right) ,$ for any domain
$\omega \subset\subset\Omega$ and $T_{1}\in\left(0,T\right) ,$ and
$C=C(N,p,q,T_{1}% ,\omega).$ For $p,q>1,$ we prove the existence of an initial
trace at time 0, which is a Borel measure on $\Omega.$ Finally we prove that
the punctual singularities at time $0$ are removable when $p,q\geqq1+2/N.
