  We consider a stochastic many-body system where a source refills uniformly
the empty sites of a hypercubic lattice, on which each particle is allowed to
jump (symmetrically) onto neighboring vacant sites. In addition, there is a
local {\it trap}, in competition with the injection reaction, which perturbs
the dynamics by removing particles from the system. In dimensions $d=1, 2$ and
3, for an ``imperfect'' and a ``perfect'' trap, the spatiotemporal effect of
the local perturbation of the dynamics is investigated by computing the exact
concentration of particles in the system and it is shown that the density
profile exhibits a depletion (in one and two dimensions) which properties
depend on the space dimension. The exact reactive (fluctuating) steady state
and the long-time behavior of the concentration of particles are explicitly
computed.
