  We analyze the time evolution of an open quantum system driven by a localized
source of bosons. We consider non-interacting identical bosons that are
injected into a single lattice site and and perform a continuous time quantum
walks on a lattice. We show that the average number of bosons grows
exponentially with time when the input rate exceeds a certain lattice-dependent
critical value. Below the threshold, the growth is quadratic in time, which is
still much faster than the naive linear in time growth. We compute the critical
input rate for hyper-cubic lattices and find that it is positive in all
dimensions $d$ except for $d=2$ where the critical input rate vanishes---the
growth is always exponential in two dimensions. To understand the exponential
growth, we construct an explicit microscopic Hamiltonian model which gives rise
to the open system dynamics once the bath is traced out. Exponential growth is
identified with a region of dynamic instability of the Hamiltonian system.
