  We consider the asymptotic behavior of the solutions of a nonlinear
Schr\"odinger (NLS) model incorporating linear and nonlinear gain/loss. First,
we describe analytically the dynamical regimes (depending on the gain/loss
strengths), for finite-time collapse, decay, and global existence of solutions
in the dynamics. Then, for all the above parametric regimes, we use direct
numerical simulations to study the dynamics corresponding to algebraically
decaying initial data. We identify crucial differences between the dynamics of
vanishing initial conditions, and those converging to a finite constant
background: in the former (latter) case we find strong (weak) collapse or
decay, when the gain/loss parameters are selected from the relevant regimes.
One of our main results, is that in all the above regimes, non-vanishing
initial data transition through spatiotemporal, algebraically decaying
waveforms. While the system is nonintegrable, the evolution of these waveforms
is reminiscent to the evolution of the Peregrine rogue wave of the integrable
NLS limit. The parametric range of gain and loss for which this phenomenology
persists is also touched upon.
