  We consider a discrete-time stochastic growth model on the $d$-dimensional
lattice with non-negative real numbers as possible values per site. The growth
model describes various interesting examples such as oriented site/bond
percolation, directed polymers in random environment, time discretizations of
the binary contact path process. We show the equivalence between the slow
population growth and a localization property in terms of "replica overlap".
The main novelty of this paper is that we obtain this equivalence even for
models with positive probability of extinction at finite time. In the course of
the proof, we characterize, in a general setting, the event on which an
exponential martingale vanishes in the limit.
