  In this kind of model, the main characteristic that determines population
viability in the long term is the stochastic growth rate (SGR) denoted
$\lambda_S$. When $\lambda_S$ is larger than one, the population grows
exponentially with probability one and when it is smaller than one, the
population goes extinct with probability one. However, even in very simple
situations it is not possible to calculate the SGR analytically. The literature
offers some approximations for the case in which environmental variability is
low, and there are also some lower and upper bounds, but there is no study of
the practical situations in which they would be tight. Some new bounds for the
SGR are built and the conditions under which each bound works best are
analyzed. These bounds are used to give some necessary and some sufficient
conditions for population explosion and extinction that are easy to check in
practice. The general results are applied to several cases, amongst them a
population structured as juveniles and adults living in an environment
switching randomly between "rich" and "poor".
