  We analyze quasi-stationary distributions
$\{\mu^{\varepsilon}\}_{\varepsilon>0}$ of a family of Markov chains
$\{X^{\varepsilon}\}_{\varepsilon>0}$ that are random perturbations of a
bounded, continuous map $F:M\to M$, where $M$ is a closed subset of
$\mathbb{R}^k$. Consistent with many models in biology, these Markov chains
have a closed absorbing set $M_0\subset M$ such that $F(M_0)=M_0$ and
$F(M\setminus M_0)=M\setminus M_0$. Under some large deviations assumptions on
the random perturbations, we show that, if there exists a positive attractor
for $F$ (i.e., an attractor for $F$ in $M\setminus M_0$), then the weak* limit
points of $\mu_{\varepsilon}$ are supported by the positive attractors of $F$.
To illustrate the broad applicability of these results, we apply them to
nonlinear branching process models of metapopulations, competing species,
host-parasitoid interactions and evolutionary games.
