  Individuals within any species exhibit differences in size, developmental
state, or spatial location. These differences coupled with environmental
fluctuations in demographic rates can have subtle effects on population
persistence and species coexistence. To understand these effects, we provide a
general theory for coexistence of structured, interacting species living in a
stochastic environment. The theory is applicable to nonlinear, multi species
matrix models with stochastically varying parameters. The theory relies on
long-term growth rates of species corresponding to the dominant Lyapunov
exponents of random matrix products. Our coexistence criterion requires that a
convex combination of these long-term growth rates is positive with probability
one whenever one or more species are at low density. When this condition holds,
the community is stochastically persistent: the fraction of time that a species
density goes below $\delta>0$ approaches zero as $\delta$ approaches zero.
Applications to predator-prey interactions in an autocorrelated environment, a
stochastic LPA model, and spatial lottery models are provided. These
applications demonstrate that positive autocorrelations in temporal
fluctuations can disrupt predator-prey coexistence, fluctuations in
log-fecundity can facilitate persistence in structured populations, and
long-lived, relatively sedentary competing populations are likely to coexist in
spatially and temporally heterogenous environments.
