  Source-sink systems are metapopulations of patches that can be of variable
habitat quality. They can be seen as graphs, where vertices represent the
patches, and the weighted oriented edges give the probability of dispersal from
one patch to another. We consider either finite or source-transitive graphs,
i.e., graphs that are identical when viewed from a(ny) source. We assume
stochastic, individual-based, density-independent reproduction and dispersal.
By studying the path of a single random disperser, we are able to display
simple criteria for persistence, either necessary and sufficient, or just
sufficient. In case of persistence, we characterize the growth rate of the
population as well as the asymptotic occupancy frequencies of the line of
ascent of a random survivor. Our method allows to decouple the roles of
reproduction and dispersal. Finally, we extend our results to the case of
periodic or random environments, where some habitats can have variable growth
rates, autocorrelated in space and possibly in time. In the whole manuscript,
special attention is given to the example of regular graphs where each pair of
adjacent sources is separated by the same number of identical sinks. In the
case of a periodic and random environment, we also display examples where all
patches are sinks when forbidding dispersal but the metapopulation survives
with positive probability in the presence of dispersal, as previously known for
a two-patch mean-field model with parent-independent dispersal.
