  This paper deals with branching processes in varying environment, namely,
whose offspring distributions depend on the generations. We provide sufficient
conditions for survival or extinction which rely only on the first and second
moments of the offspring distributions. These results are then applied to
branching processes in varying environment with selection where every particle
has a real-valued label and labels can only increase along genealogical
lineages; we obtain analogous conditions for survival or extinction. These last
results can be interpreted in terms of accessibility percolation on
Galton-Watson trees, which represents a relevant tool for modeling the
evolution of biological populations.
