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This paper deals with branching processes in varying environment, namely,whose offspring distributions depend on the generations. We provide sufficientconditions for survival or extinction which rely only on the first and secondmoments of the offspring distributions. These results are then applied tobranching processes in varying environment with selection where every particlehas a real-valued label and labels can only increase along genealogicallineages; we obtain analogous conditions for survival or extinction. These lastresults can be interpreted in terms of accessibility percolation onGalton-Watson trees, which represents a relevant tool for modeling theevolution of biological populations.

Galton-Watson processes in varying environment and accessibility percolation