  Earth-like planets orbiting M-dwarfs are prominent future targets when
searching for life outside the solar system. We apply our newly developed
Coupled Atmosphere Biogeochemistry model to investigate the coupling between
the biosphere, geosphere and atmosphere to gain deeper insight into the
atmospheric evolution of Earth-like planets orbiting M-dwarfs. Our main goal is
to understand better atmospheric processes affecting biosignatures and climate
on such worlds. Furthermore, this is the first study to our knowledge which
applies an automated chemical pathway analysis quantifying the production and
destruction pathways of O$_2$ for an Earth-like planet with an Archean O$_2$
abundance orbiting in the habitable zone of the M-dwarf AD Leo. Results suggest
that the main production arises in the upper atmosphere from CO$_2$ photolysis
followed by catalytic HO$_x$ reactions. The strongest destruction does not take
place in the troposphere, as was the case in Gebauer et al. (2017) for an
early-Earth analog planet around the Sun, but instead in the middle atmosphere
where H$_2$O photolysis is the strongest. This result was driven by the strong
Lyman-$\alpha$-radiation output of AD Leo, which efficiently photolyzes H$_2$O.
Results further suggest that early Earth-like atmospheres of planets orbiting
an M-dwarf like AD Leo are in absolute terms less destructive for atmospheric
O$_2$ than for early-Earth analog planets around the Sun despite higher
concentrations of reduced gases such as e.g. H$_2$, CH$_4$ and CO. Hence the
net primary productivity (NPP) required to produce the same amount of
atmospheric O$_2$ at the surface is reduced. This implies that a possible Great
Oxidation event, analogous to that on Earth, would have occurred earlier in
time in analog atmospheres around M-dwarfs.
