The impact of the metallicity and star formation rate on the
  time-dependent galaxy-wide stellar initial mass function
  The stellar initial mass function (IMF) is commonly assumed to be an
invariant probability density distribution function of initial stellar masses
being represented by the canonical IMF. As a consequence the galaxy-wide IMF
(gwIMF), defined as the sum of the IMFs of all star forming regions, should
also be invariant. Recent observational and theoretical results challenge the
hypothesis that the gwIMF is invariant. In order to study the possible reasons
for this variation we use the IMF determined in resolved star clusters and
apply the IGIMF-theory to calculate a grid of gwIMF models for metallicities,
-3<[Fe/H]<1, and galaxy-wide star formation rates,
$10^{-5}$<SFR<$10^{5}\,\mathrm{M_{\odot}/yr}$. For a galaxy with metallicy
[Fe/H]$<0$ and SFR$\,> 1\,M_\odot$/yr, which is a common condition in the early
Universe, we find that the gwIMF is top-heavy (more massive stars), when
compared to the canonical IMF. For a SFR $< 1\,\mathrm{M_{\odot}/yr}$ the gwIMF
becomes top-light regardless of the metallicity. For metallicities
$\mathrm{[Fe/H]} > 0$ the gwIMF can become bottom-heavy regardless of the SFR.
The IGIMF models predict that massive elliptical galaxies should have formed
with a gwIMF that is top-heavy within the first few hundred Myr of the galaxy's
life and that it evolves into a bottom-heavy gwIMF in the metal-enriched
galactic center. We study the SFR$-$H$\alpha$ relation, its dependency on
metallicity and the SFR, the correction factors to the Kennicutt SFR$_{\rm
K}-$H$\alpha$ relation, and provide new fitting functions Late-type dwarf
galaxies show significantly higher SFRs with respect to Kennicutt SFRs, while
star forming massive galaxies have significantly lower SFRs than hitherto
thought. This has implications for the gas-consumption time scales and for the
main sequence of galaxies. The Leo P and ultra-faint dwarf galaxies are
discussed explicitly. [abridged]
