  We investigate the scaling limit of the seed bank diffusion when reproduction
and migration (to and from the seed bank) happen on different time-scales. More
precisely, we consider the case when migration is `slow' and reproduction is
`standard' (in the original time-scale) and then switch to a new, accelerated
time-scale, where migration is `standard' and reproduction is `fast'. This is
motivated by models for bacterial dormancy, where periods of quiescence can be
orders of magnitude larger than reproductive times, and where it is expected to
find non-trivial degenerate genealogies on the evolutionary time-scale.
  However, the above scaling regime is not only interesting from a biological
perspective, but also from a mathematical point of view, since it provides a
prototypical example where the expected scaling limit of a continuous diffusion
should (and will be) a jump-diffusion. For this situation, standard convergence
results often seem to fail in multiple ways. For example, since the set of
continuous paths from a closed subset of the c\`adl\`ag paths in each of the
classical Skorohod topologies $J_1, J_2, M_1$ and $M_2$, none of them can be
employed for tightness on path-space. Further, a na\"ive direct rescaling of
the Markov generator corresponding to the continuous diffusion immediately
leads to a blow-up of the diffusion coefficient. Still, one can identify a
well-defined limit via duality in a surprisingly non-technical way. Indeed, we
show that a certain duality relation is in some sense stable under passage to
the limit and allows an identification of the limit, avoiding all
technicalities related to the blow-up in the classical generator. The result
then boils down to a convergence criterion for time-continuous Markov chains in
a separation of time-scales regime, which is of independent interest.
