  We consider a multi-colony version of the Wright-Fisher model with seed-bank
that was recently introduced by Blath et al. Individuals live in colonies and
change type via resampling and mutation. Each colony contains a seed-bank that
acts as a genetic reservoir. Individuals can enter the seed-bank and become
dormant or can exit the seed-bank and become active. In each colony at each
generation a fixed fraction of individuals swap state, drawn randomly from the
active and the dormant population. While dormant, individuals suspend their
resampling. While active, individuals resample from their own colony, but also
from different colonies according to a random walk transition kernel
representingmigration. Both active and dormant individuals mutate.
  We derive a formula for the probability that two individuals drawn randomly
from two given colonies are identical by descent, i.e., share a common
ancestor. This formula, which is formulated in Fourier language, is valid when
the colonies form a discrete torus. We consider the special case of a symmetric
slow seed-bank, for which in each colony half of the individuals are in the
seed-bank and at each generation the fraction of individuals that swap state is
small. This leads to a simpler formula, from which we are able to deduce how
the probability to be identical by descent depends on the distance between the
two colonies and various relevant parameters. Through an analysis of random
walk Green functions, we are able to derive explicit scaling expressions when
mutation is slower than migration. We also compute the spatial second moment of
the probability to be identical by descent for all parameters when the torus
becomes large. For the special case of a symmetric slow seed-bank, we again
obtain explicit scaling expressions.
