Multi-colony Wright-Fisher with seed-bank

We are interested in the probability that two individuals drawn randomly from two given colonies are identical by descent, i.e., share a common ancestor. This probability, which depends on the locations of the two colonies, is a measure for the inbreeding coefficient of the population. We derive a formula for this probability that 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.