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Multi-colony Wright-Fisher with seed-bank

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.