  We consider a population constituted by two types of individuals; each of
them can produce offspring in two different islands (as a particular case the
islands can be interpreted as active or dormant individuals). We model the
evolution of the population of each type using a two-type Feller diffusion with
immigration, and we study the frequency of one of the types, in each island,
when the total population size in each island is forced to be constant at a
dense set of times. This leads to the solution of a SDE which we call the
asymmetric two-island frequency process. We derive properties of this process
and obtain a large population limit when the total size of each island tends to
infinity. Additionally, we compute the fluctuations of the process around its
deterministic limit. We establish conditions under which the asymmetric
two-island frequency process has a moment dual. The dual is a continuous-time
two-dimensional Markov chain that can be interpreted in terms of mutation,
branching, pairwise branching, coalescence, and a novel mixed
selection-migration term. Also, we conduct a stability analysis of the limiting
deterministic dynamical system and present some numerical results to study
fixation and a new form of balancing selection. When restricting to the
seedbank model, we observe that some combinations of the parameters lead to
balancing selection. Besides finding yet another way in which genetic
reservoirs increase the genetic variability, we find that if a population that
sustains a seedbank competes with one that does not, the seed producers will
have a selective advantage if they reproduce faster, but will not have a
selective disadvantage if they reproduce slower: their worst case scenario is
balancing selection.
