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We consider a system of interacting Moran models with seed-banks. Individualslive in colonies and are subject to resampling and migration as long as theyare $active$. Each colony has a seed-bank into which individuals can retreat tobecome $dormant$, suspending their resampling and migration until they becomeactive again. The colonies are labelled by $\mathbb{Z}^d$, $d \geq 1$, playingthe role of a $geographic\, space$. The sizes of the active and the dormantpopulation are $finite$ and depend on the $location$ of the colony. Migrationis driven by a random walk transition kernel. Our goal is to study theequilibrium behaviour of the system as a function of the underlying modelparameters.  In the present paper we show that, under mild condition on the sizes of theactive population, the system is well-defined and has a dual. The dual consistsof a system of $interacting$ coalescing random walks in an $inhomogeneous$environment that switch between active and dormant. We analyse the dichotomy of$coexistence$ (= multi-type equilibria) versus $clustering$ (= mono-typeequilibria), and show that clustering occurs if and only if two random walks inthe dual starting from arbitrary states eventually coalesce with probabilityone. The presence of the seed-bank $enhances\, genetic\, diversity$. In thedual this is reflected by the presence of time lapses during which the randomwalks are dormant and do not move.

Spatially Inhomogeneous Populations with Seed-banks: Duality, Existence and Clustering