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

We consider a system of interacting Moran models with seed-banks. Individuals live in colonies and are subject to resampling and migration as long as they are $active$. Each colony has a seed-bank into which individuals can retreat to become $dormant$, suspending their resampling and migration until they become active again. The colonies are labelled by $\mathbb{Z}^d$, $d \geq 1$, playing the role of a $geographic\, space$. The sizes of the active and the dormant population are $finite$ and depend on the $location$ of the colony. Migration is driven by a random walk transition kernel. Our goal is to study the equilibrium behaviour of the system as a function of the underlying model parameters. In the present paper we show that, under mild condition on the sizes of the active population, the system is well-defined and has a dual. The dual consists of 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-type equilibria), and show that clustering occurs if and only if two random walks in the dual starting from arbitrary states eventually coalesce with probability one. The presence of the seed-bank $enhances\, genetic\, diversity$. In the dual this is reflected by the presence of time lapses during which the random walks are dormant and do not move.