Spatially Inhomogeneous Populations with Seed-banks: II. Clustering Regime

We consider a spatial version of the classical Moran model with seed-banks where the constituent populations have finite sizes. Individuals live in colonies labelled by $\mathbb{Z}^d$, $d\geq 1$, playing the role of a geographic space, carry one of the two $types$: $\heartsuit$ or $\spadesuit$, and change type via $resampling$ as long as they are $active$. Each colony contains a seed-bank into which individuals can enter to become $dormant$, suspending their resampling until they exit the seed-bank and become active again. Individuals resample not only from their own colony, but also from other colonies according to a symmetric random walk transition kernel. The latter is referred to as $migration$. The sizes of the populations vary across colonies and remain constant in time. It was shown in Hollander and Nandan (2021) that the system is well-defined, admits a family of equilibria parametrized by the initial density of type $\heartsuit$, and exhibits a dichotomy between $clustering$ (mono-type equilibrium) and $coexistence$ (multi-type equilibrium). This dichotomy is determined by a clustering criterion that is given in terms of a dual of the system, which consists of a system of $interacting$ coalescing random walks. In this paper we provide an alternative clustering criterion, given in terms of an auxiliary dual that is simpler than the original dual, and identify a range of parameters for which the criterion is met, which we refer to as the $clustering$ $regime$. It turns out that if the sizes of the active populations are non-clumping (i.e., do not take arbitrarily large values in finite regions of the geographic space) and the relative strengths of the seed-banks (i.e., the ratio of the sizes of dormant and active population in each colony) are bounded uniformly over the geographic space, then clustering prevails if and only if the (symmetrised) migration kernel is recurrent.