Spatially inhomogeneous populations with seed-banks: II. Clustering regime

Abstract

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 Zd, d≥1, playing the role of a geographic space, carry one of two types, ♡ or ♠, 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 active and the dormant populations depend on the colony and remain constant throughout the evolution.It was shown in den Hollander and Nandan (2021) that the spatial system is well-defined, admits a family of equilibria parametrised by the initial density of type ♡, 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 the 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 the dormant and the 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.The spatial system is hard to analyse because of the interaction in the original dual and the inhomogeneity of the colony sizes. By comparing the auxiliary dual with a non-interacting two-particle system, we are able to control the correlations that are caused by the interactions. The work in den Hollander and Nandan (2021) and the present paper is part of a broader attempt to include dormancy into interacting particle systems.