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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 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...
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.
- All authors
- Hollander, W.T.F. den; Nandan, S.
- Date
- 2022-04-28
- Volume
- 150
- Pages
- 116 - 146
Funding
- Sponsorship
- NWO
- Grant number
- TOP1.17.019