We consider the spatially inhomogeneous Moran model with seed-banks introduced in den Hollander and Nandan (2021). Populations comprising active and dormant individuals are structured in colonies labelled by Zd, d≥1. The population sizes are drawn from an ergodic, translation-invariant, uniformly elliptic field that form a random environment. Individuals carry one of two types: ♡, ♠. Dormant individual resides in what is called a seed-bank. Active individuals exchange type from seed-bank of their own colony and resample type by choosing parent from the active populations according to a symmetric migration kernel. In den Hollander and Nandan (2021) by using a dual (an interacting coalescing particle system), we showed that the spatial system exhibits a dichotomy between clustering (mono-type equilibrium) and coexistence (multi-type equilibrium). In this paper we identify the domain of attraction for each mono-type equilibrium in the clustering regime for a fixed environment. We also show that when the migration kernel is recurrent, for a.e. realization of the environment, the system with an initially consistent type distribution converges weakly to a mono-type equilibrium in which the fixation probability to type-♡ configuration does not depend on the environment. A formula for the fixation probability is given in terms of an annealed average of type-♡ densities in dormant and active population biased by ratio of the two population sizes at the target colony.
Primary techniques employed in the proofs include stochastic duality and the environment process viewed from particle, introduced in Dolgopyat and Goldsheid (2019) for random walk in random environment on a strip. A spectral analysis of Markov operator yields quenched weak convergence of the environment process associated with the single-particle dual process to a reversible ergodic distribution, which we transfer to the spatial system of populations by using duality.
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