Spatial populations with seed-banks in random environment: III. Convergence towards mono-type equilibrium

We consider the spatially inhomogeneous Moran model with seed-banks introduced in [20]. Populations comprising active and dormant individuals are structured colonies labeled by Zd,d≥1. The population sizes sampled from a translation-invariant, ergodic, uniformly elliptic field that constitutes static random environment. Individuals carry one of two types: ♡ ♠. Dormant individual resides what is called seed-bank. Active exchange type seed-bank their own colony, resample choosing parent at distinct populations according to symmetric migration kernel. In [20] exploiting dual process given an interacting coalescing particle system, we showed spatial system exhibits dichotomy between clustering (mono-type equilibrium) coexistence (multi-type equilibrium). this paper, identify domain attraction for each mono-type equilibrium regime arbitrary fixed Furthermore, show dimensions d≤2, when kernel recurrent, almost all realization environment, initially consistent type-distribution converges weakly which probability fixation type-♡ configuration does not depend on An explicit formula terms annealed average densities population, biased ratio target colony. Primary techniques employed proofs include stochastic duality environment viewed particle, [24] walk strip. A spectral analysis Markov operator yields quenched weak convergence associated single-particle reversible ergodic distribution, transfer using duality.