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2022 Spatial populations with seed-bank: well-posedness, duality and equilibrium
Andreas Greven, Frank den Hollander, Margriet Oomen
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Electron. J. Probab. 27: 1-88 (2022). DOI: 10.1214/21-EJP728


We consider a system of interacting Fisher-Wright diffusions with seed-bank. Individuals live in colonies and are subject to resampling and migration as long as they are active. Each colony has a structured seed-bank into which individuals can retreat to become dormant, suspending their resampling and migration until they become active again. As geographic space labeling the colonies we consider a countable Abelian group G endowed with the discrete topology. The key example of interest is the Euclidean lattice G=Zd, dN. Our goal is to classify the long-time behaviour of the system in terms of the underlying model parameters. In particular, we want to understand in what way the seed-bank enhances genetic diversity. We introduce three models of increasing generality, namely, individuals become dormant: (1) in the seed-bank of their colony; (2) in the seed-bank of their colony while adopting a random colour that determines their wake-up time; (3) in the seed-bank of a random colony while adopting a random colour. The extension in (2) allows us to model wake-up times with fat tails while preserving the Markov property of the evolution. The extension in (3) allows us to place individuals in different colony when they become dormant. For each of the three models we show that the system of continuum stochastic differential equations, describing the population in the large-colony-size limit, has a unique strong solution. We also show that the system converges to a unique equilibrium depending on a single density parameter that is determined by the initial state, and exhibits a dichotomy of coexistence (= locally multi-type equilibrium) versus clustering (= locally mono-type equilibrium) depending on the parameters controlling the migration and the seed-bank. The seed-bank slows down the loss of genetic diversity. In model (1), the dichotomy between clustering and coexistence is determined by migration only. In particular, clustering occurs for recurrent migration and coexistence occurs for transient migration, as for the system without seed-bank. In models (2) and (3), an interesting interplay between migration and seed-bank occurs. In particular, the dichotomy is affected by the seed-bank when the wake-up time has infinite mean. For instance, for critically recurrent migration the system exhibits clustering for finite mean wake-up time and coexistence for infinite mean wake-up time. Hence, at the critical dimension for the system without seed-bank, new universality classes appear when the seed-bank is added. If the wake-up time has a sufficiently fat tail, then the seed-bank determines the dichotomy and migration has no effect at all. The presence of the seed-bank makes the proof of convergence to a unique equilibrium a conceptually delicate issue. By combining duality arguments with coupling techniques, we show that our results also hold when we replace the Fisher-Wright diffusion function by a more general diffusion function, drawn from an appropriate class.

Funding Statement

Andreas Greven was supported by the Deutsche Forschungsgemeinschaft (through grant DFG-GR 876/16-2 of SPP-1590), Frank den Hollander and Margriet Oomen were supported by the Netherlands Organisation for Scientific Research (through NWO Gravitation Grant NETWORKS-024.002.003). Frank den Hollander was also supported by the Alexander von Humboldt Foundation (during extended visits to Bonn and Erlangen in the Fall of 2019).


Download Citation

Andreas Greven. Frank den Hollander. Margriet Oomen. "Spatial populations with seed-bank: well-posedness, duality and equilibrium." Electron. J. Probab. 27 1 - 88, 2022.


Received: 29 April 2020; Accepted: 2 December 2021; Published: 2022
First available in Project Euclid: 3 February 2022

arXiv: 2004.14137
Digital Object Identifier: 10.1214/21-EJP728

Primary: 60J70 , 60K35
Secondary: 92D25

Keywords: coexistence versus clustering , Duality , Equilibrium , Fisher-Wright diffusion , migration , Resampling , seed-bank


Vol.27 • 2022
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