# Spatial populations with seed-bank: finite-systems scheme

Greven A, Den Hollander F (2023)

**Publication Type:** Journal article

**Publication year:** 2023

### Journal

**Book Volume:** 28

**Article Number:** 84

**DOI:** 10.1214/23-EJP974

### Abstract

This is the third in a series of four papers in which we consider a system of interacting Fisher-Wright diffusions with seed-bank. Individuals carry type ♥ or ♦, 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 labelling the colonies we consider a countable Abelian group G endowed with the discrete topology. Our goal is to understand in what way the seed-bank enhances genetic diversity and causes new phenomena. In [GHO22b] we showed that the system of continuum stochastic differential equations, describing the population in the large-colony-size limit, has a unique strong solution. We further showed that if the system starts from an initial law that is invariant and ergodic under translations with a density of ♥ that is equal to θ, then it converges to an equilibrium νθ whose density of ♥ also is equal to θ. Moreover, νθ exhibits a dichotomy of coexistence (= locally multi-type equilibrium) versus clustering (= locally mono-type equilibrium). We identified the parameter regimes for which these two phases occur, and found that these regimes are different when the mean wake-up time of a dormant individual is finite or infinite. The goal of the present paper is to establish the finite-systems scheme, i.e., identify how a finite truncation of the system (both in the geographic space and in the seed-bank) behaves as both the time and the truncation level tend to infinity, properly tuned together. Since the finite system exhibits clustering, we focus on the regime where the infinite system exhibits coexistence, which consists of two sub-regimes. If the wake-up time has finite mean, then the scaling time turns out to be proportional to the volume of the truncated geographic space, and there is a single universality class for the scaling limit, namely, the system moves through a succession of equilibria of the infinite system with a density of ♥ that evolves according to a renormalised Fisher-Wright diffusion and ultimately gets trapped in either 0 or 1. On the other hand, if the wake-up time has infinite mean, then the scaling time turns out to grow faster than the volume of the truncated geographic space, and there are two universality classes depending on how fast the truncation level of the seed-bank grows compared to the truncation level of the geographic space. For slow growth the scaling limit is the same as when the wake-up time has finite mean, while for fast growth the scaling limit is different, namely, the density of ♥ initially remains fixed at θ, afterwards makes random switches between 0 and 1 on a range of different time scales, driven by individuals in deep seed-banks that wake up, until it finally gets trapped in either 0 or 1 on the time scale where the individuals in the deepest seed-banks wake up. Thus, the system evolves through a sequence of partial clusterings (or partial fixations) before it reaches complete clustering (or complete fixation).

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### How to cite

**APA:**

Greven, A., & Den Hollander, F. (2023). Spatial populations with seed-bank: finite-systems scheme. *Electronic Journal of Probability*, *28*. https://doi.org/10.1214/23-EJP974

**MLA:**

Greven, Andreas, and Frank Den Hollander. "Spatial populations with seed-bank: finite-systems scheme." *Electronic Journal of Probability* 28 (2023).

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