Year: 2025
Type: article
Abstract: We consider a system of interacting diffusions labeled by a geographic space that is given by the hierarchical group Ω N \Omega _N of order N ∈ N N\in \mathbb {N} . Individuals live in colonies and are subject to resampling and migration as long as they are active. Each colony has a seed-bank into which individuals can retreat to become dormant, suspending their resampling and migration until they become active again. The migration kernel has a hierarchical structure: individuals hop between colonies at a rate that depends on the hierarchical distance between the colonies. The seed-bank has a layered structure: when individuals become dormant they acquire a colour that determines the rate at which they become active again. The latter allows us to model seed-banks whose wake-up times have a fat tail. We analyse a system of coupled stochastic differential equations that describes the population in the large-colony-size limit. For fixed N ∈ N N\in \mathbb {N} , the system exhibits a dichotomy between coexistence (= locally multi-type equilibria) and clustering (= locally mono-type equilibria). To identify the clustering regime, i.e., the range of parameters controlling the migration and the seed-bank for which clustering prevails, we apply a necessary and sufficient criterion derived in (Electron. J. Probab. 27 (2022), Paper No. 18, 88, DOI 10.1214/21-ejp728. MR4377129) that is valid for any geographic space given by a countable Abelian group endowed with the discrete topology. The diffusion function controlling the resampling does not play a role in this criterion. We carry out a multi-scale renormalisation analysis in the hierarchical mean-field limit N → ∞ N\to \infty . In particular, we show that block averages on hierarchical space-time scale k ∈ N k \in \mathbb {N} perform a diffusion with a renormalised diffusion function that depends on k k . In the clustering regime, after an appropriate scaling with k k , this diffusion function converges to the Fisher-Wright diffusion function as k → ∞ k\to \infty , irrespective of the diffusion function controlling the resampling. Thus, the system exhibits full universality on large space-time scales in terms of the scaling limit. For several subclasses of parameters we identify the speed at which the scaled renormalised diffusion function converges to the Fisher-Wright diffusion as k → ∞ k\to \infty . This speed in turn determines the speed at which mono-type clusters grow in space and time. We show that the seed-bank reduces the speed compared to the model without seed-bank. This reduction is the result of a delicate interplay between migration and seed-bank, which in the limit as N → ∞ N\to \infty can be worked out explicitly.
DOI: 10.1090/memo/1553
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4409784525