[Paper Review] Spatial populations with seed-bank: well-posedness, duality and equilibrium

Andreas Greven, Franciscus den Hollander+1 more|arXiv (Cornell University)|Apr 29, 2020

Mathematical and Theoretical Epidemiology and Ecology Models4 citations

TL;DR

This paper studies spatial populations with seed-banks using interacting Fisher-Wright diffusions on a countable Abelian group, introducing three models with increasing complexity in dormancy mechanisms. It establishes well-posedness, duality, and convergence to equilibrium, showing that the seed-bank preserves genetic diversity by enabling coexistence even in recurrent migration regimes when wake-up times have infinite mean, thus revealing new universality classes beyond classical models.

ABSTRACT

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 labelling the colonies we consider a countable Abelian group $\mathbb{G}$ endowed with the discrete topology. The key example of interest is the Euclidean lattice $\mathbb{G}=\mathbb{Z}^d$. 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. For each of the three models we 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 dichotomy between clustering and coexistence in model 1 is determined by migration only. In models (2) and (3), when the wake-up time has infinite mean, the dichotomy is determined by both the exchange with the seed-bank and migration. It turns out that the seed-bank affects the long-time behaviour both quantitatively and qualitatively.

Motivation & Objective

• To understand how seed-banks in spatial populations affect long-term genetic diversity and equilibrium behavior.
• To classify the long-time behavior of interacting diffusions with seed-banks in terms of migration and dormancy parameters.
• To extend classical Fisher-Wright models to spatial settings with structured seed-banks, including random colony dormancy and colored wake-up times.
• To establish mathematical rigor via strong solutions, duality, and coupling techniques for systems with complex dormancy dynamics.
• To identify new universality classes in population genetics where seed-bank effects dominate over migration in critical dimensions.

Proposed method

• Models are constructed on a countable Abelian group $\mathbb{G}$, with $\mathbb{Z}^d$ as the primary example, to represent spatial colonies with migration.
• Three models are introduced: (1) dormancy in own colony, (2) dormancy with random color determining wake-up time, (3) dormancy in random colony with colored wake-up.
• Continuum stochastic differential equations describe the system in the large-colony-size limit, with well-posedness proven via strong solution existence.
• Duality is established between the forward process and a dual branching system, enabling analysis of long-time behavior.
• Coupling techniques are used to prove convergence to a unique equilibrium, even with fat-tailed wake-up times.
• Generalized diffusion functions are allowed via duality, extending results beyond the standard Fisher-Wright case.

Experimental results

Research questions

• RQ1 How does the inclusion of a seed-bank alter the dichotomy between coexistence and clustering in spatial population models with migration?
• RQ2 What role does the tail behavior of wake-up time distributions play in determining long-term genetic diversity?
• RQ3 Can the seed-bank induce coexistence in regimes where classical models (without seed-bank) would exhibit clustering, such as under critical or recurrent migration?
• RQ4 How does the interplay between migration and seed-bank dynamics affect the equilibrium structure in multi-colony systems?
• RQ5 Under what conditions does the seed-bank dominate over migration in determining the system’s long-time behavior?

Key findings

• In Model 1, the dichotomy between clustering and coexistence depends solely on migration: clustering occurs for recurrent migration, coexistence for transient migration, mirroring the non-seed-bank case.
• In Models 2 and 3, when wake-up times have infinite mean, the seed-bank can induce coexistence even under critically recurrent migration, where classical models would cluster.
• For infinite mean wake-up times with sufficiently fat tails, the seed-bank determines the dichotomy, rendering migration irrelevant to the long-term outcome.
• The system converges to a unique equilibrium determined by a single density parameter derived from the initial state, regardless of model complexity.
• Duality and coupling techniques allow the results to be extended to a general class of diffusion functions beyond the standard Fisher-Wright form.
• The addition of a seed-bank introduces new universality classes in critical dimensions, particularly when wake-up times have heavy tails.