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Spatially Inhomogeneous Populations with Seed-Banks: I. Duality, Existence and Clustering

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  • Frank Hollander

    (Universiteit Leiden)

  • Shubhamoy Nandan

    (Universiteit Leiden)

Abstract

We consider a system of interacting Moran models with seed-banks. 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 colonies are labelled by $${\mathbb {Z}}^d$$ Z d , $$d \ge 1$$ d ≥ 1 , playing the role of a geographic space. The sizes of the active and the dormant population are finite and depend on the location of the colony. Migration is driven by a random walk transition kernel. Our goal is to study the equilibrium behaviour of the system as a function of the underlying model parameters. In the present paper, under a mild condition on the sizes of the active populations, the system is well defined and has a dual. The dual consists of a system of interacting coalescing random walks in an inhomogeneous environment that switch between an active state and a dormant state. We analyse the dichotomy of coexistence (= multi-type equilibria) versus clustering (= mono-type equilibria) and show that clustering occurs if and only if two random walks in the dual starting from arbitrary states eventually coalesce with probability one. The presence of the seed-bank enhances genetic diversity. In the dual this is reflected by the presence of time lapses during which the random walks are dormant and do not move.

Suggested Citation

  • Frank Hollander & Shubhamoy Nandan, 2022. "Spatially Inhomogeneous Populations with Seed-Banks: I. Duality, Existence and Clustering," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1795-1841, September.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:3:d:10.1007_s10959-021-01119-z
    DOI: 10.1007/s10959-021-01119-z
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    References listed on IDEAS

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    1. Carinci, Gioia & Giardinà, Cristian & Giberti, Claudio & Redig, Frank, 2015. "Dualities in population genetics: A fresh look with new dualities," Stochastic Processes and their Applications, Elsevier, vol. 125(3), pages 941-969.
    2. Etheridge, A.M. & Griffiths, R.C., 2009. "A coalescent dual process in a Moran model with genic selection," Theoretical Population Biology, Elsevier, vol. 75(4), pages 320-330.
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