IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v35y2022i3d10.1007_s10959-021-01119-z.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-021-01119-z
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-021-01119-z?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Ezoe, Ayana & Morimoto, Saori & Tanaka, Yuya & Katori, Makoto & Nishimori, Hiraku, 2024. "Switching particle systems for foraging ants showing phase transitions in path selections," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 643(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kon Kam King, Guillaume & Pandolfi, Andrea & Piretto, Marco & Ruggiero, Matteo, 2024. "Approximate filtering via discrete dual processes," Stochastic Processes and their Applications, Elsevier, vol. 168(C).
    2. Mikula, Lynette Caitlin & Vogl, Claus, 2024. "The expected sample allele frequencies from populations of changing size via orthogonal polynomials," Theoretical Population Biology, Elsevier, vol. 157(C), pages 55-85.
    3. Vogl, Claus & Bergman, Juraj, 2015. "Inference of directional selection and mutation parameters assuming equilibrium," Theoretical Population Biology, Elsevier, vol. 106(C), pages 71-82.
    4. Vogl, Claus & Clemente, Florian, 2012. "The allele-frequency spectrum in a decoupled Moran model with mutation, drift, and directional selection, assuming small mutation rates," Theoretical Population Biology, Elsevier, vol. 81(3), pages 197-209.
    5. Desai, Michael M. & Nicolaisen, Lauren E. & Walczak, Aleksandra M. & Plotkin, Joshua B., 2012. "The structure of allelic diversity in the presence of purifying selection," Theoretical Population Biology, Elsevier, vol. 81(2), pages 144-157.
    6. Burden, Conrad J. & Tang, Yurong, 2017. "Rate matrix estimation from site frequency data," Theoretical Population Biology, Elsevier, vol. 113(C), pages 23-33.
    7. Malaguti, Giulia & Singh, Param Priya & Isambert, Hervé, 2014. "On the retention of gene duplicates prone to dominant deleterious mutations," Theoretical Population Biology, Elsevier, vol. 93(C), pages 38-51.
    8. Anja Sturm & Jan M. Swart, 2018. "Pathwise Duals of Monotone and Additive Markov Processes," Journal of Theoretical Probability, Springer, vol. 31(2), pages 932-983, June.
    9. Corujo, Josué, 2021. "Dynamics of a Fleming–Viot type particle system on the cycle graph," Stochastic Processes and their Applications, Elsevier, vol. 136(C), pages 57-91.
    10. Der, Ricky & Epstein, Charles L. & Plotkin, Joshua B., 2011. "Generalized population models and the nature of genetic drift," Theoretical Population Biology, Elsevier, vol. 80(2), pages 80-99.
    11. Belitsky, V. & Schütz, G.M., 2018. "Self-duality and shock dynamics in the n-species priority ASEP," Stochastic Processes and their Applications, Elsevier, vol. 128(4), pages 1165-1207.
    12. Griffiths, Robert C. & Jenkins, Paul A. & Lessard, Sabin, 2016. "A coalescent dual process for a Wright–Fisher diffusion with recombination and its application to haplotype partitioning," Theoretical Population Biology, Elsevier, vol. 112(C), pages 126-138.
    13. Griffiths, Robert C. & Jenkins, Paul A. & Spanò, Dario, 2018. "Wright–Fisher diffusion bridges," Theoretical Population Biology, Elsevier, vol. 122(C), pages 67-77.
    14. Etheridge, Alison M. & Griffiths, Robert C. & Taylor, Jesse E., 2010. "A coalescent dual process in a Moran model with genic selection, and the lambda coalescent limit," Theoretical Population Biology, Elsevier, vol. 78(2), pages 77-92.
    15. Jan Niklas Latz & Jan M. Swart, 2023. "Commutative Monoid Duality," Journal of Theoretical Probability, Springer, vol. 36(2), pages 1088-1115, June.
    16. Lenz, Ute & Kluth, Sandra & Baake, Ellen & Wakolbinger, Anton, 2015. "Looking down in the ancestral selection graph: A probabilistic approach to the common ancestor type distribution," Theoretical Population Biology, Elsevier, vol. 103(C), pages 27-37.
    17. Vogl, Claus & Mikula, Lynette C. & Burden, Conrad J., 2020. "Maximum likelihood estimators for scaled mutation rates in an equilibrium mutation–drift model," Theoretical Population Biology, Elsevier, vol. 134(C), pages 106-118.
    18. Jüri Lember & Chris Watkins, 2022. "An Evolutionary Model that Satisfies Detailed Balance," Methodology and Computing in Applied Probability, Springer, vol. 24(1), pages 1-37, March.
    19. Möhle, Martin, 2024. "On multi-type Cannings models and multi-type exchangeable coalescents," Theoretical Population Biology, Elsevier, vol. 156(C), pages 103-116.
    20. Vogl, Claus & Mikula, Lynette Caitlin, 2021. "A nearly-neutral biallelic Moran model with biased mutation and linear and quadratic selection," Theoretical Population Biology, Elsevier, vol. 139(C), pages 1-17.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:35:y:2022:i:3:d:10.1007_s10959-021-01119-z. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.