---
_id: '12787'
abstract:
- lang: eng
  text: "Populations evolve in spatially heterogeneous environments. While a certain
    trait might bring a fitness advantage in some patch of the environment, a different
    trait might be advantageous in another patch. Here, we study the Moran birth–death
    process with two types of individuals in a population stretched across two patches
    of size N, each patch favouring one of the two types. We show that the long-term
    fate of such populations crucially depends on the migration rate μ\r\n between
    the patches. To classify the possible fates, we use the distinction between polynomial
    (short) and exponential (long) timescales. We show that when μ is high then one
    of the two types fixates on the whole population after a number of steps that
    is only polynomial in N. By contrast, when μ is low then each type holds majority
    in the patch where it is favoured for a number of steps that is at least exponential
    in N. Moreover, we precisely identify the threshold migration rate μ⋆ that separates
    those two scenarios, thereby exactly delineating the situations that support long-term
    coexistence of the two types. We also discuss the case of various cycle graphs
    and we present computer simulations that perfectly match our analytical results."
acknowledgement: J.S. and K.C. acknowledge support from the ERC CoG 863818 (ForM-SMArt)
article_number: '20220685'
article_processing_charge: No
article_type: original
author:
- first_name: Jakub
  full_name: Svoboda, Jakub
  id: 130759D2-D7DD-11E9-87D2-DE0DE6697425
  last_name: Svoboda
  orcid: 0000-0002-1419-3267
- first_name: Josef
  full_name: Tkadlec, Josef
  id: 3F24CCC8-F248-11E8-B48F-1D18A9856A87
  last_name: Tkadlec
  orcid: 0000-0002-1097-9684
- first_name: Kamran
  full_name: Kaveh, Kamran
  last_name: Kaveh
- first_name: Krishnendu
  full_name: Chatterjee, Krishnendu
  id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87
  last_name: Chatterjee
  orcid: 0000-0002-4561-241X
citation:
  ama: 'Svoboda J, Tkadlec J, Kaveh K, Chatterjee K. Coexistence times in the Moran
    process with environmental heterogeneity. <i>Proceedings of the Royal Society
    A: Mathematical, Physical and Engineering Sciences</i>. 2023;479(2271). doi:<a
    href="https://doi.org/10.1098/rspa.2022.0685">10.1098/rspa.2022.0685</a>'
  apa: 'Svoboda, J., Tkadlec, J., Kaveh, K., &#38; Chatterjee, K. (2023). Coexistence
    times in the Moran process with environmental heterogeneity. <i>Proceedings of
    the Royal Society A: Mathematical, Physical and Engineering Sciences</i>. The
    Royal Society. <a href="https://doi.org/10.1098/rspa.2022.0685">https://doi.org/10.1098/rspa.2022.0685</a>'
  chicago: 'Svoboda, Jakub, Josef Tkadlec, Kamran Kaveh, and Krishnendu Chatterjee.
    “Coexistence Times in the Moran Process with Environmental Heterogeneity.” <i>Proceedings
    of the Royal Society A: Mathematical, Physical and Engineering Sciences</i>. The
    Royal Society, 2023. <a href="https://doi.org/10.1098/rspa.2022.0685">https://doi.org/10.1098/rspa.2022.0685</a>.'
  ieee: 'J. Svoboda, J. Tkadlec, K. Kaveh, and K. Chatterjee, “Coexistence times in
    the Moran process with environmental heterogeneity,” <i>Proceedings of the Royal
    Society A: Mathematical, Physical and Engineering Sciences</i>, vol. 479, no.
    2271. The Royal Society, 2023.'
  ista: 'Svoboda J, Tkadlec J, Kaveh K, Chatterjee K. 2023. Coexistence times in the
    Moran process with environmental heterogeneity. Proceedings of the Royal Society
    A: Mathematical, Physical and Engineering Sciences. 479(2271), 20220685.'
  mla: 'Svoboda, Jakub, et al. “Coexistence Times in the Moran Process with Environmental
    Heterogeneity.” <i>Proceedings of the Royal Society A: Mathematical, Physical
    and Engineering Sciences</i>, vol. 479, no. 2271, 20220685, The Royal Society,
    2023, doi:<a href="https://doi.org/10.1098/rspa.2022.0685">10.1098/rspa.2022.0685</a>.'
  short: 'J. Svoboda, J. Tkadlec, K. Kaveh, K. Chatterjee, Proceedings of the Royal
    Society A: Mathematical, Physical and Engineering Sciences 479 (2023).'
date_created: 2023-04-02T22:01:09Z
date_published: 2023-03-29T00:00:00Z
date_updated: 2025-07-14T09:09:51Z
day: '29'
ddc:
- '000'
department:
- _id: KrCh
doi: 10.1098/rspa.2022.0685
ec_funded: 1
external_id:
  isi:
  - '000957125500002'
file:
- access_level: open_access
  checksum: 13953d349fbefcb5d21ccc6b303297eb
  content_type: application/pdf
  creator: dernst
  date_created: 2023-04-03T06:25:29Z
  date_updated: 2023-04-03T06:25:29Z
  file_id: '12796'
  file_name: 2023_ProceedingsRoyalSocietyA_Svoboda.pdf
  file_size: 827784
  relation: main_file
  success: 1
file_date_updated: 2023-04-03T06:25:29Z
has_accepted_license: '1'
intvolume: '       479'
isi: 1
issue: '2271'
language:
- iso: eng
month: '03'
oa: 1
oa_version: Published Version
project:
- _id: 0599E47C-7A3F-11EA-A408-12923DDC885E
  call_identifier: H2020
  grant_number: '863818'
  name: 'Formal Methods for Stochastic Models: Algorithms and Applications'
publication: 'Proceedings of the Royal Society A: Mathematical, Physical and Engineering
  Sciences'
publication_identifier:
  eissn:
  - 1471-2946
  issn:
  - 1364-5021
publication_status: published
publisher: The Royal Society
quality_controlled: '1'
related_material:
  link:
  - relation: research_data
    url: https://doi.org/10.6084/m9.figshare.21261771.v1
scopus_import: '1'
status: public
title: Coexistence times in the Moran process with environmental heterogeneity
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 479
year: '2023'
...