---
_id: '4263'
abstract:
- lang: eng
  text: 'We introduce a general recursion for the probability of identity in state
    of two individuals sampled from a population subject to mutation, migration, and
    random drift in a two-dimensional continuum. The recursion allows for the interactions
    induced by density-dependent regulation of the population, which are inevitable
    in a continuous population. We give explicit series expansions for large neighbourhood
    size and for low mutation rates respectively and investigate the accuracy of the
    classical Malécot formula for these general models. When neighbourhood size is
    small, this formula does not give the identity even over large scales. However,
    for large neighbourhood size, it is an accurate approximation which summarises
    the local population structure in terms of three quantities: the effective dispersal
    rate, σe; the effective population density, ρe; and a local scale, κ, at which
    local interactions become significant. The results are illustrated by simulations.'
acknowledgement: This work was supported by grants from the EPSRC (GR/L10048 and an
  advanced fellowship for A.M.E.) and NERC (GR3/11635) and by the Darwin Trust of
  Edinburgh. We thank Anja Sturm for her assistance with the project and anonymous
  reviewers for helpful comments. This paper is dedicated to Charlotte, A.M.E.’s daughter
  born during the gestation of the manuscript.
article_processing_charge: No
article_type: original
author:
- first_name: Nicholas H
  full_name: Barton, Nicholas H
  id: 4880FE40-F248-11E8-B48F-1D18A9856A87
  last_name: Barton
  orcid: 0000-0002-8548-5240
- first_name: Frantz
  full_name: Depaulis, Frantz
  last_name: Depaulis
- first_name: Alison
  full_name: Etheridge, Alison
  last_name: Etheridge
citation:
  ama: Barton NH, Depaulis F, Etheridge A. Neutral evolution in spatially continuous
    populations. <i>Theoretical Population Biology</i>. 2002;61(1):31-48. doi:<a href="https://doi.org/10.1006/tpbi.2001.1557">10.1006/tpbi.2001.1557</a>
  apa: Barton, N. H., Depaulis, F., &#38; Etheridge, A. (2002). Neutral evolution
    in spatially continuous populations. <i>Theoretical Population Biology</i>. Academic
    Press. <a href="https://doi.org/10.1006/tpbi.2001.1557">https://doi.org/10.1006/tpbi.2001.1557</a>
  chicago: Barton, Nicholas H, Frantz Depaulis, and Alison Etheridge. “Neutral Evolution
    in Spatially Continuous Populations.” <i>Theoretical Population Biology</i>. Academic
    Press, 2002. <a href="https://doi.org/10.1006/tpbi.2001.1557">https://doi.org/10.1006/tpbi.2001.1557</a>.
  ieee: N. H. Barton, F. Depaulis, and A. Etheridge, “Neutral evolution in spatially
    continuous populations,” <i>Theoretical Population Biology</i>, vol. 61, no. 1.
    Academic Press, pp. 31–48, 2002.
  ista: Barton NH, Depaulis F, Etheridge A. 2002. Neutral evolution in spatially continuous
    populations. Theoretical Population Biology. 61(1), 31–48.
  mla: Barton, Nicholas H., et al. “Neutral Evolution in Spatially Continuous Populations.”
    <i>Theoretical Population Biology</i>, vol. 61, no. 1, Academic Press, 2002, pp.
    31–48, doi:<a href="https://doi.org/10.1006/tpbi.2001.1557">10.1006/tpbi.2001.1557</a>.
  short: N.H. Barton, F. Depaulis, A. Etheridge, Theoretical Population Biology 61
    (2002) 31–48.
date_created: 2018-12-11T12:07:55Z
date_published: 2002-02-01T00:00:00Z
date_updated: 2023-06-06T09:57:49Z
day: '01'
doi: 10.1006/tpbi.2001.1557
extern: '1'
external_id:
  pmid:
  - '11895381'
intvolume: '        61'
issue: '1'
language:
- iso: eng
month: '02'
oa_version: None
page: 31 - 48
pmid: 1
publication: Theoretical Population Biology
publication_identifier:
  issn:
  - 0040-5809
publication_status: published
publisher: Academic Press
publist_id: '1830'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Neutral evolution in spatially continuous populations
type: journal_article
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 61
year: '2002'
...
---
_id: '4272'
abstract:
- lang: eng
  text: 'Analysis of multilocus evolution is usually intractable for more than n ~
    10 genes, because the frequencies of very large numbers of genotypes must be followed.
    An exact analysis of up to n ~ 100 loci is feasible for a symmetrical model, in
    which a set of unlinked loci segregate for two alleles (labeled ''0'' and ''1'')
    with interchangeable effects on fitness. All haploid genotypes with the same number
    of 1 alleles can then remain equally frequent. However, such a symmetrical solution
    may be unstable: for example, under stabilizing selection, populations tend to
    fix any one genotype which approaches the optimum. Here, we show how the 2'' x
    2'' stability matrix can be decomposed into a set of matrices, each no larger
    than n x n. This allows the stability of symmetrical solutions to be determined.
    We apply the method to stabilizing and disruptive selection in a single deme and
    to selection against heterozygotes in a linear cline. (C) 2000 Academic Press.'
article_processing_charge: No
article_type: original
author:
- first_name: Nicholas H
  full_name: Barton, Nicholas H
  id: 4880FE40-F248-11E8-B48F-1D18A9856A87
  last_name: Barton
  orcid: 0000-0002-8548-5240
- first_name: Max
  full_name: Shpak, Max
  last_name: Shpak
citation:
  ama: Barton NH, Shpak M. The stability of symmetrical solutions to polygenic models.
    <i>Theoretical Population Biology</i>. 2000;57(3):249-263. doi:<a href="https://doi.org/10.1006/tpbi.2000.1455">10.1006/tpbi.2000.1455</a>
  apa: Barton, N. H., &#38; Shpak, M. (2000). The stability of symmetrical solutions
    to polygenic models. <i>Theoretical Population Biology</i>. Academic Press. <a
    href="https://doi.org/10.1006/tpbi.2000.1455">https://doi.org/10.1006/tpbi.2000.1455</a>
  chicago: Barton, Nicholas H, and Max Shpak. “The Stability of Symmetrical Solutions
    to Polygenic Models.” <i>Theoretical Population Biology</i>. Academic Press, 2000.
    <a href="https://doi.org/10.1006/tpbi.2000.1455">https://doi.org/10.1006/tpbi.2000.1455</a>.
  ieee: N. H. Barton and M. Shpak, “The stability of symmetrical solutions to polygenic
    models,” <i>Theoretical Population Biology</i>, vol. 57, no. 3. Academic Press,
    pp. 249–263, 2000.
  ista: Barton NH, Shpak M. 2000. The stability of symmetrical solutions to polygenic
    models. Theoretical Population Biology. 57(3), 249–263.
  mla: Barton, Nicholas H., and Max Shpak. “The Stability of Symmetrical Solutions
    to Polygenic Models.” <i>Theoretical Population Biology</i>, vol. 57, no. 3, Academic
    Press, 2000, pp. 249–63, doi:<a href="https://doi.org/10.1006/tpbi.2000.1455">10.1006/tpbi.2000.1455</a>.
  short: N.H. Barton, M. Shpak, Theoretical Population Biology 57 (2000) 249–263.
date_created: 2018-12-11T12:07:58Z
date_published: 2000-05-01T00:00:00Z
date_updated: 2023-04-19T12:36:39Z
day: '01'
doi: 10.1006/tpbi.2000.1455
extern: '1'
external_id:
  pmid:
  - '10828217'
intvolume: '        57'
issue: '3'
language:
- iso: eng
month: '05'
oa_version: None
page: 249 - 263
pmid: 1
publication: Theoretical Population Biology
publication_identifier:
  issn:
  - 0040-5809
publication_status: published
publisher: Academic Press
publist_id: '1820'
quality_controlled: '1'
scopus_import: '1'
status: public
title: The stability of symmetrical solutions to polygenic models
type: journal_article
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 57
year: '2000'
...
---
_id: '3649'
abstract:
- lang: eng
  text: Selection on polygenic characters is generally analyzed by statistical methods
    that assume a Gaussian (normal) distribution of breeding values. We present an
    alternative analysis based on multilocus population genetics. We use a general
    representation of selection, recombination, and drift to analyze an idealized
    polygenic system in which all genetic effects are additive (i.e., both dominance
    and epistasis are absent), but no assumptions are made about the distribution
    of breeding values or the numbers of loci or alleles. Our analysis produces three
    results. First, our equations reproduce the standard recursions for the mean and
    additive variance if breeding values are Gaussian; but they also reveal how non-Gaussian
    distributions of breeding values will alter these dynamics. Second, an approximation
    valid for weak selection shows that even if genetic variance is attributable to
    an effectively infinite number of loci with only additive effects, selection will
    generally drive the distribution of breeding values away from a Gaussian distribution
    by creating multilocus linkage disequilibria. Long-term dynamics of means can
    depart substantially from the predictions of the standard selection recursions,
    but the discrepancy may often be negligible for short-term selection. Third, by
    including mutation, we show that, for realistic parameter values, linkage disequilibrium
    has little effect on the amount of additive variance maintained at an equilibrium
    between stabilizing selection and mutation. Each of these analytical results is
    supported by numerical calculations.
acknowledgement: 'We thank R. Burger, J. A. Coyne, W. G. Hill, A. A. Hoffmann, J.
  H. Gillespie, M. Slatkin, T. Nagylaki and Z.-B. Zeng for helpful discussions and
  comments on earlier drafts. Our research is supported by grants from the National
  Science Foundation (BSR-8866548), the Science and Engineering Research Council,
  and the Institute of Theoretical Dynamics at UCD. '
article_processing_charge: No
article_type: original
author:
- first_name: Michael
  full_name: Turelli, Michael
  last_name: Turelli
- first_name: Nicholas H
  full_name: Barton, Nicholas H
  id: 4880FE40-F248-11E8-B48F-1D18A9856A87
  last_name: Barton
  orcid: 0000-0002-8548-5240
citation:
  ama: Turelli M, Barton NH. Dynamics of polygenic characters under selection. <i>Theoretical
    Population Biology</i>. 1990;38(1):1-57. doi:<a href="https://doi.org/10.1016/0040-5809(90)90002-D">10.1016/0040-5809(90)90002-D</a>
  apa: Turelli, M., &#38; Barton, N. H. (1990). Dynamics of polygenic characters under
    selection. <i>Theoretical Population Biology</i>. Academic Press. <a href="https://doi.org/10.1016/0040-5809(90)90002-D">https://doi.org/10.1016/0040-5809(90)90002-D</a>
  chicago: Turelli, Michael, and Nicholas H Barton. “Dynamics of Polygenic Characters
    under Selection.” <i>Theoretical Population Biology</i>. Academic Press, 1990.
    <a href="https://doi.org/10.1016/0040-5809(90)90002-D">https://doi.org/10.1016/0040-5809(90)90002-D</a>.
  ieee: M. Turelli and N. H. Barton, “Dynamics of polygenic characters under selection,”
    <i>Theoretical Population Biology</i>, vol. 38, no. 1. Academic Press, pp. 1–57,
    1990.
  ista: Turelli M, Barton NH. 1990. Dynamics of polygenic characters under selection.
    Theoretical Population Biology. 38(1), 1–57.
  mla: Turelli, Michael, and Nicholas H. Barton. “Dynamics of Polygenic Characters
    under Selection.” <i>Theoretical Population Biology</i>, vol. 38, no. 1, Academic
    Press, 1990, pp. 1–57, doi:<a href="https://doi.org/10.1016/0040-5809(90)90002-D">10.1016/0040-5809(90)90002-D</a>.
  short: M. Turelli, N.H. Barton, Theoretical Population Biology 38 (1990) 1–57.
date_created: 2018-12-11T12:04:26Z
date_published: 1990-01-01T00:00:00Z
date_updated: 2022-02-23T14:48:49Z
day: '01'
doi: 10.1016/0040-5809(90)90002-D
extern: '1'
intvolume: '        38'
issue: '1'
language:
- iso: eng
main_file_link:
- url: https://www.sciencedirect.com/science/article/pii/004058099090002D?via%3Dihub
month: '01'
oa_version: None
page: 1 - 57
publication: Theoretical Population Biology
publication_identifier:
  issn:
  - 0040-5809
publication_status: published
publisher: Academic Press
publist_id: '2734'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Dynamics of polygenic characters under selection
type: journal_article
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 38
year: '1990'
...
---
_id: '3657'
abstract:
- lang: eng
  text: Shifts between adaptive peaks, caused by sampling drift, are involved in both
    speciation and adaptation via Wright's “shiftingbalance.” We use techniques from
    statistical mechanics to calculate the rate of such transitions for apopulation
    in a single panmictic deme and for apopulation which is continuously distributed
    over one- and two-dimensional regions. This calculation applies in the limit where
    transitions are rare. Our results indicate that stochastic divergence is feasible
    despite free gene flow, provided that neighbourhood size is low enough. In two
    dimensions, the rate of transition depends primarily on neighbourhood size N and
    only weakly on selection pressure (≈sk exp(− cN)), where k is a number determined
    by the local population structure, in contrast with the exponential dependence
    on selection pressure in one dimension (≈exp(− cN √s)) or in a single deme (≈exp(−
    cNs)). Our calculations agree with simulations of a single deme and a one-dimensional
    population.
acknowledgement: "We thank M. Shaw, J. Felsenstein, M. Kirkpatrick, S. Via, J. S.
  Jones, M. Slatkin, J. Mallet, and B. Charlesworth for their helpful comments. This
  work was supported by grants from the SERC (GR/C/91529), the University of London
  Central Research Fund, and the Nufield Foundation. \r\n"
article_processing_charge: No
article_type: original
author:
- first_name: Shahin
  full_name: Rouhani, Shahin
  last_name: Rouhani
- first_name: Nicholas H
  full_name: Barton, Nicholas H
  id: 4880FE40-F248-11E8-B48F-1D18A9856A87
  last_name: Barton
  orcid: 0000-0002-8548-5240
citation:
  ama: Rouhani S, Barton NH. Speciation and the &#38;quot;shifting balance&#38;quot;
    in a continuous population. <i>Theoretical Population Biology</i>. 1987;31(3):465-492.
    doi:<a href="https://doi.org/10.1016/0040-5809(87)90016-5">10.1016/0040-5809(87)90016-5</a>
  apa: Rouhani, S., &#38; Barton, N. H. (1987). Speciation and the &#38;quot;shifting
    balance&#38;quot; in a continuous population. <i>Theoretical Population Biology</i>.
    Elsevier. <a href="https://doi.org/10.1016/0040-5809(87)90016-5">https://doi.org/10.1016/0040-5809(87)90016-5</a>
  chicago: Rouhani, Shahin, and Nicholas H Barton. “Speciation and the &#38;quot;Shifting
    Balance&#38;quot; in a Continuous Population.” <i>Theoretical Population Biology</i>.
    Elsevier, 1987. <a href="https://doi.org/10.1016/0040-5809(87)90016-5">https://doi.org/10.1016/0040-5809(87)90016-5</a>.
  ieee: S. Rouhani and N. H. Barton, “Speciation and the &#38;quot;shifting balance&#38;quot;
    in a continuous population,” <i>Theoretical Population Biology</i>, vol. 31, no.
    3. Elsevier, pp. 465–492, 1987.
  ista: Rouhani S, Barton NH. 1987. Speciation and the &#38;quot;shifting balance&#38;quot;
    in a continuous population. Theoretical Population Biology. 31(3), 465–492.
  mla: Rouhani, Shahin, and Nicholas H. Barton. “Speciation and the &#38;quot;Shifting
    Balance&#38;quot; in a Continuous Population.” <i>Theoretical Population Biology</i>,
    vol. 31, no. 3, Elsevier, 1987, pp. 465–92, doi:<a href="https://doi.org/10.1016/0040-5809(87)90016-5">10.1016/0040-5809(87)90016-5</a>.
  short: S. Rouhani, N.H. Barton, Theoretical Population Biology 31 (1987) 465–492.
date_created: 2018-12-11T12:04:28Z
date_published: 1987-06-01T00:00:00Z
date_updated: 2022-02-04T12:30:10Z
day: '01'
doi: 10.1016/0040-5809(87)90016-5
extern: '1'
intvolume: '        31'
issue: '3'
language:
- iso: eng
main_file_link:
- url: https://www.sciencedirect.com/science/article/pii/0040580987900165?via%3Dihub
month: '06'
oa_version: None
page: 465 - 492
publication: Theoretical Population Biology
publication_identifier:
  eissn:
  - 1096-0325
  issn:
  - 0040-5809
publication_status: published
publisher: Elsevier
publist_id: '2726'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Speciation and the &quot;shifting balance&quot; in a continuous population
type: journal_article
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 31
year: '1987'
...
---
_id: '3662'
abstract:
- lang: eng
  text: The evolution of the probabilities of genetic identity within and between
    tandemly repeated loci of a multigene family is investigated analytically and
    numerically. Unbiased intrachromosomal gene conversion, equal crossing over, random
    genetic drift, and mutation to new alleles are incorporated. Generations are discrete
    and nonoverlapping; the diploid, monoecious population mates at random. Under
    the restriction that there is at most one crossover in the multigene family per
    individual per generation, the dependence on location of the probabilities of
    identity is treated exactly. In the “homogeneous” approximation to this “exact”
    model, end effects are disregarded; in the “exchangeable” approximation, to which
    all previous work was confined, all position dependence is neglected. Numerical
    results indicate that (i) the exchangeable and homogeneous models are both qualitatively
    correct, (ii) the exchangeable model is sometimes too inaccurate for quantitative
    conclusions, and (iii) the homogeneous model is always more accurate than the
    exchangeable one and is always sufficiently accurate for quantitative conclusions.
acknowledgement: Supported by National Science Foundation Grant DEB81-03530
article_processing_charge: No
article_type: original
author:
- first_name: Thomas
  full_name: Nagylaki, Thomas
  last_name: Nagylaki
- first_name: Nicholas H
  full_name: Barton, Nicholas H
  id: 4880FE40-F248-11E8-B48F-1D18A9856A87
  last_name: Barton
  orcid: 0000-0002-8548-5240
citation:
  ama: Nagylaki T, Barton NH. Intrachromosomal gene conversion, linkage, and the evolution
    of multigene families. <i>Theoretical Population Biology</i>. 1986;29(3):407-437.
    doi:<a href="https://doi.org/10.1016/0040-5809(86)90017-1">10.1016/0040-5809(86)90017-1</a>
  apa: Nagylaki, T., &#38; Barton, N. H. (1986). Intrachromosomal gene conversion,
    linkage, and the evolution of multigene families. <i>Theoretical Population Biology</i>.
    Academic Press. <a href="https://doi.org/10.1016/0040-5809(86)90017-1">https://doi.org/10.1016/0040-5809(86)90017-1</a>
  chicago: Nagylaki, Thomas, and Nicholas H Barton. “Intrachromosomal Gene Conversion,
    Linkage, and the Evolution of Multigene Families.” <i>Theoretical Population Biology</i>.
    Academic Press, 1986. <a href="https://doi.org/10.1016/0040-5809(86)90017-1">https://doi.org/10.1016/0040-5809(86)90017-1</a>.
  ieee: T. Nagylaki and N. H. Barton, “Intrachromosomal gene conversion, linkage,
    and the evolution of multigene families,” <i>Theoretical Population Biology</i>,
    vol. 29, no. 3. Academic Press, pp. 407–437, 1986.
  ista: Nagylaki T, Barton NH. 1986. Intrachromosomal gene conversion, linkage, and
    the evolution of multigene families. Theoretical Population Biology. 29(3), 407–437.
  mla: Nagylaki, Thomas, and Nicholas H. Barton. “Intrachromosomal Gene Conversion,
    Linkage, and the Evolution of Multigene Families.” <i>Theoretical Population Biology</i>,
    vol. 29, no. 3, Academic Press, 1986, pp. 407–37, doi:<a href="https://doi.org/10.1016/0040-5809(86)90017-1">10.1016/0040-5809(86)90017-1</a>.
  short: T. Nagylaki, N.H. Barton, Theoretical Population Biology 29 (1986) 407–437.
date_created: 2018-12-11T12:04:30Z
date_published: 1986-06-01T00:00:00Z
date_updated: 2022-02-01T15:50:10Z
day: '01'
doi: 10.1016/0040-5809(86)90017-1
extern: '1'
intvolume: '        29'
issue: '3'
language:
- iso: eng
month: '06'
oa_version: None
page: 407 - 437
publication: Theoretical Population Biology
publication_identifier:
  eissn:
  - 1096-0325
  issn:
  - 0040-5809
publication_status: published
publisher: Academic Press
publist_id: '2721'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Intrachromosomal gene conversion, linkage, and the evolution of multigene families
type: journal_article
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 29
year: '1986'
...