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Stringer EJ, Gruber B, Sarre SD, Wardle GM, Edwards SV, Dickman CR, Greenville AC, Duncan RP. Boom-bust population dynamics drive rapid genetic change. Proc Natl Acad Sci U S A 2024; 121:e2320590121. [PMID: 38621118 PMCID: PMC11067018 DOI: 10.1073/pnas.2320590121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/22/2023] [Accepted: 03/06/2024] [Indexed: 04/17/2024] Open
Abstract
Increasing environmental threats and more extreme environmental perturbations place species at risk of population declines, with associated loss of genetic diversity and evolutionary potential. While theory shows that rapid population declines can cause loss of genetic diversity, populations in some environments, like Australia's arid zone, are repeatedly subject to major population fluctuations yet persist and appear able to maintain genetic diversity. Here, we use repeated population sampling over 13 y and genotype-by-sequencing of 1903 individuals to investigate the genetic consequences of repeated population fluctuations in two small mammals in the Australian arid zone. The sandy inland mouse (Pseudomys hermannsburgensis) experiences marked boom-bust population dynamics in response to the highly variable desert environment. We show that heterozygosity levels declined, and population differentiation (FST) increased, during bust periods when populations became small and isolated, but that heterozygosity was rapidly restored during episodic population booms. In contrast, the lesser hairy-footed dunnart (Sminthopsis youngsoni), a desert marsupial that maintains relatively stable population sizes, showed no linear declines in heterozygosity. These results reveal two contrasting ways in which genetic diversity is maintained in highly variable environments. In one species, diversity is conserved through the maintenance of stable population sizes across time. In the other species, diversity is conserved through rapid genetic mixing during population booms that restores heterozygosity lost during population busts.
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Affiliation(s)
- Emily J. Stringer
- Centre for Conservation Ecology and Genomics, Institute for Applied Ecology, University of Canberra, CanberraACT2617, Australia
| | - Bernd Gruber
- Centre for Conservation Ecology and Genomics, Institute for Applied Ecology, University of Canberra, CanberraACT2617, Australia
| | - Stephen D. Sarre
- Centre for Conservation Ecology and Genomics, Institute for Applied Ecology, University of Canberra, CanberraACT2617, Australia
| | - Glenda M. Wardle
- Desert Ecology Research Group, School of Life and Environmental Sciences, The University of Sydney, SydneyNSW2006, Australia
| | - Scott V. Edwards
- Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA02138
| | - Christopher R. Dickman
- Desert Ecology Research Group, School of Life and Environmental Sciences, The University of Sydney, SydneyNSW2006, Australia
| | - Aaron C. Greenville
- Desert Ecology Research Group, School of Life and Environmental Sciences, The University of Sydney, SydneyNSW2006, Australia
| | - Richard P. Duncan
- Centre for Conservation Ecology and Genomics, Institute for Applied Ecology, University of Canberra, CanberraACT2617, Australia
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2
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Hössjer O, Laikre L, Ryman N. Assessment of the Global Variance Effective Size of Subdivided Populations, and Its Relation to Other Effective Sizes. Acta Biotheor 2023; 71:19. [PMID: 37458852 PMCID: PMC10352448 DOI: 10.1007/s10441-023-09470-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/08/2021] [Accepted: 06/28/2023] [Indexed: 07/20/2023]
Abstract
The variance effective population size ([Formula: see text]) is frequently used to quantify the expected rate at which a population's allele frequencies change over time. The purpose of this paper is to find expressions for the global [Formula: see text] of a spatially structured population that are of interest for conservation of species. Since [Formula: see text] depends on allele frequency change, we start by dividing the cause of allele frequency change into genetic drift within subpopulations (I) and a second component mainly due to migration between subpopulations (II). We investigate in detail how these two components depend on the way in which subpopulations are weighted as well as their dependence on parameters of the model such a migration rates, and local effective and census sizes. It is shown that under certain conditions the impact of II is eliminated, and [Formula: see text] of the metapopulation is maximized, when subpopulations are weighted proportionally to their long term reproductive contributions. This maximal [Formula: see text] is the sought for global effective size, since it approximates the gene diversity effective size [Formula: see text], a quantifier of the rate of loss of genetic diversity that is relevant for conservation of species and populations. We also propose two novel versions of [Formula: see text], one of which (the backward version of [Formula: see text]) is most stable, exists for most populations, and is closer to [Formula: see text] than the classical notion of [Formula: see text]. Expressions for the optimal length of the time interval for measuring genetic change are developed, that make it possible to estimate any version of [Formula: see text] with maximal accuracy.
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Affiliation(s)
- Ola Hössjer
- Division of Mathematical Statistics, Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden
| | - Linda Laikre
- Division of Population Genetics, Department of Zoology, Stockholm University, 106 91 Stockholm, Sweden
| | - Nils Ryman
- Division of Population Genetics, Department of Zoology, Stockholm University, 106 91 Stockholm, Sweden
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3
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Kurland S, Ryman N, Hössjer O, Laikre L. Effects of subpopulation extinction on effective size (Ne) of metapopulations. CONSERV GENET 2023. [DOI: 10.1007/s10592-023-01510-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/29/2023]
Abstract
AbstractPopulation extinction is ubiquitous in all taxa. Such extirpations can reduce intraspecific diversity, but the extent to which genetic diversity of surviving populations are affected remains largely unclear. A key concept in this context is the effective population size (Ne), which quantifies the rate at which genetic diversity within populations is lost. Ne was developed for single, isolated populations while many natural populations are instead connected to other populations via gene flow. Recent analytical approaches and software permit modelling of Ne of interconnected populations (metapopulations). Here, we apply such tools to investigate how extinction of subpopulations affects Ne of the metapopulation (NeMeta) and of separate surviving subpopulations (NeRx) under different rates and patterns of genetic exchange between subpopulations. We assess extinction effects before and at migration-drift equilibrium. We find that the effect of extinction on NeMeta increases with reduced connectivity, suggesting that stepping stone models of migration are more impacted than island-migration models when the same number of subpopulations are lost. Furthermore, in stepping stone models, after extinction and before a new equilibrium has been reached, NeRx can vary drastically among surviving subpopulations and depends on their initial spatial position relative to extinct ones. Our results demonstrate that extinctions can have far more complex effects on the retention of intraspecific diversity than typically recognized. Metapopulation dynamics need heightened consideration in sustainable management and conservation, e.g., in monitoring genetic diversity, and are relevant to a wide range of species in the ongoing extinction crisis.
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Nadachowska‐Brzyska K, Konczal M, Babik W. Navigating the temporal continuum of effective population size. Methods Ecol Evol 2021. [DOI: 10.1111/2041-210x.13740] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022]
Affiliation(s)
| | | | - Wieslaw Babik
- Jagiellonian University in Kraków Faculty of Biology Institute of Environmental Sciences Kraków Poland
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5
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Ryman N, Laikre L, Hössjer O. Do estimates of contemporary effective population size tell us what we want to know? Mol Ecol 2019; 28:1904-1918. [PMID: 30663828 PMCID: PMC6850010 DOI: 10.1111/mec.15027] [Citation(s) in RCA: 33] [Impact Index Per Article: 6.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2018] [Revised: 01/14/2019] [Accepted: 01/15/2019] [Indexed: 12/25/2022]
Abstract
Estimation of effective population size (Ne) from genetic marker data is a major focus for biodiversity conservation because it is essential to know at what rates inbreeding is increasing and additive genetic variation is lost. But are these the rates assessed when applying commonly used Ne estimation techniques? Here we use recently developed analytical tools and demonstrate that in the case of substructured populations the answer is no. This is because the following: Genetic change can be quantified in several ways reflecting different types of Ne such as inbreeding (NeI), variance (NeV), additive genetic variance (NeAV), linkage disequilibrium equilibrium (NeLD), eigenvalue (NeE) and coalescence (NeCo) effective size. They are all the same for an isolated population of constant size, but the realized values of these effective sizes can differ dramatically in populations under migration. Commonly applied Ne‐estimators target NeV or NeLD of individual subpopulations. While such estimates are safe proxies for the rates of inbreeding and loss of additive genetic variation under isolation, we show that they are poor indicators of these rates in populations affected by migration. In fact, both the local and global inbreeding (NeI) and additive genetic variance (NeAV) effective sizes are consistently underestimated in a subdivided population. This is serious because these are the effective sizes that are relevant to the widely accepted 50/500 rule for short and long term genetic conservation. The bias can be infinitely large and is due to inappropriate parameters being estimated when applying theory for isolated populations to subdivided ones.
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Affiliation(s)
- Nils Ryman
- Department of Zoology, Division of Population Genetics, Stockholm University, Stockholm, Sweden
| | - Linda Laikre
- Department of Zoology, Division of Population Genetics, Stockholm University, Stockholm, Sweden
| | - Ola Hössjer
- Department of Mathematics, Stockholm University, Stockholm, Sweden
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6
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Kopatz A, Eiken HG, Schregel J, Aspi J, Kojola I, Hagen SB. Genetic substructure and admixture as important factors in linkage disequilibrium-based estimation of effective number of breeders in recovering wildlife populations. Ecol Evol 2017; 7:10721-10732. [PMID: 29299252 PMCID: PMC5743533 DOI: 10.1002/ece3.3577] [Citation(s) in RCA: 39] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/21/2017] [Revised: 10/03/2017] [Accepted: 10/04/2017] [Indexed: 01/18/2023] Open
Abstract
The number of effective breeders (Nb ) and effective population size (Ne ) are population parameters reflective of evolutionary potential, susceptibility to stochasticity, and viability. We have estimated these parameters using the linkage disequilibrium-based approach with LDNE through the latest phase of population recovery of the brown bears (Ursus arctos) in Finland (1993-2010; N = 621). This phase of the recovery was recently documented to be associated with major changes in genetic composition. In particular, differentiation between the northern and the southern genetic cluster declined rapidly within 1.5 generations. Based on this, we have studied effects of the changing genetic structure on Nb and Ne , by comparing estimates for whole Finland with the estimates for the two genetic clusters. We expected a potentially strong relationship between estimate sizes and genetic differentiation, which should disappear as the population recovers and clusters merge. Consistent with this, our estimates for whole Finland were lower than the sum of the estimates of the two genetic clusters and both approaches produced similar estimates in the end. Notably, we also found that admixed genotypes strongly increased the estimates. In all analyses, our estimates for Ne were larger than Nb and likely reflective for brown bears of the larger region of Finland and northwestern Russia. Conclusively, we find that neglecting genetic substructure may lead to a massive underestimation of Nb and Ne . Our results also suggest the need for further empirical analysis focusing on individuals with admixed genotypes and their potential high influence on Nb and Ne .
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Affiliation(s)
| | - Hans Geir Eiken
- NIBIO—Norwegian Institute of Bioeconomy ResearchSvanvikNorway
| | - Julia Schregel
- NIBIO—Norwegian Institute of Bioeconomy ResearchSvanvikNorway
| | - Jouni Aspi
- Department of BiologyUniversity of OuluOuluFinland
| | - Ilpo Kojola
- Natural Resources Institute Finland (Luke)RovaniemiFinland
| | - Snorre B. Hagen
- NIBIO—Norwegian Institute of Bioeconomy ResearchSvanvikNorway
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7
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Olsson F, Laikre L, Hössjer O, Ryman N. gesp: A computer program for modelling genetic effective population size, inbreeding and divergence in substructured populations. Mol Ecol Resour 2017; 17:1378-1384. [PMID: 28339169 PMCID: PMC5724513 DOI: 10.1111/1755-0998.12673] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/11/2016] [Revised: 03/05/2017] [Accepted: 03/07/2017] [Indexed: 12/01/2022]
Abstract
The genetically effective population size (Ne) is of key importance for quantifying rates of inbreeding and genetic drift and is often used in conservation management to set targets for genetic viability. The concept was developed for single, isolated populations and the mathematical means for analysing the expected Ne in complex, subdivided populations have previously not been available. We recently developed such analytical theory and central parts of that work have now been incorporated into a freely available software tool presented here. gesp (Genetic Effective population size, inbreeding and divergence in Substructured Populations) is R‐based and designed to model short‐ and long‐term patterns of genetic differentiation and effective population size of subdivided populations. The algorithms performed by gesp allow exact computation of global and local inbreeding and eigenvalue effective population size, predictions of genetic divergence among populations (GST) as well as departures from random mating (FIS, FIT) while varying (i) subpopulation census and effective size, separately or including trend of the global population size, (ii) rate and direction of migration between all pairs of subpopulations, (iii) degree of relatedness and divergence among subpopulations, (iv) ploidy (haploid or diploid) and (v) degree of selfing. Here, we describe gesp and exemplify its use in conservation genetics modelling.
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Affiliation(s)
- Fredrik Olsson
- Department of Mathematics, Division of Mathematical Statistics, Stockholm University, Stockholm, Sweden
| | - Linda Laikre
- Department of Zoology, Division of Population Genetics, Stockholm University, Stockholm, Sweden
| | - Ola Hössjer
- Department of Mathematics, Division of Mathematical Statistics, Stockholm University, Stockholm, Sweden
| | - Nils Ryman
- Department of Zoology, Division of Population Genetics, Stockholm University, Stockholm, Sweden
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8
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Hössjer O, Laikre L, Ryman N. Effective sizes and time to migration–drift equilibrium in geographically subdivided populations. Theor Popul Biol 2016; 112:139-156. [DOI: 10.1016/j.tpb.2016.09.001] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/07/2016] [Revised: 08/23/2016] [Accepted: 09/03/2016] [Indexed: 10/21/2022]
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Laikre L, Olsson F, Jansson E, Hössjer O, Ryman N. Metapopulation effective size and conservation genetic goals for the Fennoscandian wolf (Canis lupus) population. Heredity (Edinb) 2016; 117:279-89. [PMID: 27328654 PMCID: PMC5026756 DOI: 10.1038/hdy.2016.44] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/14/2015] [Revised: 04/26/2016] [Accepted: 05/08/2016] [Indexed: 11/17/2022] Open
Abstract
The Scandinavian wolf population descends from only five individuals, is isolated, highly inbred and exhibits inbreeding depression. To meet international conservation goals, suggestions include managing subdivided wolf populations over Fennoscandia as a metapopulation; a genetically effective population size of Ne⩾500, in line with the widely accepted long-term genetic viability target, might be attainable with gene flow among subpopulations of Scandinavia, Finland and Russian parts of Fennoscandia. Analytical means for modeling Ne of subdivided populations under such non-idealized situations have been missing, but we recently developed new mathematical methods for exploring inbreeding dynamics and effective population size of complex metapopulations. We apply this theory to the Fennoscandian wolves using empirical estimates of demographic parameters. We suggest that the long-term conservation genetic target for metapopulations should imply that inbreeding rates in the total system and in the separate subpopulations should not exceed Δf=0.001. This implies a meta-Ne of NeMeta⩾500 and a realized effective size of each subpopulation of NeRx⩾500. With current local effective population sizes and one migrant per generation, as recommended by management guidelines, the meta-Ne that can be reached is ~250. Unidirectional gene flow from Finland to Scandinavia reduces meta-Ne to ~130. Our results indicate that both local subpopulation effective sizes and migration among subpopulations must increase substantially from current levels to meet the conservation target. Alternatively, immigration from a large (Ne⩾500) population in northwestern Russia could support the Fennoscandian metapopulation, but immigration must be substantial (5–10 effective immigrants per generation) and migration among Fennoscandian subpopulations must nevertheless increase.
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Affiliation(s)
- L Laikre
- Department of Zoology, Division of Population Genetics, Stockholm University, Stockholm, Sweden
| | - F Olsson
- Department of Mathematics, Stockholm University, Stockholm, Sweden
| | - E Jansson
- Department of Zoology, Division of Population Genetics, Stockholm University, Stockholm, Sweden
| | - O Hössjer
- Department of Mathematics, Stockholm University, Stockholm, Sweden
| | - N Ryman
- Department of Zoology, Division of Population Genetics, Stockholm University, Stockholm, Sweden
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10
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Sundqvist L, Keenan K, Zackrisson M, Prodöhl P, Kleinhans D. Directional genetic differentiation and relative migration. Ecol Evol 2016; 6:3461-3475. [PMID: 27127613 PMCID: PMC4842207 DOI: 10.1002/ece3.2096] [Citation(s) in RCA: 151] [Impact Index Per Article: 18.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/18/2016] [Revised: 03/01/2016] [Accepted: 03/03/2016] [Indexed: 12/18/2022] Open
Abstract
Understanding the population structure and patterns of gene flow within species is of fundamental importance to the study of evolution. In the fields of population and evolutionary genetics, measures of genetic differentiation are commonly used to gather this information. One potential caveat is that these measures assume gene flow to be symmetric. However, asymmetric gene flow is common in nature, especially in systems driven by physical processes such as wind or water currents. As information about levels of asymmetric gene flow among populations is essential for the correct interpretation of the distribution of contemporary genetic diversity within species, this should not be overlooked. To obtain information on asymmetric migration patterns from genetic data, complex models based on maximum-likelihood or Bayesian approaches generally need to be employed, often at great computational cost. Here, a new simpler and more efficient approach for understanding gene flow patterns is presented. This approach allows the estimation of directional components of genetic divergence between pairs of populations at low computational effort, using any of the classical or modern measures of genetic differentiation. These directional measures of genetic differentiation can further be used to calculate directional relative migration and to detect asymmetries in gene flow patterns. This can be done in a user-friendly web application called divMigrate-online introduced in this study. Using simulated data sets with known gene flow regimes, we demonstrate that the method is capable of resolving complex migration patterns under a range of study designs.
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Affiliation(s)
- Lisa Sundqvist
- Department of Marine SciencesUniversity of GothenburgSE‐405 30GothenburgSweden
| | - Kevin Keenan
- School of Biological SciencesInstitute for Global Food SecurityQueen's University BelfastBelfastBT9 7BLUK
| | - Martin Zackrisson
- Department for Chemistry and Molecular BiologyUniversity of GothenburgSE‐405 30GothenburgSweden
| | - Paulo Prodöhl
- School of Biological SciencesInstitute for Global Food SecurityQueen's University BelfastBelfastBT9 7BLUK
| | - David Kleinhans
- ForWind Center for Wind Energy ResearchInstitute of PhysicsCarl von Ossietzky UniversityDE‐26129OldenburgGermany
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11
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Hössjer O, Olsson F, Laikre L, Ryman N. Metapopulation inbreeding dynamics, effective size and subpopulation differentiation--A general analytical approach for diploid organisms. Theor Popul Biol 2015; 102:40-59. [PMID: 25875853 DOI: 10.1016/j.tpb.2015.03.006] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/22/2014] [Revised: 03/18/2015] [Accepted: 03/23/2015] [Indexed: 11/26/2022]
Abstract
Motivated by problems in conservation biology we study genetic dynamics in structured populations of diploid organisms (monoecious or dioecious). Our analysis provides an analytical framework that unifies substantial parts of previous work in terms of exact identity by descent (IBD) and identity by state (IBS) recursions. We provide exact conditions under which two structured haploid and diploid populations are equivalent, and some sufficient conditions under which a dioecious diploid population can be treated as a monoecious diploid one. The IBD recursions are used for computing local and metapopulation inbreeding and coancestry effective population sizes and for predictions of several types of fixation indices over different time horizons.
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Affiliation(s)
- Ola Hössjer
- Department of Mathematics, Division of Mathematical Statistics, Stockholm University, SE 106 91 Stockholm, Sweden.
| | - Fredrik Olsson
- Department of Mathematics, Division of Mathematical Statistics, Stockholm University, SE 106 91 Stockholm, Sweden.
| | - Linda Laikre
- Department of Zoology, Division of Population Genetics, Stockholm University, SE 106 91 Stockholm, Sweden.
| | - Nils Ryman
- Department of Zoology, Division of Population Genetics, Stockholm University, SE 106 91 Stockholm, Sweden.
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12
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Estimation of the variance effective population size in age structured populations. Theor Popul Biol 2015; 101:9-23. [PMID: 25703299 DOI: 10.1016/j.tpb.2015.02.003] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/07/2015] [Revised: 01/29/2015] [Accepted: 02/10/2015] [Indexed: 11/20/2022]
Abstract
The variance effective population size for age structured populations is generally hard to estimate and the temporal method often gives biased estimates. Here, we give an explicit expression for a correction factor which, combined with estimates from the temporal method, yield approximately unbiased estimates. The calculation of the correction factor requires knowledge of the age specific offspring distribution and survival probabilities as well as possible correlation between survival and reproductive success. In order to relax these requirements, we show that only first order moments of these distributions need to be known if the time between samples is large, or individuals from all age classes which reproduce are sampled. A very explicit approximate expression for the asymptotic coefficient of standard deviation of the estimator is derived, and it can be used to construct confidence intervals and optimal ways of weighting information from different markers. The asymptotic coefficient of standard deviation can also be used to design studies and we show that in order to maximize the precision for a given sample size, individuals from older age classes should be sampled since their expected variance of allele frequency change is higher and easier to estimate. However, for populations with fluctuating age class sizes, the accuracy of the method is reduced when samples are taken from older age classes with high demographic variation. We also present a method for simultaneous estimation of the variance effective and census population size.
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13
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Hössjer O. On the eigenvalue effective size of structured populations. J Math Biol 2014; 71:595-646. [PMID: 25230676 PMCID: PMC4532751 DOI: 10.1007/s00285-014-0832-5] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2013] [Revised: 08/16/2014] [Indexed: 11/26/2022]
Abstract
A general theory is developed for the eigenvalue effective size (N(e)E) of structured populations in which a gene with two alleles segregates in discrete time. Generalizing results of Ewens (Theor Popul Biol 21:373-378, 1982), we characterize N(e)E in terms of the largest non-unit eigenvalue of the transition matrix of a Markov chain of allele frequencies. We use Perron-Frobenius Theorem to prove that the same eigenvalue appears in a linear recursion of predicted gene diversities between all pairs of subpopulations. Coalescence theory is employed in order to characterize this recursion, so that explicit novel expressions for N(e)E can be derived. We then study N(e)E asymptotically, when either the inverse size and/or the overall migration rate between subpopulations tend to zero. It is demonstrated that several previously known results can be deduced as special cases. In particular when the coalescence effective size N(e)C exists, it is an asymptotic version of N(e)E in the limit of large populations.
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Affiliation(s)
- Ola Hössjer
- Divsion of Mathematical Statistics, Department of Mathematics, Stockholm University, Stockholm, Sweden,
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