The effective number of a population that varies cyclically in size. II. Overlapping generations

We consider haploid and dioecious age-structured populations that vary over time in cycles of length k. Results are obtained for both autosomal and sex-linked loci if the population is dioecious. It is assumed that k is small in comparison with numbers of haploid individuals (or of numbers of males and females) in any generation of a cycle. The inbreeding effective population size N(e) is then approximately given by the expression [T summation operator (k-1)(j=0)1/[N(e)(j)T(j)]](-1), where N(e)(j) and T(j) are, respectively, the effective population size and generation interval that would hold if the population was at all times generated in the same way as at time j. The constant T, which is the effective overall generation interval, is defined to be k times the harmonic mean of the quantities T(j). Our expressions for T and N(e), in terms of N(e)(j) and T(j), are general, but the N(e)(j)s are derived under the assumption that offspring are produced according to Poisson distributions.     

Coalescent process with fluctuating population size and its effective size

We consider a Wright-Fisher model whose population size is a finite Markov chain. We introduce a sequence of two-dimensional discrete time Markov chains whose components describe the coalescent process and the fluctuation of population size. For the limiting process of the sequence of Markov chains, the relationship of the expectation of coalescence time to the harmonic and the arithmetic means of population sizes is shown, and the Laplace transform of the distribution of coalescence time is calculated. We define the coalescence effective population size (cEPS) by the expectation of coalescence time. We show that cEPS is strictly larger (resp. smaller) than the harmonic (resp. arithmetic) mean. As the population size fluctuates more quickly (resp. slowly), cEPS is closer to the harmonic (resp. arithmetic) mean. For the case of a two-valued Markov chain, we show the explicit expression of cEPS and its dependency on the sample size.     

Implications of the variance effective population size on the genetic conservation of monoecious species

The concept of variance effective population size [Ne_(v)] and other expressions are reviewed and described for specific sampling steps in germplasm collection and regeneration of monoecious species. Special attention is given to procedures for computing the variance of the number of contributed gametes [V(k)] to the next generation. Drift, as it occurs between generations, was considered to contain a component due to the sampling of parents and a subsequent component due to the sampling of gametes. This demonstrates that drift, caused by reduction of seed viability, damages the genetic integrity of accessions stored in germplasm banks. The study shows how mating designs, such as plant-to-plant or chain crossings with additional female gametic control, can partially alleviate this problem. Optimal procedures for increasing Ne_(v) when collecting germplasm in the field are also discussed. The effect of different female and male gametic control strategies on Ne_(v) is considered under several situations. Practical examples illustrating the use of V(k) and Ne_(v) expressions are given.     

The influence of persistent individual differences and age at maturity on effective population size

Ratios of effective populations size, N(e), to census population size, N, are used as a measure of genetic drift in populations. Several life-history parameters have been shown to affect these ratios, including mating system and age at sexual maturation. Using a stochastic matrix model, we examine how different levels of persistent individual differences in mating success among males may affect N(e)/N, and how this relates to generation time. Individual differences of this type are shown to cause a lower N(e)/N ratio than would be expected when mating is independent among seasons. Examining the way in which age at maturity affects N(e)/N, we find that both the direction and magnitude of the effect depends on the survival rate of juveniles in the population. In particular, when maturation is delayed, lowered juvenile survival causes higher levels of genetic drift. In addition, predicted shifts in N(e)/N with changing age at maturity are shown to be dependent on which of the commonly used definitions of census population size, N, is employed. Our results demonstrate that patterns of mating success, as well as juvenile survival probabilities, have substantial effects on rates of genetic drift.     

Allele frequency covariance among cohorts and its use in estimating effective size of age-structured populations

A general expression for the covariance of allele frequencies among cohorts in age-structured populations is derived. The expression is used to extend the so-called temporal method for estimating effective population size from allele frequency shifts among samples from cohorts born any number of years apart. Computer simulations are used to check on the accuracy and precision of the method, and an application to coastal Atlantic cod is presented.     

How to compute the effective size of spatiotemporally structured populations using separation of time scales

Calculations to derive effective population size become highly complicated when complex population structure is considered. We provide an easy method of computing the effective size of a subdivided population with overlapping generations (a spatiotemporally structured population) using an approximation based on separation of time scales. We also numerically compute the effective size to verify the accuracy of the derived formula. Various interesting quantities, including moments of coalescent time, are readily derived using this approach.     

Estimating the ratio of effective to actual size of an age‐structured population from individual demographic data

The effective population size is a central concept for understanding evolutionary processes in a finite population. We employ Fisher's reproductive value to estimate the ratio of effective to actual population size for an age‐structured population with two sexes using random samples of individual vital rates. The population may be subject to environmental stochasticity affecting the vital rates. When the mean sex ratio at birth is known, improved efficiency is obtained by utilizing the records of total number of offspring rather than considering separately female and male offspring. We also show how to incorporate uncertain paternity.     

GONe: software for estimating effective population size in species with generational overlap

GONe is a user-friendly, Windows-based program for estimating effective size (N(e) ) in populations with overlapping generations. It uses the Jorde-Ryman modification to the temporal method to account for age structure in populations. This method requires estimates of age-specific survival and birth rate and allele frequencies measured in two or more consecutive cohorts. Allele frequencies are acquired by reading in genotypic data from files formatted for either GENEPOP or TEMPOFS. For each interval between consecutive cohorts, N(e) is estimated at each locus and over all loci. Furthermore, N(e) estimates are output for three different genetic drift estimators (F(s) , F(c) and F(k) ). Confidence intervals are derived from a chi-square distribution with degrees of freedom equal to the number of independent alleles. GONe has been validated over a wide range of N(e) values, and for scenarios where survival and birth rates differ between sexes, sex ratios are unequal and reproductive variances differ. GONe is freely available for download at https://bcrc.bio.umass.edu/pedigreesoftware/.     

Generation time and effective population size in Polar Eskimos

North Greenland Polar Eskimos are the only hunter-gatherer population, to our knowledge, who can offer precise genealogical records spanning several generations. This is the first report from Eskimos on two key parameters in population genetics, namely, generation time (T) and effective population size (Ne). The average mother-daughter and father-son intervals were 27 and 32 years, respectively, roughly similar to the previously published generation times obtained from recent agricultural societies across the world. To gain an insight for the generation time in our distant ancestors, we calculated maternal generation time for two wild chimpanzee populations. We also provide the first comparison among three distinct approaches (genealogy, variance and life table methods) for calculating Ne, which resulted in slightly differing values for the Eskimos. The ratio of the effective to the census population size is estimated as 0.6-0.7 for autosomal and X-chromosomal DNA, 0.7-0.9 for mitochondrial DNA and 0.5 for Y-chromosomal DNA. A simulation of alleles along the genealogy suggested that Y-chromosomal DNA may drift a little faster than mitochondrial DNA in this population, in contrast to agricultural Icelanders. Our values will be useful not only in prehistoric population inference but also in understanding the shaping of our genome today.     

Small effective population size in the long-toed salamander

The effective population sizes (Ne) of six populations of the long-toed salamander (Ambystoma macrodactylum) from Montana and Idaho, USA were estimated from allozyme data from samples collected in 1978, 1996 and 1997 using the temporal allele frequency method. Five of the six estimates ranged from 23 to 207 (mean = 123 +/- 79); one estimate was indistinguishable from infinity. In order to infer the actual Ne of salamander populations, we compared the frequency distribution of our observed Ne estimates with distributions obtained from simulated populations of known Ne. Our observed Ne estimate distribution was consistent with distributions from simulated populations with Ne values of 10, 25, and 50, suggesting an actual Ne for each of the six salamander populations of less than 100. This Ne estimate agrees with most other Ne estimates for amphibians. We conclude by discussing the conservation implications of small Ne values in amphibians in the context of increasing isolation of populations due to habitat fragmentation.     

An empirical test of the estimation of historical effective population size using Drosophila melanogaster

The availability of a large number of high-density markers (SNPs) allows the estimation of historical effective population size (N_e) from linkage disequilibrium between loci. A recent refinement of methods to estimate historical N_e from the recent past has been shown to be rather accurate with simulation data. The method has also been applied to real data for numerous species. However, the simulation data cannot encompass all the complexities of real genomes, and the performance of any estimation method with real data is always uncertain, as the true demography of the populations is not known. Here, we carried out an experimental design with Drosophila melanogaster to test the method with real data following a known demographic history. We used a population maintained in the laboratory with a constant census size of about 2800 individuals and subjected the population to a drastic decline to a size of 100 individuals. After a few generations, the population was expanded back to the previous size and after a few further generations again expanded to twice the initial size. Estimates of historical N_e were obtained with the software GONE both for autosomal and X chromosomes from samples of 17 individuals sequenced for the whole genome. Estimates of the historical effective size were able to infer the patterns of changes that occurred in the populations showing generally good performance of the method. We discuss the limitations of the method and the application of the software carried out so far.     

Gene genealogies when the sample size exceeds the effective size of the population

We study the properties of gene genealogies for large samples using a continuous approximation introduced by R. A. Fisher. We show that the major effect of large sample size, relative to the effective size of the population, is to increase the proportion of polymorphisms at which the mutant type is found in a single copy in the sample. We derive analytical expressions for the expected number of these singleton polymorphisms and for the total number of polymorphic, or segregating, sites that are valid even when the sample size is much greater than the effective size of the population. We use simulations to assess the accuracy of these predictions and to investigate other aspects of large-sample genealogies. Lastly, we apply our results to some data from Pacific oysters sampled from British Columbia. This illustrates that, when large samples are available, it is possible to estimate the mutation rate and the effective population size separately, in contrast to the case of small samples in which only the product of the mutation rate and the effective population size can be estimated.     

Estimating effective population size using RADseq: Effects of SNP selection and sample size

Effective population size (N_e) is a key parameter of population genetics. However, N_e remains challenging to estimate for natural populations as several factors are likely to bias estimates. These factors include sampling design, sequencing method, and data filtering. One issue inherent to the restriction site‐associated DNA sequencing (RADseq) protocol is missing data and SNP selection criteria (e.g., minimum minor allele frequency, number of SNPs). To evaluate the potential impact of SNP selection criteria on N_e estimates (Linkage Disequilibrium method) we used RADseq data for a nonmodel species, the thornback ray. In this data set, the inbreeding coefficient F_IS was positively correlated with the amount of missing data, implying data were missing nonrandomly. The precision of N_eestimates decreased with the number of SNPs. Mean N_e estimates (averaged across 50 random data sets with2000 SNPs) ranged between 237 and 1784. Increasing the percentage of missing data from 25% to 50% increased N_e estimates between 82% and 120%, while increasing the minor allele frequency (MAF) threshold from 0.01 to 0.1 decreased estimates between 71% and 75%. Considering these effects is important when interpreting RADseq data‐derived estimates of effective population size in empirical studies.     

Monitoring the effective population size of a brown bear (Ursus arctos) population using new single-sample approaches

The effective population size (N(e) ) could be the ideal parameter for monitoring populations of conservation concern as it conveniently summarizes both the evolutionary potential of the population and its sensitivity to genetic stochasticity. However, tracing its change through time is difficult in natural populations. We applied four new methods for estimating N(e) from a single sample of genotypes to trace temporal change in N(e) for bears in the Northern Dinaric Mountains. We genotyped 510 bears using 20 microsatellite loci and determined their age. The samples were organized into cohorts with regard to the year when the animals were born and yearly samples with age categories for every year when they were alive. We used the Estimator by Parentage Assignment (EPA) to directly estimate both N(e) and generation interval for each yearly sample. For cohorts, we estimated the effective number of breeders (N(b) ) using linkage disequilibrium, sibship assignment and approximate Bayesian computation methods and extrapolated these estimates to N(e) using the generation interval. The N(e) estimate by EPA is 276 (183-350 95% CI), meeting the inbreeding-avoidance criterion of N(e) > 50 but short of the long-term minimum viable population goal of N(e) > 500. The results obtained by the other methods are highly consistent with this result, and all indicate a rapid increase in N(e) probably in the late 1990s and early 2000s. The new single-sample approaches to the estimation of N(e) provide efficient means for including N(e) in monitoring frameworks and will be of great importance for future management and conservation.     

Multiple estimates of effective population size for monitoring a long‐lived vertebrate: an application to Yellowstone grizzly bears

Effective population size (N_e) is a key parameter for monitoring the genetic health of threatened populations because it reflects a population's evolutionary potential and risk of extinction due to genetic stochasticity. However, its application to wildlife monitoring has been limited because it is difficult to measure in natural populations. The isolated and well‐studied population of grizzly bears (Ursus arctos) in the Greater Yellowstone Ecosystem provides a rare opportunity to examine the usefulness of different N_e estimators for monitoring. We genotyped 729 Yellowstone grizzly bears using 20 microsatellites and applied three single‐sample estimators to examine contemporary trends in generation interval (GI), effective number of breeders (N_b) and N_e during 1982–2007. We also used multisample methods to estimate variance (N_eV) and inbreeding N_e (N_eI). Single‐sample estimates revealed positive trajectories, with over a fourfold increase in N_e (≈100 to 450) and near doubling of the GI (≈8 to 14) from the 1980s to 2000s. N_eV (240–319) and N_eI (256) were comparable with the harmonic mean single‐sample N_e (213) over the time period. Reanalysing historical data, we found N_eV increased from ≈80 in the 1910s–1960s to ≈280 in the contemporary population. The estimated ratio of effective to total census size (N_e/N_c) was stable and high (0.42–0.66) compared to previous brown bear studies. These results support independent demographic evidence for Yellowstone grizzly bear population growth since the 1980s. They further demonstrate how genetic monitoring of N_e can complement demographic‐based monitoring of N_c and vital rates, providing a valuable tool for wildlife managers.     

The genetic effective and adult census size of an Australian population of tiger prawns (Penaeus esculentus)

This study compares estimates of the census size of the spawning population with genetic estimates of effective current and long-term population size for an abundant and commercially important marine invertebrate, the brown tiger prawn (Penaeus esculentus). Our aim was to focus on the relationship between genetic effective and census size that may provide a source of information for viability analyses of naturally occurring populations. Samples were taken in 2001, 2002 and 2003 from a population on the east coast of Australia and temporal allelic variation was measured at eight polymorphic microsatellite loci. Moments-based and maximum-likelihood estimates of current genetic effective population size ranged from 797 to 1304. The mean long-term genetic effective population size was 9968. Although small for a large population, the effective population size estimates were above the threshold where genetic diversity is lost at neutral alleles through drift or inbreeding. Simulation studies correctly predicted that under these experimental conditions the genetic estimates would have non-infinite upper confidence limits and revealed they might be overestimates of the true size. We also show that estimates of mortality and variance in family size may be derived from data on average fecundity, current genetic effective and census spawning population size, assuming effective population size is equivalent to the number of breeders. This work confirms that it is feasible to obtain accurate estimates of current genetic effective population size for abundant Type III species using existing genetic marker technology.     

Sensitivity analysis of effective population size to demographic parameters in house sparrow populations

The ratio between the effective and the census population size, , is an important measure of the long‐term viability and sustainability of a population. Understanding which demographic processes that affect most will improve our understanding of how genetic drift and the probability of fixation of alleles is affected by demography. This knowledge may also be of vital importance in management of endangered populations and species. Here, we use data from 13 natural populations of house sparrow (Passer domesticus) in Norway to calculate the demographic parameters that determine . Using the global variance‐based Sobol’ method for the sensitivity analyses, we found that was most sensitive to demographic variance, especially among older individuals. Furthermore, the individual reproductive values (that determine the demographic variance) were most sensitive to variation in fecundity. Our results draw attention to the applicability of sensitivity analyses in population management and conservation. For population management aiming to reduce the loss of genetic variation, a sensitivity analysis may indicate the demographic parameters towards which resources should be focused. The result of such an analysis may depend on the life history and mating system of the population or species under consideration, because the vital rates and sex–age classes that is most sensitive to may change accordingly.     

Effects of population characteristics and structure on estimates of effective population size in a house sparrow metapopulation

Effective population size (N_e) is a key parameter to understand evolutionary processes and the viability of endangered populations as it determines the rate of genetic drift and inbreeding. Low N_e can lead to inbreeding depression and reduced population adaptability. In this study, we estimated contemporary N_e using genetic estimators (LDNE, ONeSAMP, MLNE and CoNe) as well as a demographic estimator in a natural insular house sparrow metapopulation. We investigated whether population characteristics (population size, sex ratio, immigration rate, variance in population size and population growth rate) explained variation within and among populations in the ratio of effective to census population size (N_e/N_c). In general, N_e/N_c ratios increased with immigration rates. Genetic N_e was much larger than demographic N_e, probably due to a greater effect of immigration on genetic than demographic processes in local populations. Moreover, although estimates of genetic N_e seemed to track N_c quite well, the genetic N_e‐estimates were often larger than N_c within populations. Estimates of genetic N_e for the metapopulation were however within the expected range (<N_c). Our results suggest that in fragmented populations, even low levels of gene flow may have important consequences for the interpretation of genetic estimates of N_e. Consequently, further studies are needed to understand how N_e estimated in local populations or the total metapopulation relates to actual rates of genetic drift and inbreeding.     

The effects of cyclic dynamics and mating system on the effective size of an island mouflon population

The Haute Island mouflon (Ovis aries) population is isolated on one small (6.5 km2) island of the remote Kerguelen archipelago. Given a promiscuous mating system, a cyclic demography and a strong female-biased sex ratio after population crashes, we expected a low effective population size (Ne). We estimated Ne using demographic and temporal genetic approaches based on genetic information at 25 microsatellite loci from 62 and 58 mouflons sampled in 1988 and 2003, respectively. Genetic Ne estimates were higher than expected, varying between 104 and 250 depending on the methods used. Both demographic and genetic approaches show the Haute Island Ne is buffered against population crashes. The unexpectedly high Ne likely results from the cyclic winter crashes that allow young males to reproduce, limiting the variance of male reproductive success. Based on individual-based simulations, we suggest that despite a strongly female-biased sex ratio, the effects of the mating system on the effective population size more closely resemble random mating or weak polygyny.     

Mature male parr contribution to the effective size of an anadromous Atlantic salmon (Salmo salar) population over 30 years

We describe temporal changes in the genetic composition of a small anadromous Atlantic salmon (Salmo salar) population from South Newfoundland, an area where salmon populations are considered threatened (COSEWIC 2010). We examined the genetic variability (13 microsatellite loci) in 869 out‐migrating smolt and post‐spawning kelt samples, collected from 1985 to 2011 for a total of 22 annual collections and a 30 year span of assigned cohorts. We estimated the annual effective number of breeders (N_b) and the generational effective population size (N_e) through genetic methods and demographically using the adult sex ratio. Comparisons between genetic and demographic estimates show that the adult spawners inadequately explain the observed N_e estimates, suggesting that mature male parr are significantly increasing N_b and N_e over the study period. Spawning as parr appears to be a viable and important strategy in the near absence of adult males.