Effective size of density‐dependent two‐sex populations: the effect of mating systems

Density dependence in vital rates is a key feature affecting temporal fluctuations of natural populations. This has important implications for the rate of random genetic drift. Mating systems also greatly affect effective population sizes, but knowledge of how mating system and density regulation interact to affect random genetic drift is poor. Using theoretical models and simulations, we compare N_e in short‐lived, density‐dependent animal populations with different mating systems. We study the impact of a fluctuating, density‐dependent sex ratio and consider both a stable and a fluctuating environment. We find a negative relationship between annual N_e/N and adult population size N due to density dependence, suggesting that loss of genetic variation is reduced at small densities. The magnitude of this decrease was affected by mating system and life history. A male‐biased, density‐dependent sex ratio reduces the rate of genetic drift compared to an equal, density‐independent sex ratio, but a stochastic change towards male bias reduces the N_e/N ratio. Environmental stochasticity amplifies temporal fluctuations in population size and is thus vital to consider in estimation of effective population sizes over longer time periods. Our results on the reduced loss of genetic variation at small densities, particularly in polygamous populations, indicate that density regulation may facilitate adaptive evolution at small population sizes.     

Effective size of harvested ungulate populations

The harvest of ungulate populations is often directed against certain sex or age classes to maximize the yield in terms of biomass, number of shot animals or number of trophies. Here we examine how such directional harvest affects the effective size of the population. We parameterize an age-specific model assumed to describe the dynamics of Fennoscandian moose. Based on expressions for the demographic variance     for a small subpopulation of heterozygotes Aa bearing a rare neutral allele a , we use this model to calculate how different harvest strategies influence the effective size of the population, given that the population remains stable after harvest. We show that the annual genetic drift, determined by     , increases with decreasing harvest rate of calves and increasing sex bias in the harvest towards bulls 1 year or older. The effective population size per generation decreased with reduced harvest of calves and increased harvest of bulls 1 year or older. The magnitude of these effects depends on the age-specific pattern of variation in reproductive success, which influences the demographic variance. This shows that the choice of harvest strategy strongly affects the genetic dynamics of harvested ungulate populations.     

Effective size of fluctuating populations with two sexes and overlapping generations

We derive formulas that can be applied to estimate the effective population size N(e) for organisms with two sexes reproducing once a year and having constant adult mean vital rates independent of age. Temporal fluctuations in population size are generated by demographic and environmental stochasticity. For populations with even sex ratio at birth, no deterministic population growth and identical mean vital rates for both sexes, the key parameter determining N(e) is simply the mean value of the demographic variance for males and females considered separately. In this case Crow and Kimura's generalization of Wright's formula for N(e) with two sexes, in terms of the effective population sizes for each sex, is applicable even for fluctuating populations with different stochasticity in vital rates for males and females. If the mean vital rates are different for the sexes then a simple linear combination of the demographic variances determines N(e), further extending Wright's formula. For long-lived species an expression is derived for N(e) involving the generation times for both sexes. In the general case with nonzero population growth and uneven sex ratio of newborns, we use the model to investigate numerically the effects of different population parameters on N(e). We also estimate the ratio of effective to actual population size in six populations of house sparrows on islands off the coast of northern Norway. This ratio showed large interisland variation because of demographic differences among the populations. Finally, we calculate how N(e) in a growing house sparrow population will change over time.     

Estimating fluctuating selection in age-structured populations

In age-structured populations, viability and fecundity selection of varying strength may occur in different age classes. On the basis of an original idea by Fisher of weighting individuals by their reproductive value, we show that the combined effect of selection on traits at different ages acts through the individual reproductive value defined as the stochastic contribution of an individual to the total reproductive value of the population the following year. The selection differential is a weighted sum of age-specific differentials that are the covariances between the phenotype and the age-specific relative fitness defined by the individual reproductive value. This enables estimation of weak selection on a multivariate quantitative character in populations with no density regulation by combinations of age-specific linear regressions of individual reproductive values on the traits. Demographic stochasticity produces random variation in fitness components in finite samples of individuals and affects the statistical inference of the temporal average directional selection as well as the magnitude of fluctuating selection. Uncertainties in parameter estimates and test power depend strongly on the demographic stochasticity. Large demographic variance results in large uncertainties in yearly estimates of selection that complicates detection of significant fluctuating selection. The method is illustrated by an analysis of age-specific selection in house sparrows on a fitness-related two-dimensional morphological trait, tarsus length and body mass of fledglings.     

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.     

Effective/census population size ratio estimation: a compendium and appraisal

With an ecological-evolutionary perspective increasingly applied toward the conservation and management of endangered or exploited species, the genetic estimation of effective population size (N_e) has proliferated. Based on a comprehensive analysis of empirical literature from the past two decades, we asked: (i) how often do studies link N_e to the adult census population size (N)? (ii) To what extent is N_e correctly linked to N? (iii) How readily is uncertainty accounted for in both N_e and N when quantifying N_e/N ratios? and (iv) how frequently and to what degree might errors in the estimation of N_e or N affect inferences of N_e/N ratios? We found that only 20% of available N_e estimates (508 of 2617; 233 studies) explicitly attempted to link N_e and N; of these, only 31% (160 of 508) correctly linked N_e and N. Moreover, only 7% (41 of 508) of N_e/N ratios (correctly linked or not) reported confidence intervals for both N_e and N; for those cases where confidence intervals were reported for N_e only, 31% of N_e/N ratios overlapped with 1, of which more than half also reached below N_e/N = 0.01. Uncertainty in N_e/N ratios thus sometimes spanned at least two orders of magnitude. We conclude that the estimation of N_e/N ratios in natural populations could be significantly improved, discuss several options for doing so, and briefly outline some future research directions.     

Life‐history evolution under fluctuating density‐dependent selection and the adaptive alignment of pace‐of‐life syndromes

We present a novel perspective on life‐history evolution that combines recent theoretical advances in fluctuating density‐dependent selection with the notion of pace‐of‐life syndromes (POLSs) in behavioural ecology. These ideas posit phenotypic co‐variation in life‐history, physiological, morphological and behavioural traits as a continuum from the highly fecund, short‐lived, bold, aggressive and highly dispersive ‘fast’ types at one end of the POLS to the less fecund, long‐lived, cautious, shy, plastic and socially responsive ‘slow’ types at the other. We propose that such variation in life histories and the associated individual differences in behaviour can be explained through their eco‐evolutionary dynamics with population density – a single and ubiquitous selective factor that is present in all biological systems. Contrasting regimes of environmental stochasticity are expected to affect population density in time and space and create differing patterns of fluctuating density‐dependent selection, which generates variation in fast versus slow life histories within and among populations. We therefore predict that a major axis of phenotypic co‐variation in life‐history, physiological, morphological and behavioural traits (i.e. the POLS) should align with these stochastic fluctuations in the multivariate fitness landscape created by variation in density‐dependent selection. Phenotypic plasticity and/or genetic (co‐)variation oriented along this major POLS axis are thus expected to facilitate rapid and adaptively integrated changes in various aspects of life histories within and among populations and/or species. The fluctuating density‐dependent selection POLS framework presented here therefore provides a series of clear testable predictions, the investigation of which should further our fundamental understanding of life‐history evolution and thus our ability to predict natural population dynamics.     

Variance effective population size is affected by census size in sub-structured populations

Measurement of allele frequency shifts between temporally spaced samples has long been used for assessment of effective population size (N_e), and this ‘temporal method’ provides estimates of N_e referred to as variance effective size (N_eV). We show that N_eV of a local population that belongs to a sub-structured population (a metapopulation) is determined not only by genetic drift and migration rate (m), but also by the census size (N_c). The realized N_eV of a local population can either increase or decrease with increasing m, depending on the relationship between N_e and N_c in isolation. This is shown by explicit mathematical expressions for the factors affecting N_eV derived for an island model of migration. We verify analytical results using high-resolution computer simulations, and show that the phenomenon is not restricted to the island model migration pattern. The effect of N_c on the realized N_eV of a local subpopulation is most pronounced at high migration rates. We show that N_c only affects local N_eV, whereas N_eV for the metapopulation as a whole, inbreeding (N_eI), and linkage disequilibrium (N_eLD) effective size are all independent of N_c. Our results provide a possible explanation to the large variation of N_e/N_c ratios reported in the literature, where N_e is frequently estimated by N_eV. They are also important for the interpretation of empirical N_e estimates in genetic management where local N_eV is often used as a substitute for inbreeding effective size, and we suggest an increased focus on metapopulation N_eV as a proxy for N_eI.     

Effective population size associated with self-fertilization: lessons from temporal changes in allele frequencies in the selfing annual Medicago truncatula

Despite its significance in evolutionary and conservation biology, few estimates of effective population size (N(e)) are available in plant species. Self-fertilization is expected to affect N(e), through both its effect on homozygosity and population dynamics. Here, we estimated N(e) using temporal variation in allele frequencies for two contrasted populations of the selfing annual Medicago truncatula: a large and continuous population and a subdivided population. Estimated N(e) values were around 5-10% of the population census size suggesting that other factors than selfing must contribute to variation in allele frequencies. Further comparisons between monolocus allelic variation and changes in the multilocus genotypic composition of the populations show that the local dynamics of inbred lines can play an important role in the fluctuations of allele frequencies. Finally, comparing N(e) estimates and levels of genetic variation suggest that H(e) is a poor estimator of the contemporaneous variance effective population size.     

Ontogenetic variation in density‐dependent growth of brown trout through habitat competition

1. Density‐dependent growth has been widely reported in freshwater fishes, but the ontogenetic evolution of competition and its subsequent effects on growth through a life span remains unclear. 2. Patterns of competition can be described by integrating population abundance data with habitat‐modelling results. Weighted usable area (WUA; m² WUA ha^?1) curves are obtained for each flow value and are then coupled with demographic data to obtain the occupancy rates (trout m^?2 WUA, the density of a given age class related to its suitable habitat) of the WUA for every age class, year and site. 3. We examined a long‐term data series searching for temporal variation in the influence of habitat occupancy rate on the growth of brown trout Salmo trutta. We tested whether (i) mean cohort mass (mean mass of the cohort during the first 3 years of life) is affected by the occupancy rate experienced across a life span; and (ii) the occupancy rate experienced at different ages influenced mean body size. 4. We observed a consistent negative power relationship between average cohort mass and mean occupancy rate through a life span, indicating that stronger cohorts were related to a reduced growth, with likely consequences for individual fitness. 5. The effects of occupancy rate on size‐at‐age were mainly detected in the size attained at the second year of life, but they were because of the competition at different times. Thus, the level of competition varied through ontogeny, in some of the rivers affecting growth since the first year of life, whereas in most of the rivers the main effects on body size resulted from the competition during the second year of life. 6. Occupancy rate appears more appropriate than density for assessing the occurrence of habitat competition in freshwater fishes, since it encompasses the differences in quantity and quality of suitable habitat for each age class. 7. Our study highlights the importance of density‐dependent growth as a key process in the dynamics of brown trout populations, its temporal variation depending on the temporal changes of density and the variation of competition associated with the habitat capacity for each life stage.     

Unexpected high genetic diversity in small populations suggests maintenance by associative overdominance

The effective population size (N_e) is a central factor in determining maintenance of genetic variation. The neutral theory predicts that loss of variation depends on N_e, with less genetic drift in larger populations. We monitored genetic drift in 42 Drosophila melanogaster populations of different adult census population sizes (10, 50 or 500) using pooled RAD sequencing. In small populations, variation was lost at a substantially lower rate than expected. This observation was consistent across two ecological relevant thermal regimes, one stable and one with a stressful increase in temperature across generations. Estimated ratios between N_e and adult census size were consistently higher in small than in larger populations. The finding provides evidence for a slower than expected loss of genetic diversity and consequently a higher than expected long‐term evolutionary potential in small fragmented populations. More genetic diversity was retained in areas of low recombination, suggesting that associative overdominance, driven by disfavoured homozygosity of recessive deleterious alleles, is responsible for the maintenance of genetic diversity in smaller populations. Consistent with this hypothesis, the X‐chromosome, which is largely free of recessive deleterious alleles due to hemizygosity in males, fits neutral expectations even in small populations. Our experiments provide experimental answers to a range of unexpected patterns in natural populations, ranging from variable diversity on X‐chromosomes and autosomes to surprisingly high levels of nucleotide diversity in small populations.     

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 habitat size and quality on equilibrium density and extinction time of Sorex araneus populations

1. The effects of changes in habitat size and quality on the expected population density and the expected time to extinction of Sorex araneus are studied by means of mathematical models that incorporate demographic stochasticity.
2. Habitat size is characterized by the number of territories, while habitat quality is represented by the expected number of offspring produced during the lifetime of an individual.
3. The expected population density of S. araneus is shown to be mainly influenced by the habitat size. The expected time to extinction of S. araneus populations due to demographic stochasticity, on the other hand, is much more affected by the habitat quality.
4. In a more general setting we demonstrate that, irrespective of the actual species under consideration, the likelihood of extinction as a consequence of demographic stochasticity is more effectively countered by increasing the reproductive success and survival of individuals then by increasing total population size.     

The extended Moran effect and large-scale synchronous fluctuations in the size of great tit and blue tit populations

1. Synchronous fluctuations of geographically separated populations are in general explained by the Moran effect, i.e. a common influence on the local population dynamics of environmental variables that are correlated in space. Empirical support for such a Moran effect has been difficult to provide, mainly due to problems separating out effects of local population dynamics, demographic stochasticity and dispersal that also influence the spatial scaling of population processes. Here we generalize the Moran effect by decomposing the spatial autocorrelation function for fluctuations in the size of great tit Parus major and blue tit Cyanistes caeruleus populations into components due to spatial correlations in the environmental noise, local differences in the strength of density regulation and the effects of demographic stochasticity. 2. Differences between localities in the strength of density dependence and nonlinearity in the density regulation had a small effect on population synchrony, whereas demographic stochasticity reduced the effects of the spatial correlation in environmental noise on the spatial correlations in population size by 21.7% and 23.3% in the great tit and blue tit, respectively. 3. Different environmental variables, such as beech mast and climate, induce a common environmental forcing on the dynamics of central European great and blue tit populations. This generates synchronous fluctuations in the size of populations located several hundred kilometres apart. 4. Although these environmental variables were autocorrelated over large areas, their contribution to the spatial synchrony in the population fluctuations differed, dependent on the spatial scaling of their effects on the local population dynamics. We also demonstrate that this effect can lead to the paradoxical result that a common environmental variable can induce spatial desynchronization of the population fluctuations. 5. This demonstrates that a proper understanding of the ecological consequences of environmental changes, especially those that occur simultaneously over large areas, will require information about the spatial scaling of their effects on local population dynamics.     

Estimating Effective Population Size from Temporally Spaced Samples with a Novel,Efficient Maximum-Likelihood Algorithm

The effective population size N_e is a key parameter in population genetics and evolutionary biology, as it quantifies the expected distribution of changes in allele frequency due to genetic drift. Several methods of estimating N_e have been described, the most direct of which uses allele frequencies measured at two or more time points. A new likelihood-based estimator NB^ for contemporary effective population size using temporal data is developed in this article. The existing likelihood methods are computationally intensive and unable to handle the case when the underlying N_e is large. This article tries to work around this problem by using a hidden Markov algorithm and applying continuous approximations to allele frequencies and transition probabilities. Extensive simulations are run to evaluate the performance of the proposed estimator NB^, and the results show that it is more accurate and has lower variance than previous methods. The new estimator also reduces the computational time by at least 1000-fold and relaxes the upper bound of N_e to several million, hence allowing the estimation of larger N_e. Finally, we demonstrate how this algorithm can cope with nonconstant N_e scenarios and be used as a likelihood-ratio test to test for the equality of N_e throughout the sampling horizon. An R package “NB” is now available for download to implement the method described in this article.     

Stochastic exits from dormancy give rise to heavy‐tailed distributions of descendants in bacterial populations

Variance in reproductive success is a major determinant of the degree of genetic drift in a population. While many plants and animals exhibit high variance in their number of progeny, far less is known about these distributions for microorganisms. Here, we used a strain barcoding approach to quantify variability in offspring number among replicate bacterial populations and developed a Bayesian method to infer the distribution of descendants from this variability. We applied our approach to measure the offspring distributions for five strains of bacteria from the genus Streptomyces after germination and growth in a homogenous laboratory environment. The distributions of descendants were heavy‐tailed, with a few cells effectively ‘winning the jackpot’ to become a disproportionately large fraction of the population. This extreme variability in reproductive success largely traced back to initial populations of spores stochastically exiting dormancy, which provided early‐germinating spores with an exponential advantage. In simulations with multiple dormancy cycles, heavy‐tailed distributions of descendants decreased the effective population size by many orders of magnitude and led to allele dynamics differing substantially from classical population genetics models with matching effective population size. Collectively, these results demonstrate that extreme variability in reproductive success can occur even in growth conditions that are far more homogeneous than the natural environment. Thus, extreme variability in reproductive success might be an important factor shaping microbial population dynamics with implications for predicting the fate of beneficial mutations, interpreting sequence variability within populations and explaining variability in infection outcomes across patients.     

Samples from subdivided populations yield biased estimates of effective size that overestimate the rate of loss of genetic variation

Many empirical studies estimating effective population size apply the temporal method that provides an estimate of the variance effective size through the amount of temporal allele frequency change under the assumption that the study population is completely isolated. This assumption is frequently violated, and the magnitude of the resulting bias is generally unknown. We studied how gene flow affects estimates of effective size obtained by the temporal method when sampling from a population system and provide analytical expressions for the expected estimate under an island model of migration. We show that the temporal method tends to systematically underestimate both local and global effective size when populations are connected by gene flow, and the bias is sometimes dramatic. The problem is particularly likely to occur when sampling from a subdivided population where high levels of gene flow obscure identification of subpopulation boundaries. In such situations, sampling in a manner that prevents biased estimates can be difficult. This phenomenon might partially explain the frequently reported unexpectedly low effective population sizes of marine populations that have raised concern regarding the genetic vulnerability of even exceptionally large populations.