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

Harvest and decimation affect genetic drift and the effective population size in wild reindeer

Harvesting and culling are methods used to monitor and manage wildlife diseases. An important consequence of these practices is a change in the genetic dynamics of affected populations that may threaten their long-term viability. The effective population size (N e) is a fundamental parameter for describing such changes as it determines the amount of genetic drift in a population. Here, we estimate N e of a harvested wild reindeer population in Norway. Then we use simulations to investigate the genetic consequences of management efforts for handling a recent spread of chronic wasting disease, including increased adult male harvest and population decimation. The N e/N ratio in this population was found to be 0.124 at the end of the study period, compared to 0.239 in the preceding 14 years period. The difference was caused by increased harvest rates with a high proportion of adult males (older than 2.5 years) being shot (15.2% in 2005-2018 and 44.8% in 2021). Increased harvest rates decreased N e in the simulations, but less sex biased harvest strategies had a lower negative impact. For harvest strategies that yield stable population dynamics, shifting the harvest from calves to adult males and females increased N e. Population decimation always resulted in decreased genetic variation in the population, with higher loss of heterozygosity and rare alleles with more severe decimation or longer periods of low population size. A very high proportion of males in the harvest had the most severe consequences for the loss of genetic variation. This study clearly shows how the effects of harvest strategies and changes in population size interact to determine the genetic drift of a managed population. The long-term genetic viability of wildlife populations subject to a disease will also depend on population impacts of the disease and how these interact with management actions.

Partitioning variance in reproductive success, within years and across lifetimes

Variance in reproductive success (s k 2 , with k = number of offspring) plays a large role in determining the rate of genetic drift and the scope within which selection acts. Various frameworks have been proposed to parse factors that contribute to s k 2 , but none has focused on age-specific values of ϕ = s k 2 / k ¯ , which indicate the degree to which reproductive skew is overdispersed (compared to the random Poisson expectation) among individuals of the same age and sex. Instead, within-age effects are generally lumped with residual variance and treated as "noise." Here, an ANOVA sums-of-squares framework is used to partition variance in annual and lifetime reproductive success into between-group and within-group components. For annual reproduction, the between-age effect depends on age-specific fecundity (b x), but relatively few empirical data are available on the within-age effect, which depends on ϕ x. By defining groups by age-at-death rather than age, the same ANOVA framework can be used to partition variance in lifetime reproductive success (LRS) into between-group and within-group components. Analytical methods are used to develop null-model expectations for random contributions to within-group and between-group components. For analysis of LRS, random variation in longevity appears as part of the between-group variance, and effects (if any) of skip breeding and persistent individual differences contribute to the within-group variance. Simulations are used to show that the methods for variance partitioning are asymptotically unbiased. Practical application is illustrated with empirical data for annual reproduction in American black bears and lifetime reproduction in Dutch great tits. Results show that overdispersed within-age variance (1) dominates annual s k 2 in both male and female black bears, (2) is the primary factor that reduces annual effective size to a fraction of the number of adults, and (3) represents most of the opportunity for selection. In contrast, about a quarter of the variance in LRS in great tits can be attributed to random variation in longevity, and most of the rest is due to modest differences in fecundity with age estimated for a single cohort of females. R code is provided that reads generic input files for annual and lifetime reproductive success and allows users to conduct variance partitioning with their own data.

Using long-term data for a reintroduced population to empirically estimate future consequences of inbreeding

Inbreeding depression is an important long-term threat to reintroduced populations. However, the strength of inbreeding depression is difficult to estimate in wild populations because pedigree data are inevitably incomplete and because good data are needed on survival and reproduction. Predicting future population consequences is especially difficult because this also requires projecting future inbreeding levels and their impacts on long-term population dynamics, which are subject to many uncertainties. We illustrate how such projections can be derived through Bayesian state-space modeling methods based on a 26-year data set for North Island Robins (Petroica longipes) reintroduced to Tiritiri Matangi Island in 1992. We used pedigree data to model increases in the average inbreeding level (F) over time based on kinship of possible breeding pairs and to estimate empirically N_e /N (effective/census population size). We used multiple imputation to model the unknown components of inbreeding coefficients, which allowed us to estimate effects of inbreeding on survival for all 1458 birds in the data set while modeling density dependence and environmental stochasticity. This modeling indicated that inbreeding reduced juvenile survival (1.83 lethal equivalents [SE 0.81]) and may have reduced subsequent adult survival (0.44 lethal equivalents [0.81]) but had no apparent effect on numbers of fledglings produced. Average inbreeding level increased to 0.10 (SE 0.001) as the population grew from 33 (0.3) to 160 (6) individuals over the 25 years, giving a Ne/N ratio of 0.56 (0.01). Based on a model that also incorporated habitat regeneration, the population was projected to reach a maximum of 331-1144 birds (median 726) in 2130, then to begin a slow decline. Without inbreeding, the population would be expected stabilize at 887-1465 birds (median 1131). Such analysis, therefore, makes it possible to empirically derive the information needed for rational decisions about inbreeding management while accounting for multiple sources of uncertainty.

Phenotypic evolution in stochastic environments: The contribution of frequency- and density-dependent selection

Understanding how environmental variation affects phenotypic evolution requires models based on ecologically realistic assumptions that include variation in population size and specific mechanisms by which environmental fluctuations affect selection. Here we generalize quantitative genetic theory for environmentally induced stochastic selection to include general forms of frequency- and density-dependent selection. We show how the relevant fitness measure under stochastic selection relates to Fisher's fundamental theorem of natural selection, and present a general class of models in which density regulation acts through total use of resources rather than just population size. In this model, there is a constant adaptive topography for expected evolution, and the function maximized in the long run is the expected factor restricting population growth. This allows us to generalize several previous results and to explain why apparently " K -selected" species with slow life histories often have low carrying capacities. Our joint analysis of density- and frequency-dependent selection reveals more clearly the relationship between population dynamics and phenotypic evolution, enabling a broader range of eco-evolutionary analyses of some of the most interesting problems in evolution in the face of environmental variation.

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

Effective population size (Ne ) 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 Ne 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 Ne estimates decreased with the number of SNPs. Mean Ne 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 Ne 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.

Decomposing demographic contributions to the effective population size with moose as a case study

Levels of random genetic drift are influenced by demographic factors, such as mating system, sex ratio and age structure. The effective population size (N_e ) is a useful measure for quantifying genetic drift. Evaluating relative contributions of different demographic factors to N_e is therefore important to identify what makes a population vulnerable to loss of genetic variation. Until recently, models for estimating N_e have required many simplifying assumptions, making them unsuitable for this task. Here, using data from a small, harvested moose population, we demonstrate the use of a stochastic demographic framework allowing for fluctuations in both population size and age distribution to estimate and decompose the total demographic variance and hence the ratio of effective to total population size (N_e /N) into components originating from sex, age, survival and reproduction. We not only show which components contribute most to N_e /N currently, but also which components have the greatest potential for changing N_e /N. In this relatively long-lived polygynous system we show that N_e /N is most sensitive to the demographic variance of older males, and that both reproductive autocorrelations (i.e., a tendency for the same individuals to be successful several years in a row) and covariance between survival and reproduction contribute to decreasing N_e /N (increasing genetic drift). These conditions are common in nature and can be caused by common hunting strategies. Thus, the framework presented here has great potential to increase our understanding of the demographic processes that contribute to genetic drift and viability of populations, and to inform management decisions.

Towards a predictive conservation biology: the devil is in the behaviour

One of the most important challenges in conservation biology is to predict the viability of populations of vulnerable and threatened species. This requires that the demographic stochasticity strongly affecting the ecological and evolutionary dynamics of especially small populations is correctly estimated and modelled. Here, we summarize theoretical evidence showing that the demographic variance in population dynamics is a key parameter determining the probability of extinction and also is directly linked to the magnitude of the genetic drift in the population. The demographic variance is dependent on the mating system, being larger in a polygynous than in monogamous populations. Understanding factors affecting intersexual differences in mating success is therefore essential in explaining variation in the demographic variance. We hypothesize that the strength of sexual selection, for example, quantified by the Bateman gradient, may be a useful predictor of the magnitude of the demographic stochasticity and hence the genetic drift in the population. We provide results from a field study of moose that support this claim. Thus, including central principles from behavioural ecology may increase the reliability of population viability analyses through an improvement of our understanding of factors affecting stochastic influences on population dynamics and evolutionary processes. This article is part of the theme issue 'Linking behaviour to dynamics of populations and communities: application of novel approaches in behavioural ecology to conservation'.

Estimating demographic contributions to effective population size in an age-structured wild population experiencing environmental and demographic stochasticity

A population's effective size (N_e ) is a key parameter that shapes rates of inbreeding and loss of genetic diversity, thereby influencing evolutionary processes and population viability. However, estimating N_e , and identifying key demographic mechanisms that underlie the N_e to census population size (N) ratio, remains challenging, especially for small populations with overlapping generations and substantial environmental and demographic stochasticity and hence dynamic age-structure. A sophisticated demographic method of estimating N_e /N, which uses Fisher's reproductive value to account for dynamic age-structure, has been formulated. However, this method requires detailed individual- and population-level data on sex- and age-specific reproduction and survival, and has rarely been implemented. Here, we use the reproductive value method and detailed demographic data to estimate N_e /N for a small and apparently isolated red-billed chough (Pyrrhocorax pyrrhocorax) population of high conservation concern. We additionally calculated two single-sample molecular genetic estimates of N_e to corroborate the demographic estimate and examine evidence for unobserved immigration and gene flow. The demographic estimate of N_e /N was 0.21, reflecting a high total demographic variance (σ2dg) of 0.71. Females and males made similar overall contributions to σ2dg. However, contributions varied among sex-age classes, with greater contributions from 3 year-old females than males, but greater contributions from ≥5 year-old males than females. The demographic estimate of N_e was ~30, suggesting that rates of increase of inbreeding and loss of genetic variation per generation will be relatively high. Molecular genetic estimates of N_e computed from linkage disequilibrium and approximate Bayesian computation were approximately 50 and 30, respectively, providing no evidence of substantial unobserved immigration which could bias demographic estimates of N_e . Our analyses identify key sex-age classes contributing to demographic variance and thus decreasing N_e /N in a small age-structured population inhabiting a variable environment. They thereby demonstrate how assessments of N_e can incorporate stochastic sex- and age-specific demography and elucidate key demographic processes affecting a population's evolutionary trajectory and viability. Furthermore, our analyses show that N_e for the focal chough population is critically small, implying that management to re-establish genetic connectivity may be required to ensure population viability.

Evidence of the phenotypic expression of a lethal recessive allele under inbreeding in a wild population of conservation concern

Deleterious recessive alleles that are masked in outbred populations are predicted to be expressed in small, inbred populations, reducing both individual fitness and population viability. However, there are few definitive examples of phenotypic expression of lethal recessive alleles under inbreeding conditions in wild populations. Studies that demonstrate the action of such alleles, and infer their distribution and dynamics, are required to understand their potential impact on population viability and inform management responses. The Scottish population of red-billed choughs (Pyrrhocorax pyrrhocorax), which currently totals <60 breeding pairs and is of major conservation concern, has recently been affected by lethal blindness in nestlings. We used family data to show that the pattern of occurrence of blindness within and across affected families that produced blind nestlings was exactly 0·25, matching that expected given a single-locus autosomal lethal recessive allele. Furthermore, the observed distribution of blind nestlings within affected families did not differ from that expected given Mendelian inheritance of such an allele. Relatedness estimates showed that individuals from affected families were not more closely related to each other than they were to individuals from unaffected families that did not produce blind nestlings. Blind individuals tended to be less heterozygous than non-blind individuals, as expected if blindness was caused by the expression of a recessive allele under inbreeding. However, there was no difference in the variance in heterozygosity estimates, suggesting that some blind individuals were relatively outbred. These results suggest carriers of the blindness allele may be widely distributed across contemporary families rather than restricted to a single family lineage, implying that the allele has persisted across multiple generations. Blindness occurred at low frequency (affecting 1·6% of observed nestlings since 1981). However, affected families had larger initial brood sizes than unaffected families. Such high fecundity of carriers of a lethal recessive allele might reflect overdominance, potentially reducing purging and increasing allele persistence probability. We thereby demonstrate the phenotypic expression of a lethal recessive allele in a wild population of conservation concern, and provide a general framework for inferring allele distribution and persistence and informing management responses.

Simple life-history traits explain key effective population size ratios across diverse taxa

Effective population size (Ne) controls both the rate of random genetic drift and the effectiveness of selection and migration, but it is difficult to estimate in nature. In particular, for species with overlapping generations, it is easier to estimate the effective number of breeders in one reproductive cycle (Nb) than Ne per generation. We empirically evaluated the relationship between life history and ratios of Ne, Nb and adult census size (N) using a recently developed model (agene) and published vital rates for 63 iteroparous animals and plants. Nb/Ne varied a surprising sixfold across species and, contrary to expectations, Nb was larger than Ne in over half the species. Up to two-thirds of the variance in Nb/Ne and up to half the variance in Ne/N was explained by just two life-history traits (age at maturity and adult lifespan) that have long interested both ecologists and evolutionary biologists. These results provide novel insights into, and demonstrate a close general linkage between, demographic and evolutionary processes across diverse taxa. For the first time, our results also make it possible to interpret rapidly accumulating estimates of Nb in the context of the rich body of evolutionary theory based on Ne per generation.

Beyond the Mean: Sensitivities of the Variance of Population Growth

Populations in variable environments are described by both a mean growth rate and a variance of stochastic population growth. Increasing variance will increase the width of confidence bounds around estimates of population size, growth, probability of and time to quasi-extinction. However, traditional sensitivity analyses of stochastic matrix models only consider the sensitivity of the mean growth rate. We derive an exact method for calculating the sensitivity of the variance in population growth to changes in demographic parameters. Sensitivities of the variance also allow a new sensitivity calculation for the cumulative probability of quasi-extinction. We apply this new analysis tool to an empirical dataset on at-risk polar bears to demonstrate its utility in conservation biology We find that in many cases a change in life history parameters will increase both the mean and variance of population growth of polar bears. This counterintuitive behaviour of the variance complicates predictions about overall population impacts of management interventions. Sensitivity calculations for cumulative extinction risk factor in changes to both mean and variance, providing a highly useful quantitative tool for conservation management. The mean stochastic growth rate and its sensitivities do not fully describe the dynamics of population growth. The use of variance sensitivities gives a more complete understanding of population dynamics and facilitates the calculation of new sensitivities for extinction processes.

Can genetic estimators provide robust estimates of the effective number of breeders in small populations?

The effective population size (N(e)) is proportional to the loss of genetic diversity and the rate of inbreeding, and its accurate estimation is crucial for the monitoring of small populations. Here, we integrate temporal studies of the gecko Oedura reticulata, to compare genetic and demographic estimators of N(e). Because geckos have overlapping generations, our goal was to demographically estimate N(bI), the inbreeding effective number of breeders and to calculate the N(bI)/N(a) ratio (N(a) =number of adults) for four populations. Demographically estimated N(bI) ranged from 1 to 65 individuals. The mean reduction in the effective number of breeders relative to census size (N(bI)/N(a)) was 0.1 to 1.1. We identified the variance in reproductive success as the most important variable contributing to reduction of this ratio. We used four methods to estimate the genetic based inbreeding effective number of breeders N(bI(gen)) and the variance effective populations size N(eV(gen)) estimates from the genotype data. Two of these methods - a temporal moment-based (MBT) and a likelihood-based approach (TM3) require at least two samples in time, while the other two were single-sample estimators - the linkage disequilibrium method with bias correction LDNe and the program ONeSAMP. The genetic based estimates were fairly similar across methods and also similar to the demographic estimates excluding those estimates, in which upper confidence interval boundaries were uninformative. For example, LDNe and ONeSAMP estimates ranged from 14-55 and 24-48 individuals, respectively. However, temporal methods suffered from a large variation in confidence intervals and concerns about the prior information. We conclude that the single-sample estimators are an acceptable short-cut to estimate N(bI) for species such as geckos and will be of great importance for the monitoring of species in fragmented landscapes.

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.