Variance In Offspring Number Research Articles

Estimating effective population size using close‐kin mark–recapture

Abstract Close‐kin mark–recapture (CKMR) is a method that allows estimating population census size, among other parameters, through the observed number of pairs that are close‐kin including parent–offspring pairs (POPs) and half‐sibling pairs (HSPs). CKMR models are capable of estimating abundance, fecundity and survival at age using POPs and HSPs from different cohorts. The link between effective population size (Ne) and the number of ‘within‐cohort’ sibling pairs has been noted before but how to actually achieve such an estimate with CKMR has not been previously demonstrated. We show it is possible to use the number of ‘within‐cohort’ sibling pairs along with POPs and ‘different‐cohort’ HSPs to estimate Ne. These can be combined with an estimate of annual variance in number of offspring to estimate the lifetime variance of total reproductive success which can be used to find Ne. We show that the variance in number of offspring produced by adults in a given year is related to the within‐cohort comparisons. Our approach is demonstrated on an individual‐based simulation where we show that the CKMR Ne estimate offers similar results to Ne estimated from methods like linkage disequilibrium. Our methods allow estimating Ne using CKMR while also estimating demographic parameters.

The Ne/N ratio in applied conservation.

Recent developments within the IUCN and the Convention on Biological Diversity have affirmed the increasingly key role that effective population size (N e) and the effective size: census size ratio (N e/N) play in applied conservation and management of global biodiversity. This paper reviews and synthesizes information regarding the definition of N e and demographic and genetic methods for estimating effective size, census size, and their ratio. Emphasis is on single-generation estimates of contemporary N e/N, which are the most informative for practical applications. It is crucial to clearly define which individuals are included in the census size (N). Defining N as the number of adults alive at a given time facilitates comparisons across species. For a wide range of applications and experimental designs, inbreeding N e is simpler to calculate and interpret than variance N e. Effects of skewed sex ratio are generally modest, so most reductions to N e/N arise from overdispersed (greater-than-Poisson) variance in offspring number (). Even when fecundity changes with age, overdispersed within-age variance generally contributes most to overall , and both random and deterministic (mediated by selection) factors can be important. Most species are age-structured, so it is important to distinguish between effective size per generation (N e) and the effective number of breeders in one season or year (N b). Both N e and N b are important for applied conservation and management. For iteroparous species, a key metric is variance in lifetime reproductive success (), which can be affected by a variety of additional factors, including variation in longevity, skip or intermittent breeding, and persistent individual differences in reproductive success. Additional factors that can be important for some species are also discussed, including mating systems, population structure, sex reversal, reproductive compensation, captive propagation, and delayed maturity.

Robustness of Hill's overlapping-generation method for calculating Ne to extreme patterns of reproductive success.

For species with overlapping generations, the most widely used method to calculate effective population size (Ne) is Hill's, the key parameter for which is lifetime variance in offspring number ([Formula: see text]). Hill's model assumes a stable age structure and constant abundance, and sensitivity to those assumptions has been evaluated previously. Here I evaluate the robustness of Hill's model to extreme patterns of reproductive success, whose effects have not been previously examined: (1) very strong reproductive skew; (2) strong temporal autocorrelations in individual reproductive success; and (3) strong covariance of individual reproduction and survival. Genetic drift (loss of heterozygosity and increase in allele frequency variance) was simulated in age-structured populations using methods that generated no autocorrelations or covariances (Model NoCor); or created strong positive (Model Positive) or strong negative (Model Negative) temporal autocorrelations in reproduction and covariances between reproduction and survival. Compared to Model NoCor, the other models led to greatly elevated or reduced [Formula: see text], and hence greatly reduced or elevated Ne, respectively. A new index is introduced (ρα,α+), which is the correlation between (1) the number of offspring produced by each individual at the age at maturity (α), and (2) the total number of offspring produced during the rest of their lifetimes. Mean ρα,α+ was ≈0 under Model NoCor, strongly positive under Model Positive, and strongly negative under Model Negative. Even under the most extreme reproductive scenarios in Models Positive and Negative, when [Formula: see text] was calculated from the realized population pedigree and used to calculate Ne in Hill's model, the result accurately predicted the rate of genetic drift in simulated populations. These results held for scenarios where age-specific reproductive skew was random (variance ≈ mean) and highly overdispersed (variance up to 20 times the mean). Collectively, these results are good news for researchers as they demonstrate the robustness of Hill's model even in extreme reproductive scenarios.

Reproductive variance can drive behavioral dynamics

The concept of fitness is central to evolution, but it quantifies only the expected number of offspring an individual will produce. The actual number of offspring is also subject to demographic stochasticity-that is, randomness associated with birth and death processes. In nature, individuals who are more fecund tend to have greater variance in their offspring number. Here, we develop a model for the evolution of two types competing in a population of nonconstant size. The fitness of each type is determined by pairwise interactions in a prisoner's dilemma game, and the variance in offspring number depends upon its mean. Although defectors are preferred by natural selection in classical population models, since they always have greater fitness than cooperators, we show that sufficiently large offspring variance can reverse the direction of evolution and favor cooperation. Large offspring variance produces qualitatively new dynamics for other types of social interactions, as well, which cannot arise in populations with a fixed size or with a Poisson offspring distribution.

What Is Ne, Anyway?

Few doubt that effective population size (Ne) is one of the most important parameters in evolutionary biology, but how many can say they really understand the concept? Ne is the evolutionary analog of the number of individuals (or adults) in the population, N. Whereas ecological consequences of population size depend on N, evolutionary consequences (rates of loss of genetic diversity and increase in inbreeding; relative effectiveness of selection) depend on Ne. Formal definitions typically relate effective size to a key population genetic parameter, such as loss of heterozygosity or variance in allele frequency. However, for practical application to real populations, it is more useful to define Ne in terms of 3 demographic parameters: number of potential parents (adult N), and mean and variance in offspring number. Defined this way, Ne determines the rate of random genetic drift across the entire genome in the offspring generation. Other evolutionary forces (mutation, migration, selection)-together with factors such as variation in recombination rate-can also affect genetic variation, and this leads to heterogeneity across the genome in observed rates of genetic change. For some, it has been convenient to interpret this heterogeneity in terms of heterogeneity in Ne, but unfortunately, this has muddled the concepts of genetic drift and effective population size. A commonly repeated misconception is that Ne is the number of parents that actually contribute genes to the next generation (NP). In reality, NP can be smaller or larger than Ne, and the NP/Ne ratio depends on the sex ratio, the mean and variance in offspring number, and whether inbreeding or variance Ne is of interest.

The arithmetic mean of what? A Cautionary Tale about the Use of the Geometric Mean as a Measure of Fitness

Showing that the arithmetic mean number of offspring for a trait type often fails to be a predictive measure of fitness was a welcome correction to the philosophical literature on fitness. While the higher mathematical moments (variance, skew, kurtosis, etc.) of a probability-weighted offspring distribution can influence fitness measurement in distinct ways, the geometric mean number of offspring is commonly singled out as the most appropriate measure. For it is well-suited to a compounding (multiplicative) process and is sensitive to variance in offspring number. The geometric mean thus proves to be a predictively efficacious measure of fitness in examples featuring discrete generations and within- or between-generation variance in offspring output. Unfortunately, this advance has subsequently led some to conclude that the arithmetic mean is never (or at best infrequently) a good measure of fitness and that the geometric mean should accordingly be the default measure of fitness. We show not only that the arithmetic mean is a perfectly reasonable measure of fitness so long as one is clear about what it refers to (in particular, when it refers to growth rate), but also that it functions as a more general measure when properly interpreted. It must suffice as a measure of fitness in any case where the geometric mean has been effectively deployed as a measure. We conclude with a discussion about why the mathematical equivalence we highlight cannot be dismissed as merely of mathematical interest.

How much is enough? Sampling intensity influences estimates of reproductive variance in an introduced population.

Conservation introductions to islands and fenced enclosures are increasing as insitu mitigations fail to keep pace with population declines. Few studies consider the potential loss of genetic diversity and increased inbreeding if released individuals breed disproportionately. As funding is limited and post-release monitoring expensive for conservation programs, understanding how sampling effort influences estimates of reproductive variance is useful. To investigate this relationship, we used a well-studied population of Tasmanian devils (Sarcophilus harrisii) introduced to Maria Island, Tasmania, Australia. Pedigree reconstruction based on molecular data revealed high variance in number of offspring per breeder and high proportions of unsuccessful individuals. Computational subsampling of 20%, 40%, 60%, and 80% of observed offspring resulted in inaccurate estimates of reproductive variance compared to the pedigree reconstructed with all sampled individuals. With decreased sampling effort, the proportion of inferred unsuccessful individuals was overestimated and the variance in number of offspring per breeder was underestimated. To accurately estimate reproductive variance, we recommend sampling as many individuals as logistically possible during the early stages of population establishment. Further, we recommend careful selection of colonizing individuals as they may be disproportionately represented in subsequent generations. Within the conservation management context, our results highlight important considerations for sample collection and post-release monitoring during population establishment.

Differences in the effective population sizes of males and females do not require differences in their distribution of offspring number

Difference in male and female effective population sizes has, at times, been attributed to both sexes having unequal variance in their number of offspring. Such difference is paralleled by the relative effective sizes of autosomes, sex chromosomes, and mitochondrial DNA. I develop a simple framework to calculate the inbreeding effective population sizes for loci with different modes of inheritance. In this framework, I separate the effects due to mating strategy and those due to genetic transmission. I then show that, in addition to differences in the variance in offspring number, skew in the male/female effective sizes can also be caused by family composition. This approach can be used to illustrate the effect of induced behaviors on the relative male and female effective population sizes. In particular, I show the impact of the one-child policy formerly implemented in the People’s Republic of China on the relative male and female effective population sizes. Furthermore, I argue that, under some strong constraints on family structure, the concepts of male and female effective population sizes are invalid.

Nucleotide variation and balancing selection at the Ckma gene in Atlantic cod: analysis with multiple merger coalescent models.

High-fecundity organisms, such as Atlantic cod, can withstand substantial natural selection and the entailing genetic load of replacing alleles at a number of loci due to their excess reproductive capacity. High-fecundity organisms may reproduce by sweepstakes leading to highly skewed heavy-tailed offspring distribution. Under such reproduction the Kingman coalescent of binary mergers breaks down and models of multiple merger coalescent are more appropriate. Here we study nucleotide variation at the Ckma (Creatine Kinase Muscle type A) gene in Atlantic cod. The gene shows extreme differentiation between the North (Canada, Greenland, Iceland, Norway, Barents Sea) and the South (Faroe Islands, North-, Baltic-, Celtic-, and Irish Seas) with FST > 0.8 between regions whereas neutral loci show no differentiation. This is evidence of natural selection. The protein sequence is conserved by purifying selection whereas silent and non-coding sites show extreme differentiation. The unfolded site-frequency spectrum has three modes, a mode at singleton sites and two high frequency modes at opposite frequencies representing divergent branches of the gene genealogy that is evidence for balancing selection. Analysis with multiple-merger coalescent models can account for the high frequency of singleton sites and indicate reproductive sweepstakes. Coalescent time scales vary with population size and with the inverse of variance in offspring number. Parameter estimates using multiple-merger coalescent models show that times scales are faster than under the Kingman coalescent.

The Natures of Selection

Elliott Sober and his defenders think of selection, drift, mutation, and migration as distinct evolutionary forces. This paper exposes an ambiguity in Sober’s account of the force of selection: sometimes he appears to equate the force of selection with variation in fitness, sometimes with ‘selection for properties’. Sober’s own account of fitness as a property analogous to life-expectancy shows how the two conceptions come apart. Cases where there is selection against variance in offspring number also show that selection and drift cannot be distinguished in the way Sober hopes for. These issues have significance beyond the parochial matter of the coherence of Sober’s system. There is no good principled answer to the question of which features of a population should count among the contributors to fitness. This means there is no non-arbitrary account of the nature of selection.

Breeding Designs for Recombinant Inbred Advanced Intercross Lines

Recombinant inbred lines derived from an advanced intercross, in which multiple generations of mating have increased the density of recombination breakpoints, are powerful tools for mapping the loci underlying complex traits. We investigated the effects of intercross breeding designs on the utility of such lines for mapping. The simplest design, random pair mating with each pair contributing exactly two offspring to the next generation, performed as well as the most extreme inbreeding avoidance scheme at expanding the genetic map, increasing fine-mapping resolution, and controlling genetic drift. Circular mating designs offer negligible advantages for controlling drift and exhibit greatly reduced map expansion. Random-mating designs with variance in offspring number are also poor at increasing mapping resolution. Given equal contributions of each parent to the next generation, the constraint of monogamy has no impact on the qualities of the final population of inbred lines. We find that the easiest crosses to perform are well suited to the task of generating populations of highly recombinant inbred lines.

Selection Against Demographic Stochasticity in Age-Structured Populations

It has been shown that differences in fecundity variance can influence the probability of invasion of a genotype in a population; i.e., a genotype with lower variance in offspring number can be favored in finite populations even if it has a somewhat lower mean fitness than a competitor. In this article, Gillespie's results are extended to population genetic systems with explicit age structure, where the demographic variance (variance in growth rate) calculated in the work of Engen and colleagues is used as a generalization of "variance in offspring number" to predict the interaction between deterministic and random forces driving change in allele frequency. By calculating the variance from the life-history parameters, it is shown that selection against variance in the growth rate will favor a genotypes with lower stochasticity in age-specific survival and fertility rates. A diffusion approximation for selection and drift in a population with two genotypes with different life-history matrices (and therefore different mean growth rates and demographic variances) is derived and shown to be consistent with individual-based simulations. It is also argued that for finite populations, perturbation analyses of both the mean and the variance in growth rate may be necessary to determine the sensitivity of fitness to changes in the life-history parameters.

The Role of Life Cycle and Migration in Selection for Variance in Offspring Number

The Role of Life Cycle and Migration in Selection for Variance in Offspring Number

Natural Selection on Fecundity Variance in Subdivided Populations: Kin Selection Meets Bet Hedging

In a series of seminal articles in 1974, 1975, and 1977, J. H. Gillespie challenged the notion that the "fittest" individuals are those that produce on average the highest number of offspring. He showed that in small populations, the variance in fecundity can determine fitness as much as mean fecundity. One likely reason why Gillespie's concept of within-generation bet hedging has been largely ignored is the general consensus that natural populations are of large size. As a consequence, essentially no work has investigated the role of the fecundity variance on the evolutionary stable state of life-history strategies. While typically large, natural populations also tend to be subdivided in local demes connected by migration. Here, we integrate Gillespie's measure of selection for within-generation bet hedging into the inclusive fitness and game theoretic measure of selection for structured populations. The resulting framework demonstrates that selection against high variance in offspring number is a potent force in large, but structured populations. More generally, the results highlight that variance in offspring number will directly affect various life-history strategies, especially those involving kin interaction. The selective pressures on three key traits are directly investigated here, namely within-generation bet hedging, helping behaviors, and the evolutionary stable dispersal rate. The evolutionary dynamics of all three traits are markedly affected by variance in offspring number, although to a different extent and under different demographic conditions.

The Role of Life Cycle and Migration in Selection for Variance in Offspring Number

For two genotypes that have the same mean number of offspring but differ in the variance in offspring number, naturalselection will favor the genotype with lower variance. In such cases, the average growth rate is not sufficient as a measure of fitness or as a predictor of fixation probability. However, the effect of variance in offspring number on the fixationprobability of mutant strategies has been calculated under several scenarios with the general conclusion that variance in offspring number reduces fitness in proportion to the inverse of the population size [Gillespie, J., Genetics 76:601-606, 1974; Proulx, S.R., Theor. Popul. Biol. 58:33-47, 2000]. This relationship becomes more complicated under a metapopulation scenario where the "effective" population size depends on migration rate, population structure, and lifecycle. It is shown that in a life cycle where reproduction and migration (the birth-migration-regulation life cycle, or BMR)occur prior to density regulation within every deme, the fitness of a strategy depends on migration rate. When migration rates are near zero, the fitness of the strategy is determined by the size of individual demes, so that the strategy favoredin small populations tends to be fixed. As migration rate increases and approaches panmixis between demes, the fitness ofa reproductive strategy approaches what its value would be in a single, panmictic deme with a population size correspondingtothe census size of the metapopulation. Interestingly, when the life cycle is characterized by having density regulation in each deme prior to migration (the BRM life cycle) the fixation probability of a strategy is independent of migration rate. These results are found to be qualitatively consistent with the individual-based simulation results in Shpak [Theor. Biosci.124:65-85, 2005].

Evolution of variance in offspring number: The effects of population size and migration

It was shown by Gillespie [1974. Am. Nat. 108, 145–151], that if two genotypes produce the same average number of offspring on but have a different variance associated within each generation, the genotype with a lower variance will have a higher effective fitness. Specifically, the effective fitness is w e = w - σ 2 / N , where w is the mean fitness, σ 2 is the variance in offspring number, and N is the total population size. The model also predicts that if a strategy has a higher arithmetic mean fitness and a higher variance than the competitor, the outcome of selection will depend on the population size (with larger population sizes favoring the high-variance, high-mean genotype). This suggests that for metapopulations with large numbers of (relatively) small demes, a strategy with lower variance and lower mean may be favored if the migration rate is low while higher migration rates (consistent with a larger effective population size) favor the opposite strategy. Individual-based simulation confirms that this is indeed the case for an island model of migration, though the effect of migration differs greatly depending on whether migration precedes or follows selection. It is noted in the appendix that while Gillespie [1974. Am. Nat. 108, 145–151] does seem to be heuristically accurate, it is not clear that the definition of effective fitness follows from his derivation.

The ESS under Spatial Variation with Applications to Sex Allocation

I derive a new approximation which uses the backward Kolmogorov equation to describe evolution when individuals have variable numbers of offspring. This approximation is based on an explicit fixed population size assumption and therefore differs from previous models. I show that for individuals to accept an increase in the variance of offspring number, they must be compensated by an increase in mean offspring number. Based on this model and any given set of feasible alleles, an evolutionary stable strategy (ESS) can be found. Four types of ESS are possible and can be discriminated by graphical methods. These ESS values depend on population size, but population size can be reinterpreted as deme size in a structured population. I adapt this theory to the problem of sex allocation under variable returns to male and female function and derive the ESS sex allocation strategy. I show that allocation to the more variable sexual function should be reduced, but that this effect decreases as population size increases and as variability decreases. These results are compared with results from exact matrix models and computer simulations, all of which show strong congruence.

Life history evolution and demographic stochasticity

Can demographic stochasticity bias the evolution of life history traits? Under a neutral version of the Cole-Charnov-Schaffer model, variance in offspring number for both annuals and perennials depends on the precise values of fitness components. Either annuals or perennials may have the larger variance (for equal λ), depending on the importance of random survivalversus fixed reproduction. By extension, the variance in offspring number should generally depend on whether λ is mainly composed of highly variable elements or elements with limited variation. Thus, data about the variability of demographic parameters may be as important as data about their mean values.

Gene flow across tribal barriers and its effect among the Amazonian Içana river Indians.

Demographic information was obtained from 622 individuals of five communities of primarily Baniwa Amerindians living near the Içana river in Brazil. Four of these populations, plus another from the same area, were also studied genetically. The latter investigation included the blood and, in some cases, saliva of 531 subjects, variously tested in relation to 40 genetic systems. Demographically these groups are characterized by young age, high intertribal admixture, low non-Indian admixture, high exogamy but low marital distance and high inbreeding, high fertility but low variance in offspring number, and relatively low mortality. Their gene pool shows a peptidase B variant (PEPB2BAN1) and "private" polymorphism of carbonic anhydrase2 (CA2BAN1) until now observed only among them. Other distinctive characteristics are the low frequencies of LNS (0.08), LNs (0.09), RZ (0.01), RO or r(0.02), ACPA (0.08), GALTD (0.01), and the relatively high prevalences of Gm (0.05) and Gc1 (0.82). TfDchi occurs with a low prevalence (0.01). Genetic distance analysis reveals that the one Baniwa sample by history comprised of minimally admixed individuals is quite similar genetically to the Wapishana, another Arawak-speaking tribe some 900 km to the east, and that the genetic distances between the Baniwa communities reflect the amount of historical admixture in a way that indicates which should be excluded from considerations of intertribal genetic distances. Finally, the genetic relation of the Baniwa to the nearby tribes is examined.

Risk avoidance and nesting strategies

Gillespie (1974,1975) has shown that selection will tend to minimize variance in offspring number. It is shown that this effect is due to selection maximizing the average number of surviving offspring when there is density-dependent survival, and that it is unnecessary to invoke principles such as minimizing the variability of fitness (Real, 1980). The effect of this principle of selecting for laying several small clutches rather than one large clutch when there is predation on whole clutches is investigated. It is found that the selection pressure is weak, contrary to the conclusion of Rubenstein (1982), and is unlikely to be of evolutionary importance.