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.     

On two-sex models leading to stable populations

A mathematical, 2-sex, stable-population model that treats sex and age simultaneously was developed. The birth function is expressed in the form of an integral formula which when solved yields the intrinsic growth rate. Some of the concepts involved include the intrinsic sex ratio, the intrinsic age-specific birthrates, and the gross and net reproduction rates for both sexes. The new model demonstrates that both sexes can coexist in a stable population, whereas in the 1-sex model the intrinsic rates are internally inconsistent in regard to sex. 7 forms of the model are discussed and applied to data for the United States from 1963.     

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.     

Relationships in a simple harmonic mean two-sex fertility model

A continuous time two-sex stable population model that does not recognize age is examined under the “harmonic mean” consistency condition of equation (2). Solutions for the stable population intrinsic growth rate r and the sex composition 5 are presented in equations (5) and (6). The process of stabilization is examined, and it is shown that, given two basic constraints, any initial population sex composition will eventually converge to the stable population value. An algebraic solution for the discrete case where the sex ratio at birth is unity is presented and used to describe the trajectory to stability of several hypothetical populations. A closed form algebraic expression for the trajectory to stability is presented for the continuous model in the special case of no mortality.     

Sex in space: population dynamic consequences

Sex, so important in the reproduction of bigametic species, is nonetheless often ignored in explorations of the dynamics of populations. Using a growth model of dispersal-coupled populations we can keep track of fluctuations in numbers of females and males. The sexes may differ from each other in their ability to disperse and their sensitivity to population density. As a further complication, the breeding system is either monogamous or polygamous. We use the harmonic mean birth function to account for sex-ratio-dependent population growth in a Moran–Ricker population renewal process. Incorporating the spatial dimension stabilizes the dynamics of populations with monogamy as the breeding system, but does not stabilize the population dynamics of polygamous species. Most notably, in populations coupled with dispersal, where the sexes differ in their dispersal ability there are rarely stable and equal sex ratios. Rather, a two-point cycle, four-point cycle and eventually complex behaviour of sex-ratio dynamics will emerge with increasing birth rates. Monogamy often leads to less noisy sex-ratio dynamics than polygamy. In our model, the sex-ratio dynamics of coupled populations differ from those of an isolated population system, where a stable 50:50 sex ratio is achievable with equal density-dependence costs for females and males. When sexes match in their dispersal ability, population dynamics and sex-ratio dynamics of coupled populations collapse to those of isolated populations.     

A predictive relationship between population and genetic sex ratios in clonal species

Sexual reproduction depends on mate availability that is reflected by local sex ratios. In species where both sexes can clonally expand, the population sex ratio describes the proportion of males, including clonally derived individuals (ramets) in addition to sexually produced individuals (genets). In contrast to population sex ratio that accounts for the overall abundance of the sexes, the genetic sex ratio reflects the relative abundance of genetically unique mates, which is critical in predicting effective population size but is difficult to estimate in the field. While an intuitive positive relationship between population (ramet) sex ratio and genetic (genet) sex ratio is expected, an explicit relationship is unknown. In this study, we determined a mathematical expression in the form of a hyperbola that encompasses a linear to a nonlinear positive relationship between ramet and genet sex ratios. As expected when both sexes clonally have equal number of ramets per genet both sex ratios are identical, and thus ramet sex ratio becomes a linear function of genet sex ratio. Conversely, if sex differences in ramet number occur, this mathematical relationship becomes nonlinear and a discrepancy between the sex ratios amplifies from extreme sex ratios values towards intermediate values. We evaluated our predictions with empirical data that simultaneously quantified ramet and genet sex ratios in populations of several species. We found that the data support the predicted positive nonlinear relationship, indicating sex differences in ramet number across populations. However, some data may also fit the null model, which suggests that sex differences in ramet number were not extensive, or the number of populations was too small to capture the curvature of the nonlinear relationship. Data with lack of fit suggest the presence of factors capable of weakening the positive relationship between the sex ratios. Advantages of this model include predicting genet sex ratio using population sex ratios given known sex differences in ramet number, and detecting sex differences in ramet number among populations.     

Analysis of variance of first-year growth in captive garter snakes (Thamnophis elegans) by family and sex

Although body size in reptiles has important consequences for their ecology, it is quite variable in most species, presumably because growth in these animals is influenced by many factors. In this study, we analyse the effects of family membership and sex on first-year growth of garter snakes (Thamnophis elegans) held under identical conditions in the laboratory. Litters, but not the two sexes, differed significantly in mean size at birth. Larger snakes ate more food and had a higher absolute (but not relative) growth rate in length than smaller snakes. Thus, there was significant variation in length increase among litters, but not between the sexes. All else being equal, variance in size at birth could contribute significantly to variance in adult body size.     

花尾榛鸡的非对称性比和成体存活率

1999至2002年,在法国东南部的阿尔卑斯山,通过无线电追踪方法研究了花尾榛鸡的性比问题。首先,我们通过直接观测无线电标记雄性的配对比例,在消除可能的捕捉误差后,发现繁殖种群性比为1．26（即未配对的雄性占21％）。然后,我们估计了标记个体的存活率和繁殖成功率。在假设窝扩散时幼体的性比平衡的前提下,通过两性和两个年龄组（幼体,〈10月龄的个体;成体〉10月龄的个体）的确定性统计模型,得到的理论性比为1．22（即未配对的雄性占18％）,这一结果与实际观测的性比十分接近。我们认为,雌性死亡率高导致的两性存活率差异可以解释雄性性比偏高现象。我们进而讨论了花尾榛鸡整个分布区内居间强度的非对称性比的适应意义,以及在衰退的花尾榛鸡种群中雌性数量极端不足的后果[动物学报52（4）：655—662,2006]。     

Adaptive sex ratio variation in pre-industrial human (Homo sapiens) populations?

Sex allocation theory predicts that in a population with a biased operational sex ratio (OSR), parents will increase their fitness by adjusting the sex ratio of their progeny towards the rarer sex, until OSR has reached a level where the overproduction of either sex no longer increases a parent''s probability of having grandchildren. Furthermore, in a monogamous mating system, a biased OSR is expected to lead to lowered mean fecundity among individuals of the more abundant sex. We studied the influence of OSR on the sex ratio of newborns and on the population birth rate using an extensive data set (n = 14,420 births) from pre-industrial (1775-1850) Finland. The overall effect of current OSR on sex ratio at birth was significant, and in the majority of the 21 parishes included in this study, more sons were produced when males were rarer than females. This suggests that humans adjusted the sex ratio of their offspring in response to the local OSR to maximize the reproductive success of their progeny. Birth rate and, presumably, also population growth rate increased when the sex ratio (males:females) among reproductive age classes approached equality. However, the strength of these patterns varied across the parishes, suggesting that factors other than OSR (e.g. socioeconomic or environmental factors may also have influenced the sex ratio at birth and the birth rate.     

Sex ratio at reproductive age: changes over the last century in the Italian population

The radical improvement in living conditions experienced in Italy during the last century caused a reduction in male extra-mortality during the prereproductive years. As a consequence, a progressive increase in the sex ratio at the beginning of the reproductive age (15-19 years) occurred, so that in recent times the sex ratio in the young adult population has approached the almost constant value of 1.06 observed at birth. We calculated that the sex composition would be the same in newborns and in young adults in about one generation: obviously, we have to assume that the sex differentials in mortality and migration are constant over time. The 1:1 equilibrium between sexes, which maximizes reproductive success, occurred in the 15-19 age group at the beginning of the century and shifted to the 30-35 age group in the 1990s. We compared the 1993-1995 sex ratios in different age groups in European Union countries and observed that in Italy as well as in other Mediterranean countries the numerical equality between sexes is reached at 30-35 years of age, while in north-central Europe it is reached later, approximately at the end of reproductive life.     

Lack of Sex-Biased Maternal Investment in spite of a Skewed Birth Sex Ratio in White-Headed Langurs (Trachypithecus leucocephalus)

Numerous hypotheses have been developed to explain sex allocation. In male-dispersing, female cooperatively breeding species, the local resource competition model predicts male-biased birth sex ratio, the local resource enhancement model predicts female-biased birth sex ratio, and the population adjustment model predicts that biased birth sex ratio should not be favored if the two sexes are equally costly to rear. The male quality model predicts that, in polygynous species, females in better physical condition will either produce more sons than daughters or invest more heavily in sons than in daughters. White-headed langurs are a female philopatry and female cooperatively breeding species. During a 11-yr study, a total of 133 births were recorded, among which birth sex ratio (M:F = 73:49) was significantly male-biased. This is consistent with the prediction of the local resource competition model. On the other hand, if mothers balanced their investment between the two sexes, according to Fisher's population adjustment model, males should be the less-costly-to-rear sex. However, we found no sex difference for infant mortality (12.3% in males and 12.2% in females), and sons induced slightly longer interbirth interval (son: 26.4 ± 1.1 mo, daughter: 24.1 ± 0.6 mo) and lactational period (son: 20.9 ± 1.0 mo, daughters: 19.6 ± 0.5 mo) for their mothers. Thus, the population adjustment model was not supported by this study. The local resource enhancement model was not supported because birth sex ratio did not bias to females who provided more reproductive assistance. On the individual level, probit regression showed no relation between birth sex ratio and group size. Because the group size was considered to be negatively related to female physical condition, our study did not support the male-quality model. We suggested several possibilities to explain these results.     

A two-generation study of human sex-ratio variation.

We report here the first vertical population study of human sex-ratio variation. Sex-ratio data for 2 generations from Akita, Japan, have been analyzed. Parental age, birth order, sequences of the sexes at birth, and generations have no statistically significant effect on sex ratio. There is a slight excess of males at birth, as is typical for human sex-ratio studies. There is evidence of sex-ratio-dependent family planning. An analysis of vertical transmission of sex-ratio modifying factors that excludes effects of birth order in both the parental and offspring generations has detected a marginally significant paternal effect. Genetic variability of the sex ratio, if present at all, is of a very minor magnitude.     

Maternal expenditure in the polygynous and monomorphic guanaco: suckling behavior, reproductive effort, yearly variation, and influence on juvenile survival

We investigated patterns of maternal expenditure and its influenceon juvenile survival in the polygynous monomorphic guanaco (Lama guanicoe)in southern Chile from 1990 to 1994. Birth weight and growth rate(until age 1) of males and females were similar. Suckling ratesof males and females were not significantly different, althoughmothers of males rejected suckling attempts more often thanmothers of females during fall and winter. Mothers with sonsterminated suckling bouts in equal proportion as did motherswith daughters. Our estimated level of reproductive effort for guanacosfalls within the range of species exhibiting no sex-biased maternal expenditureon offspring. Mean yearly birth weight was negatively correlated withpopulation density. Mean suckling time throughout the year differedamong cohorts, as did the mean number of suckling attempts andrejected suckling attempts per hour throughout the year. Juvenilesurvival was estimated until age 1. Of the model with five covariatesincluding juvenile sex, birth weight, adult female aggressiontoward taggers, mean suckling time, and population density,only mean suckling time and population density were significantly relatedto survival. The risk ratio for mean suckling time indicatesthat the risk of mortality increases as suckling time increases,whereas the risk ratio for population density indicates thatthe risk of mortality decreases as population density increases.Under some conditions increasing population density may be correlatedwith lower offspring birth weight, yet enhanced juvenile survival.This effect on survival was possibly associated with the numberof predators on the study area from year to year.     

An evolutionary model of stature, age at first birth and reproductive success in Gambian women

We have built a model to predict optimal age at first birth for women in a natural fertility population. The only existing fully evolutionary model, based on Ache hunter-gatherers, argues that as women gain weight, their fertility (rate of giving birth) increases-thus age at first birth represents a trade-off between time allocated to weight gain and greater fertility when mature. We identify the life-history implications of female age at first birth in a Gambian population, using uniquely detailed longitudinal data collected from 1950 to date. We use height rather than weight as an indicator of growth as it is more strongly correlated with age at first birth. Stature does not greatly influence fertility in this population but has a significant effect on offspring mortality. We model age at first reproduction as a trade-off between the time spent growing and reduced infant mortality after maturation. Parameters derived from this population are fitted to show that the predicted optimal mean age of first birth, which maximizes reproductive success, is 18 years, very close to that observed. The reaction norm associated with variation in growth rate during childhood also satisfactorily predicts the variation in age at first birth.     

No differential consequences of reproduction according to sex in Juniperus communis var. depressa (Cupressaceae)

In dioecious plant species, males and females are thought to have dissimilar allocation patterns. Females are believed to invest more in reproduction and less in growth and maintenance than males. This differential investment between sexes could result in distinct growth patterns and contrasting survival rates, thereby affecting the sex ratio of a population and the age and size distribution of males and females, possibly leading to habitat segregation according to sex. These effects might become more apparent under particularly limiting conditions, such as in nutrient-deficient soils or in climatically stressed environments. To verify these predictions, growth patterns, microsite characteristics, and age and size distribution of male and female individuals were compared, and population sex ratio was determined in three populations of the dioecious shrub Juniperus communis var. depressa (Cupressaceae, Pinophyta) along a short latitudinal gradient on the eastern coast of Hudson Bay (Northern Québec, Canada). We found that the northernmost population had a male-biased sex ratio, but that the southernmost one had a higher proportion of females. Our results failed to reveal any significant differences in radial growth patterns, mean sensitivity, annual elongation of the main axis, and size and age frequency distribution between males and females in any population. Furthermore, there was no evidence of microhabitat segregation according to sex as indicated by the lack of differences in the physicochemical characteristics of the substrate under males and females. Clearly, the expected ecological consequences of a presumed greater investment of females in reproduction were not apparent even under the very stressful conditions prevailing on subarctic dunes. Many factors could reduce differences in the cost of reproduction between males and females, such as the number and quality of reproductive structures produced annually by individuals of each sex, the possible photosynthetic activity of the immature female cones, and the complexity of the source/sink relationship within individuals. Alternatively, there may be no differences between sexes in their reproductive investment.     

Parental age,birth order and tertiary sex ratio in the human population of Punjab,Pakistan

The data of this study, an extension of a previous study on secondary sex ratio in the human population of Muridke, Punjab, Pakistan, are based on the population of Muridke, 27 km north of Lahore, Punjab, Pakistan. Records of deaths of children, at later stages of birth, for different birth ranks, and that of maternal and paternal ages were made. 1000 families were scored for this study. Families providing the required information were included. Data for paternal age and maternal age combination consisted of 4807 total number of children of which 2586 were male. Paternal age and birth order combination was comprised of a total of 4405 children, containing 2316 males. Maternal age and birth order combination consisted of 4658 children, of which 2458 were males. The discrepancy in the number of children in the 3 types of combinations was due to the lack of required information in different groups. Sex ratio based on total number of males in relation to paternal age and maternal age was 0.54. Younger fathers (15-19 years) showed higher sex ratio (0.69). This dropped in paternal age groups 20-24 years (0.59) and 25-29 years (0.51). Younger mothers (15-19 years) showed higher sex ratio (0.62), declines in the age groups 20-24 years (0.52) and 25-29 years (0.51) and rise in age groups 35-39 years (0.55) and 40-44 years (0.54). Chi-square tests were carried out to compare the number of male and female offspring in the paternal age groups 15-19, 20-24, and 25-29 years. These showed highly significant deviation from the expected number. The higher age groups showed nonsignificant differences in the number of male and female offspring. Maternal age groups 15-19, 20-24, and 25-29 years showed highly significant differences in the male and female offspring and nonsignificant results in the higher age groups. Maternal age in relation to paternal age showed positive simple and partial correlations. Sex ratio for the total number of males based on paternal age and birth order was 0.52. 1st birth order showed higher sex ratio (0.55) and decreased in the 2nd (0.50) and 3rd birth orders (0.51), showed increase in the 4th birth order (0.53) and declines in the higher birth ranks. The number of male and female offspring in the birth orders 1, 2, and 3 showed significant differences, but in higher birth ranks the difference was insignificant. Paternal age and birth order indicated positive simple and partial correlations. Higher sex ratio (0.58) was seen in the 1st birth order and then it decreased in the 2nd (0.50) and 3rd (0.51) birth order. Chi-square tests carried out to compare the number of male and female offspring in borth orders 1, 2, and 3 showed highly significant differences but in higher birth ranks the difference was insignificant.     

Characteristics and environmental determinants of natality, growth and maturity in a natural population of the desert scorpion, Paruroctonus mesaensis (Scorpionida: Vaejovidae)

The scorpion, Paruroctonus mesaensis (Vaejovidae) has seven instar stages and moults six times. There is no post-reproductive moult. The average growth ratio between successive instars is 1–38. The average rate of weight increase is 000249/day. This rate fluctuates seasonally: it is highest in spring and lowest in winter and summer. Growth rate is a function of ambient burrow temperature and prey abundance. Body size is determinate. The growth rate of the pectines is significantly greater in males as compared with females. Analysis indicates that the pectines are a secondary sexual trait in males.
Animals mature from 19 to 24 months of age. Matings were observed from May through October. Gestation lasts 10–14 months. First surface appearance of newborn (instar II) is synchronous throughout the population and occurred in early August for five consecutive years. Some females were observed to be gravid for at least three consecutive years. Gravid females carry an average of 48–8 embryos. However, on the average only 19–9 young/female survive to appear on the surface. Factors which may be responsible for this observed decrease include resorption of embryos; death at birth, eclosion or first moult; and maternal and sibling cannibalism. The influence of various environmental factors on reproduction were analyzed. The number of newborn appearing on the surface and the population biomass of newborn showed a high positive correlation with precipitation and vegetation cover.     

Prospective cohort study of factors influencing the relative weights of the placenta and the newborn infant.

OBJECTIVES: To determine the demographic, environmental, and medical factors that influence the relative weights of the newborn infant and the placenta and compare this ratio with other factors known to predispose to adult ill health. DESIGN: Prospective cohort study. SETTING: The tertiary referral centre for perinatal care in Perth, Western Australia. SUBJECTS: 2507 pregnant women who delivered a single live infant at term. MAIN OUTCOME MEASURES: Placental weight, birth weight, and the ratio of placental weight to birth weight. RESULTS: By multiple regression analysis the placental weight to birthweight ratio was significantly and positively associated with gestational age, female sex, Asian parentage, increasing maternal body mass index, increased maternal weight at booking, lower socioeconomic status, maternal anaemia, and increasing number of cigarettes smoked daily. There were no consistent relations between the placental weight to birthweight ratio and measures of newborn size. CONCLUSIONS: The ratio of placental weight to birth weight is not an accurate marker of fetal growth. In its role as a predictor of adult disease the ratio may be acting as a surrogate for other factors which are already known to influence health and may act before or after birth. Determining the role that relative growth rates of the fetus and placenta have in predisposing to adult disease requires prospective study to account for the many confounding variables which complicate this hypothesis.     

Breeding durations as estimators of adult sex ratios and population size

Adult sex ratios (ASRs) and population size are two of the most fundamental parameters in population biology, as they are the main determinants of genetic and demographic viability, and vulnerability of a population to stochastic events. Underpinning the application of population viability analysis for predicting the extinction risk of populations is the need to accurately estimate parameters that determine the viability of populations (i.e. the ASR and population size). Here we demonstrate that a lack of temporal information can confound estimation of both parameters. Using acoustic telemetry, we compared differences in breeding durations of both sexes for a giant Australian cuttlefish Sepia apama breeding aggregation to the strongly male-biased operational sex ratio (4:1), in order to estimate the population ASR. The ratio of breeding durations between sexes was equal to the operational sex ratio, suggesting that the ASR is not strongly male-biased, but balanced. Furthermore, the short residence times of individuals at the breeding aggregation suggests that previous density-based abundance estimates have significantly underestimated population size. With the current wide application of population viability analysis for predicting the extinction risk of populations, tools to improve the accuracy of such predictions are vital. Here we provide a new approach to estimating the fundamental ASR parameter, and call for temporal considerations when estimating population size.     

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/.