Estimating Density Dependence, Environmental Variance, and Long-Term Selection on a Stage-Structured Life History.

AbstractA method for analyzing long-term demographic data on density-dependent stage-structured populations in a stochastic environment is derived to facilitate comparison of populations and species with different life histories. We assume that a weighted sum of stage abundances, N, exerts density dependence on stage-specific vital rates of survival and reproduction and that N has a small or moderate coefficient of variation. The dynamics of N are approximated as a univariate stochastic process governed by three key parameters: the density-independent growth rate, the net density dependence, and environmental variance in the life history. We show how to estimate the relative weighs of stages in N and the key parameters. Life history evolution represents a stochastic maximization of a simple function of the key parameters. The long-term selection gradient on the life history can be expressed as a vector of sensitivities of this function with respect to density-independent, density-dependent, and stochastic components of the vital rates. To illustrate the method, we analyze 38 years of demographic data on a great tit population, estimating the key parameters, which accurately predict the observed mean, coefficient of variation, and fluctuation rate of N; we also evaluate the long-term selection gradient on the life history.

Parasite prevalence increases with temperature in an avian metapopulation in northern Norway.

Climate and weather conditions may have substantial effects on the ecology of both parasites and hosts in natural populations. The strength and shape of the effects of weather on parasites and hosts are likely to change as global warming affects local climate. These changes may in turn alter fundamental elements of parasite-host dynamics. We explored the influence of temperature and precipitation on parasite prevalence in a metapopulation of avian hosts in northern Norway. We also investigated if annual change in parasite prevalence was related to winter climate, as described by the North Atlantic Oscillation (NAO). We found that parasite prevalence increased with temperature within-years and decreased slightly with increasing precipitation. We also found that a mild winter (positive winter NAO index) was associated with higher mean parasite prevalence the following year. Our results indicate that both local and large scale weather conditions may affect the proportion of hosts that become infected by parasites in natural populations. Understanding the effect of climate and weather on parasite-host relationships in natural populations is vital in order to predict the full consequence of global warming.

Insights into the genetic architecture of morphological traits in two passerine bird species.

Knowledge about the underlying genetic architecture of phenotypic traits is needed to understand and predict evolutionary dynamics. The number of causal loci, magnitude of the effects and location in the genome are, however, still largely unknown. Here, we use genome-wide single-nucleotide polymorphism (SNP) data from two large-scale data sets on house sparrows and collared flycatchers to examine the genetic architecture of different morphological traits (tarsus length, wing length, body mass, bill depth, bill length, total and visible badge size and white wing patches). Genomic heritabilities were estimated using relatedness calculated from SNPs. The proportion of variance captured by the SNPs (SNP-based heritability) was lower in house sparrows compared with collared flycatchers, as expected given marker density (6348 SNPs in house sparrows versus 38 689 SNPs in collared flycatchers). Indeed, after downsampling to similar SNP density and sample size, this estimate was no longer markedly different between species. Chromosome-partitioning analyses demonstrated that the proportion of variance explained by each chromosome was significantly positively related to the chromosome size for some traits and, generally, that larger chromosomes tended to explain proportionally more variation than smaller chromosomes. Finally, we found two genome-wide significant associations with very small-effect sizes. One SNP on chromosome 20 was associated with bill length in house sparrows and explained 1.2% of phenotypic variation (VP), and one SNP on chromosome 4 was associated with tarsus length in collared flycatchers (3% of VP). Although we cannot exclude the possibility of undetected large-effect loci, our results indicate a polygenic basis for morphological traits.

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 Ne 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 Ne /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 Ne /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.

Spatial variation in senescence rates in a bird metapopulation.

Investigating factors which affect the decline in survival with age, i.e. actuarial senescence, is important in order to understand how demographic rates vary in wild populations. Although the evidence for the occurrence of actuarial senescence in wild populations is growing, very few studies have compared actuarial senescence rates between wild populations of the same species. We used data from a long-time study of demography of house sparrows (Passer domesticus) to investigate differences in rates of actuarial senescence between habitats and sub-populations. We also investigated whether rates of actuarial senescence differed between males and females. We found that rates of actuarial senescence showed large spatial variation. We also found that the onset of actuarial senescence varied between sub-populations. However, these differences were not significantly explained by a general difference in habitat type. We also found no significant difference in actuarial senescence rates between males and females. This study shows that senescence rates in natural populations may vary significantly between sub-populations and that failing to account for such differences may give a biased estimate of senescence rates of a species.

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.

Sexual variation in heritability and genetic correlations of morphological traits in house sparrow (Passer domesticus).

Estimates of genetic components are important for our understanding of how individual characteristics are transferred between generations. We show that the level of heritability varies between 0.12 and 0.68 in six morphological traits in house sparrows (Passer domesticus L.) in northern Norway. Positive and negative genetic correlations were present among traits, suggesting evolutionary constraints on the evolution of some of these characters. A sexual difference in the amount of heritable genetic variation was found in tarsus length, wing length, bill depth and body condition index, with generally higher heritability in females. In addition, the structure of the genetic variance-covariance matrix for the traits differed between the sexes. Genetic correlations between males and females for the morphological traits were however large and not significantly different from one, indicating that sex-specific responses to selection will be influenced by intersexual differences in selection differentials. Despite this, some traits had heritability above 0.1 in females, even after conditioning on the additive genetic covariance between sexes and the additive genetic variances in males. Moreover, a meta-analysis indicated that higher heritability in females than in males may be common in birds. Thus, this indicates sexual differences in the genetic architecture of birds. Consequently, as in house sparrows, the evolutionary responses to selection will often be larger in females than males. Hence, our results suggest that sex-specific additive genetic variances and covariances, although ignored in most studies, should be included when making predictions of evolutionary changes from standard quantitative genetic models.

Estimating density dependence in time-series of age-structured populations.

For a life history with age at maturity alpha, and stochasticity and density dependence in adult recruitment and mortality, we derive a linearized autoregressive equation with time-lags of from 1 to alpha years. Contrary to current interpretations, the coefficients for different time-lags in the autoregressive dynamics do not simply measure delayed density dependence, but also depend on life-history parameters. We define a new measure of total density dependence in a life history, D, as the negative elasticity of population growth rate per generation with respect to change in population size, D = - partial differential lnlambda(T)/partial differential lnN, where lambda is the asymptotic multiplicative growth rate per year, T is the generation time and N is adult population size. We show that D can be estimated from the sum of the autoregression coefficients. We estimated D in populations of six avian species for which life-history data and unusually long time-series of complete population censuses were available. Estimates of D were in the order of 1 or higher, indicating strong, statistically significant density dependence in four of the six species.

Estimating density dependence from population time series using demographic theory and life-history data.

For populations with a density-dependent life history reproducing at discrete annual intervals, we analyze small or moderate fluctuations in population size around a stable equilibrium, which is applicable to many vertebrate populations. Using a life history having age at maturity alpha, with stochasticity and density dependence in adult recruitment and mortality, we derive a linearized autoregressive equation with time lags from 1 to alpha yr. Contrary to current interpretations, the coefficients corresponding to different time lags in the autoregressive dynamics are not simply measures of delayed density dependence but also depend on life-history parameters. The theory indicates that the total density dependence in a life history, D, should be defined as the negative elasticity of population growth rate per generation with respect to change in population size, [Formula: see text], where lambda is the asymptotic multiplicative growth rate per year, T is the generation time, and N is adult population size. The total density dependence in the life history, D, can be estimated from the sum of the autoregression coefficients. We estimate D in populations of seven vertebrate species for which life-history studies and unusually long time series of complete population censuses are available. Estimates of D were statistically significant and large, on the order of 1 or higher, indicating strong density dependence in five of the seven species. We also show that life history can explain the qualitative features of population autocorrelation functions and power spectra and observations of increasing empirical variance in population size with increasing length of time series.

Predicting the time to quasi-extinction for populations far below their carrying capacity.

Populations threatened by extinction are often far below their carrying capacity. A population collapse or quasi-extinction is defined to occur when the population size reaches some given lower density. If this density is chosen to be large enough for the demographic stochasticity to be ignored compared to environmental stochasticity, then the logarithm of the population size may be modelled by a Brownian motion until quasi-extinction occurs. The normal-gamma mixture of inverse Gaussian distributions can then be applied to define prediction intervals for the time to quasi-extinction in such processes. A similar mixture is used to predict the population size at a finite time for the same process provided that quasi-extinction has not occurred before that time. Stochastic simulations indicate that the coverage of the prediction interval is very close to the probability calculated theoretically. As an illustration, the method is applied to predict the time to extinction of a declining population of white stork in southwestern Germany.

Ecology. Weather ruins winter vacations.

There has been increasing concern over the decline in many migratory bird species. As Saether discusses in his Perspective, evidence is accumulating (Sillett et al.) that climate change resulting from the El Niño Southern Oscillation affects both the survival rate of adult birds at tropical wintering sites and their reproductive rate at summer breeding grounds in the Northern Hemisphere.

Estimating the time to extinction in an island population of song sparrows.

We estimated and modelled how uncertainties in stochastic population dynamics and biases in parameter estimates affect the accuracy of the projections of a small island population of song sparrows which was enumerated every spring for 24 years. The estimate of the density regulation in a theta-logistic model (theta = 1.09 suggests that the dynamics are nearly logistic, with specific growth rate r1 = 0.99 and carrying capacity K = 41.54. The song sparrow population was strongly influenced by demographic (sigma2(d) = 0.66) and environmental (sigma2(d) = 0.41) stochasticity. Bootstrap replicates of the different parameters revealed that the uncertainties in the estimates of the specific growth rate r1 and the density regulation theta were larger than the uncertainties in the environmental variance sigma2(e) and the carrying capacity K. We introduce the concept of the population prediction interval (PPI), which is a stochastic interval which includes the unknown population size with probability (1 - alpha). The width of the PPI increased rapidly with time because of uncertainties in the estimates of density regulation as well as demographic and environmental variance in the stochastic population dynamics. Accepting a 10% probability of extinction within 100 years, neglecting uncertainties in the parameters will lead to a 33% overestimation of the time it takes for the extinction barrier (population size X = 1) to be included into the PPI. This study shows that ignoring uncertainties in population dynamics produces a substantial underestimation of the extinction risk.

Environmental stochasticity and extinction risk in a population of a small songbird, the great tit.

Using a long-term demographic data set, we estimated the separate effects of demographic and environmental stochasticity in the growth rate of the great tit population in Wytham Wood, United Kingdom. Assuming logistic density regulation, both the demographic (sigma2d = 0.569) and environmental (sigma2e = 0.0793) variance, with interactions included, were significantly greater than zero. The estimates of the demographic variance seemed to be relatively insensitive to the length of the study period, whereas reliable estimates of the environmental variance required long time series (at least 15 yr of data). The demographic variance decreased significantly with increasing population density. These estimates are used in a quantitative analysis of the demographic factors affecting the risk of extinction of this population. The very long expected time to extinction of this population (approximately 10(19) yr) was related to a relatively large population size (>/=120 pairs during the study period). However, for a given population size, the expected time to extinction was sensitive to both variation in population growth rate and environmental stochasticity. Furthermore, the form of the density regulation strongly affected the expected time to extinction. Time to extinction decreased when the maximum density regulation approached K. This suggests that estimates of viability of small populations should be given both with and without inclusion of density dependence.

Environmental stochasticity and population dynamics of large herbivores: a search for mechanisms.

Recently, the results from several long-term individual-based population studies of ungulates have been published. One major conclusion is that the population dynamics of ungulates in predator-free environments is strongly influenced by a combination of stochastic variation in the environment, and population density. Both density dependence and environmental stochasticity operate through changes in life history traits, correlated with variation in body weight. This generates delays in the response of the population to changes in environment. In the absence of predation, a stable equilibrium is therefore unlikely to exist between an ungulate population and its food resources. This thorough understanding of the mechanisms generating population fluctuations suggests that studies of ungulates will provide an important source for examining effects of long-term changes in the environment, for instance, resulting from a climatic change.

The adaptive significance of reproductive strategies in ungulates.

We examine the relation between litter size, gestation length, neonate mass and growth rate among ungulates. By using a recent method for analysing comparative data, we show that ungulates can be divided along a slow-fast continuum, even after accounting for the effects of maternal body mass and common ancestry. Some species produce many small offspring during a short period, whereas others take a long time to raise a single large offspring. These differences in life-history strategy are associated with diet, i.e. browsers have relatively larger litters and smaller neonates than grazers.

Pattern of covariation between life-history traits of European birds.

A large amount of variation is found in most reproductive traits of birds. Clutch size for instance, can vary from 1 to 15 between species of similar body weight. The adaptive significance of this variation is only poorly understood. According to life-history theory, large clutch size and early onset of reproduction are expected when the chances of survival are low. There is some support for the existence of such a relationship from studies of single species. Here I present evidence that, in European birds, clutch size is increased, and onset of reproduction occurs earlier in life, when the probability of survival is low.

The evolutionary significance of sexual selection.

A model for the joint evolution of a secondary sexual male trait Z and a female mating preference Y is discussed. Recurrence relations for the moments of (Z, Y) are given under the assumption that the traits are binormally distributed. It is shown that female preference for a male character can lead to an equilibrium distribution of the male trait with non-zero variances. The conditions under which the distribution is stable, are given. Unstable situations, in which a continued exaggeration of the male trait occurs, are described. It is demonstrated that the effect of sexual selection on the evolution of the male trait depends on the intensity of natural selection, i.e. the effect of the sexual selection increases when the intensity of natural selection is reduced. The effect of the female preference on the male trait also increases with increasing availability of males. This provides a link to several ecological conditions which have generally been known to be correlated with the degree of sexual selection. Furthermore, it is demonstrated that perturbations away from the equilibrium may cause rapid evolution of the male character, eventually leading to speciation.