Demographic stochasticity and temporal autocorrelation in the dynamics of structured populations

Populations can show temporal autocorrelation in the dynamics arising from different mechanisms, including fluctuations in the demographic structure. This autocorrelation is often treated as a complicating factor in the analyses of stochastic population growth and extinction risk. However, it also reflects important information about the demographic structure. Here, we consider how temporal autocorrelation is related to demographic stochasticity in structured populations. Demographic stochasticity arises from inherent randomness in the demographic processes of individuals, like survival and reproduction, and the resulting impact on population growth is measured by the demographic variance. Earlier studies have shown that population structure have positive or negative effects on the demographic variance compared to a model where the structure is ignored. Here, we derive a new expression for the demographic variance of a structured population, using the temporal autocorrelation function of the population growth rate. We show that the relative difference in demographic variance when the structure is included or ignored (the effect of structure on demographic variance) is approximately twice the sum of the autocorrelations. We demonstrate the result for a simple hypothetical example, as well as a set of empirical examples using age‐structured models of 24 mammals from the demographic database COMADRE. In the empirical examples, the sum of the autocorrelation function was negative in all cases, indicating that age structure generally has a negative effect on the demographic variance (i.e. the demographic variance is lower compared to that of a model where the structure is ignored). Other kinds of structure, such as spatial heterogeneity affecting fecundity, can have positive effects on the demographic variance, and the sum of the autocorrelations will then be positive. These results yield new insights into the complex interplay between population structure, demographic variance, and temporal autocorrelation, that shapes the population dynamics and extinction risk of populations.     

Extinction in relation to demographic and environmental stochasticity in age-structured models

The demographic variance of an age-structured population is defined. This parameter is further split into components generated by demographic stochasticity in each vital rate. The applicability of these parameters are investigated by checking how an age-structured population process can be approximated by a diffusion with only three parameters. These are the deterministic growth rate computed from the expected projection matrix and the environmental and demographic variances. We also consider age-structured populations where the fecundity at any stage is either zero or one, and there is neither environmental stochasticity nor dependence between individual fecundity and survival. In this case the demographic variance is uniquely determined by the vital rates defining the projection matrix. The demographic variance for a long-lived bird species, the wandering albatross in the southwestern part of the Indian Ocean, is estimated. We also compute estimates of the age-specific contributions to the total demographic variance from survival, fecundity and the covariance between survival and fecundity.     

An integrated population model sheds light on the complex population dynamics of a unique colonial breeder

Climate models forecast increasing climatic variation and more extreme events, which could increase the variability in animal demographic rates. More variable demographic rates generally lead to lower population growth and can be detrimental to wild populations, especially if the particular demographic rates affected are those to which population growth is most sensitive. We investigated the population dynamics of a metapopulation of 25 colonies of a semi-arid bird species, the sociable weaver Philetairus socius, and how it was influenced by seasonal weather during 1993–2014. We constructed an integrated population model which estimated population sizes similar to observed population counts, and allowed us to estimate annual fecundity and recruitment. Variance in fecundity contributed most to variance in population growth, which showed no trend over time. No weather variables explained overall demographic variation at the population level. However, a separate analysis of the largest colony showed a clear decline with a high extinction probability (0.05 to 0.33) within 5 years after the study period. In this colony, juvenile survival was lower when summers were hot, and adult survival was lower when winters were cold. Rainfall was also negatively correlated with adult survival. These weather effects could be due to increased physiological demands of thermoregulation and rainfall-induced breeding activity. Our results suggest that the dynamics of the population on the whole are buffered against current weather variation, as individual colonies apparently react in different ways. However, if more and increasingly extreme weather events synchronize colony dynamics, they are likely to have negative effects.     

A DEMOGRAPHIC TRANSITION ALTERED THE STRENGTH OF SELECTION FOR FITNESS AND AGE‐SPECIFIC SURVIVAL AND FERTILITY IN A 19TH CENTURY AMERICAN POPULATION

Modernization has increased longevity and decreased fertility in many human populations, but it is not well understood how or to what extent these demographic transitions have altered patterns of natural selection. I integrate individual‐based multivariate phenotypic selection approaches with evolutionary demographic methods to demonstrate how a demographic transition in 19th century female populations of Utah altered relationships between fitness and age‐specific survival and fertility. Coincident with this demographic transition, natural selection for fitness, as measured by the opportunity for selection, increased by 13% to 20% over 65 years. Proportional contributions of age‐specific survival to total selection (the complement to age‐specific fertility) diminished from approximately one third to one seventh following a marked increase in infant survival. Despite dramatic reductions in age‐specific fertility variance at all ages, the absolute magnitude of selection for fitness explained by age‐specific fertility increased by approximately 45%. I show that increases in the adaptive potential of fertility traits followed directly from decreased population growth rates. These results suggest that this demographic transition has increased the adaptive potential of the Utah population, intensified selection for reproductive traits, and de‐emphasized selection for survival‐related traits.     

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.     

Drivers of megaherbivore demographic fluctuations: inference from elephants

Environmentally induced variation in survival and fecundity generates demographic fluctuations that affect population growth rate. However, a general pattern of the comparative influence of variation in fecundity and juvenile survival on elephant population dynamics has not been investigated at a broad scale. We evaluated the relative importance of conception, gestation, first year survival and subsequent survivorship for controlling demographic variation by exploring the relationship between past environmental conditions determined by integrated normalized difference vegetation index (INDVI) and the shape of age distributions at 17 sites across Africa. We showed that, generally, INDVI during gestation best explained anomalies in age structure. However, in areas with low mean annual rainfall, INDVI during the first year of life was critical. The results challenge Eberhardt's paradigm for population analysis that suggests that populations respond to limited resource availability through a sequential decrease in juvenile survival, reproductive rate and adult survival. Contrastingly, elephants appear to respond first through a reduction in reproductive rate. We conclude that this discrepancy is likely due to the evolutionary significance of extremely large body size – an adaptation that increases survival rate but decreases reproductive potential. Other megaherbivores may respond similarly to resource limitation due to similarities in population dynamics. Knowing how vital rates vary with changing environmental conditions will permit better forecasts of the trajectories of megaherbivore populations.     

Demographic variance in heterogeneous populations: matrix models and sensitivity analysis

The demographic consequences of stochasticity in processes such as survival and reproduction are modulated by the heterogeneity within the population. Therefore, to study effects of stochasticity on population growth and extinction risk, it is critical to use structured population models in which the most important sources of heterogeneity (e.g. age, size, developmental stage) are incorporated as i‐state variables. Demographic stochasticity in heterogeneous populations has often been studied using one of two approaches: multitype branching processes and diffusion approximations. Here, we link these approaches, through the demographic stochasticity in age‐ or stage‐structured matrix population models. We derive the demographic variance, σ²_d, which measures the per capita contribution to the variance in population growth increment, and we show how it can be decomposed into contributions from transition probabilities and fertility across ages or stages. Furthermore, using matrix calculus we derive the sensitivity of σ²_d to age‐ or stage‐specific mortality and fertility. We apply the methods to an extensive set of data from age‐classified human populations (long‐term time‐series for Sweden, Japan and the Netherlands; two hunter–gatherer populations, and the high‐fertility Hutterites), and to a size‐classified population of the herbaceous plant Calathea ovandensis. For the human populations our analysis reveals substantial temporal changes in the demographic variance as well as its main components across age. These new methods provide a powerful approach for calculating the demographic variance for any structured model, and for analyzing its main components and sensitivities. This will make possible new analyses of demographic variance across different kinds of heterogeneity in different life cycles, which will in turn improve our understanding of mechanisms underpinning extinction risk and other important biological outcomes.     

Linking demographic responses and life history tactics from longitudinal data in mammals

In stochastic environments, a change in a demographic parameter can influence the population growth rate directly or via a resulting impact on age structure. Stochastic elasticity of the long‐run stochastic growth rate λ_s to a demographic parameter offers a suitable way to measure the overall demographic response because it includes both the direct effect of changing the demographic parameter and its indirect effect through changes in the age structure. From 25 mammalian populations with contrasting life histories, we investigated how pace of life and population growth rate influence the demographic responses (measured as the relative contributions of the direct and indirect components of stochastic elasticity on λ_s). We found that in short‐lived species, the change in population structure resulting from an increase in yearling survival leads to an additional increase in λ_s, whereas in long‐lived species, the same change in population structure leads to a decrease. Short‐lived species thus display a boom‐bust life history strategy contrary to long‐lived species, for which the long lifespan dampens the demographic consequences of changing age structure. Irrespective of the species’ life history strategy, the change in population age structure resulting from an increase in adult survival leads to an additional increase in λ_s due to an increase of the proportion of mature individuals in the population. On the contrary, a change in population age structure resulting from an increase of reproductive performance leads to a decrease in λ_s that is due to the increase of the proportion of immature individuals in the population. Our comparative analysis of stochastic elasticity patterns in mammals shows the existence of different demographic responses to changes in age structure between short‐ and long‐lived species, which improves our understanding of population dynamics in variable environments in relation to the species‐specific pace of life.     

Local population dynamics and the impact of scale and isolation: a study on different little owl populations

The understanding of how variation of demographic rates translates into variation of population growth is a central aim in population ecology. Besides stochastic and deterministic factors, the spatial extent and the isolation of a local population may have an impact on the contribution of the different demographic components. Using long-term demographic data we performed retrospective population analyses of four little owl ( Athene noctua ) populations with differential spatial extent and degree of isolation to assess the contribution of demographic rates to the variation of the growth rate (λ) of each local population and to the difference of λ among populations. In all populations variation of fecundity contributed least to variation of λ, and variation of adult survival contributed most to variation of λ in three of four populations. Between population comparisons revealed that differences mainly stem from differences of immigration and juvenile local survival. The relative importance of immigration to λ tended to decrease with increasing spatial extent and isolation of the local populations. None of the four local populations was self-sustainable. Because the local populations export and import individuals, they can be considered as open recruitment systems in which part of the recruited breeding birds are not produced locally. The spatial extent and the degree of isolation of a local population have an impact on local population dynamics; hence these factors need to be considered in studies about local population dynamics and for deriving conservation measures.     

Density‐dependent regulation of the critically endangered black rhinoceros population in Ithala Game Reserve,South Africa

Ensuring the persistence of populations of endangered species requires an understanding of, and response to, the causes of population declines. Species occurring in small populations are vulnerable to stochastic problems that are environmental, demographic, or genetic in nature and can reduce survival as much as the deterministic threats of habitat degradation. Critically endangered black rhinoceros (Diceros bicornis) populations declined throughout Africa since 1960, with numbers steadily increasing at a continental level, but remaining lower than three generations ago. However, size, demographics, trends, and factors affecting these, are poorly known. We used 18 years (1990–2008) of long‐term sightings data from Ithala Game Reserve, KwaZulu‐Natal, South Africa, to determine population estimates, growth rate and fecundity over time, as well as sex and age structure and age‐specific probabilities of survival. Calf survivorship between the ages of 0 and 1 year was 74% for females and 94% for males. Age‐specific survivorship for both sexes was significantly higher from yearling to subadult age‐classes (1–6 years) than for adults (7–30 years). The most frequent cause of mortality was attributed to unknown causes while fighting injuries was recorded as the second most common cause of mortality, particularly among subadult and adult males. There was no significant difference in the sex ratios at birth, although the proportion of females in the population was 0.58. There was strong evidence for density‐dependent regulation, with density in conception year a key driver of population performance (birth rate). The population does not appear to be at ecological carrying capacity; however, social effects are delaying conception. To mitigate density‐dependent social effects, we recommend an adaptive management strategy of pre‐selecting individuals for removal from the reserve, so as to maintain stability in the social organization of the population.     

Accounting for stochasticity in demographic compensation along the elevational range of an alpine plant

Demographic compensation arises when vital rates change in opposite directions across populations, buffering the variation in population growth rates, and is a mechanism often invoked to explain the stability of species geographic ranges. However, studies on demographic compensation have disregarded the effects of temporal variation in vital rates and their temporal correlations, despite theoretical evidence that stochastic dynamics can affect population persistence in temporally varying environments. We carried out a seven‐year‐long demographic study on the perennial plant Arabis alpina (L.) across six populations encompassing most of its elevational range. We discovered demographic compensation in the form of negative correlations between the means of plant vital rates, but also between their temporal coefficients of variation, correlations and elasticities. Even if their contribution to demographic compensation was small, this highlights a previously overlooked, but potentially important, role of stochastic processes in stabilising population dynamics at range margins.     

Evolutionary tracking is determined by differential selection on demographic rates and density dependence

Recent ecological forecasts predict that ~25% of species worldwide will go extinct by 2050. However, these estimates are primarily based on environmental changes alone and fail to incorporate important biological mechanisms such as genetic adaptation via evolution. Thus, environmental change can affect population dynamics in ways that classical frameworks can neither describe nor predict. Furthermore, often due to a lack of data, forecasting models commonly describe changes in population demography by summarizing changes in fecundity and survival concurrently with the intrinsic growth rate (r). This has been shown to be an oversimplification as the environment may impose selective pressure on specific demographic rates (birth and death) rather than directly on r (the difference between the birth and death rates). This differential pressure may alter population response to density, in each demographic rate, further diluting the information combined to produce r. Thus, when we consider the potential for persistence via adaptive evolution, populations with the same r can have different abilities to persist amidst environmental change. Therefore, we cannot adequately forecast population response to climate change without accounting for demography and selection on density dependence. Using a continuous‐time Markov chain model to describe the stochastic dynamics of the logistic model of population growth and allow for trait evolution via mutations arising during birth events, we find persistence via evolutionary tracking more likely when environmental change alters birth rather than the death rate. Furthermore, species that evolve responses to changes in the strength of density dependence due to environmental change are less vulnerable to extinction than species that undergo selection independent of population density. By incorporating these key demographic considerations into our predictive models, we can better understand how species will respond to climate change.     

Quantifying the importance of geographic replication and representativeness when estimating demographic rates,using a coastal species as a case study

Demographic rates are rarely estimated over an entire species range, limiting empirical tests of ecological patterns and theories, and raising questions about the representativeness of studies that use data from a small part of a range. The uncertainty that results from using demographic rates from just a few sites is especially pervasive in population projections, which are critical for a wide range of questions in ecology and conservation. We developed a simple simulation to quantify how this lack of geographic representativeness can affect inferences about the global mean and variance of growth rates, which has implications for the robust design of a wide range of population studies. Using a coastal songbird, saltmarsh sparrow Ammodramus caudacutus, as a case study, we first estimated survival, fecundity, and population growth rates at 21 sites distributed across much of their breeding range. We then subsampled this large, representative dataset according to five sampling scenarios in order to simulate a variety of geographic biases in study design. We found spatial variation in demographic rates, but no large systematic patterns. Estimating the global mean and variance of growth rates using subsets of the data suggested that at least 10–15 sites were required for reasonably unbiased estimates, highlighting how relying on demographic data from just a few sites can lead to biased results when extrapolating across a species range. Sampling at the full 21 sites, however, offered diminishing returns, raising the possibility that for some species accepting some geographical bias in sampling can still allow for robust range‐wide inferences. The subsampling approach presented here, while conceptually simple, could be used with both new and existing data to encourage efficiency in the design of long‐term or large‐scale ecological studies.     

Spatial and temporal patterns in maternal energetic traits of yellow perch (Perca flavescens) in Lake Erie

1. For many fish species, survival during early life stages is linked to the size and energetic condition of females prior to reproduction. For example, females in good energetic condition are often more fecund and produce larger eggs and offspring than those in poor condition. 2. We measured the characteristics of female yellow perch (Perca flavescens) that may influence annual population fluctuations. From 2005 to 2007, we measured spatial variation in female reproductive traits, such as age, length, mass and energy density (J g^?1) of somatic tissues and ovaries among four spawning aggregations of yellow perch in western and central Lake Erie. 3. Maternal traits, such as somatic energy density and spawner age distribution, differed between the western and central basin, whereas reproductive traits, such as fecundity and ovarian energy density, differed across years. 4. To understand the implications of observed differences in demographic rates (growth and mortality rates) between basins, we developed a deterministic model to simulate the total egg production in the western and central basins under different scenarios of fishing mortality. 5. High growth rates and low mortality rates combined to produce higher modelled estimates of total egg production in the central than in the western basin, and a larger proportion of eggs were produced by old age classes in the central basin than in the western basin. 6. Our results demonstrate that changing harvest levels for populations with different demographic rates can influence total reproductive output through complex interactions between age‐specific mortality, growth and size‐specific fecundity, which has implications for the population dynamics of yellow perch and related species across a broad geographic range.     

Can life history predict the effect of demographic stochasticity on extinction risk?

Demographic stochasticity is important in determining extinction risks of small populations, but it is largely unknown how its effect depends on the life histories of species. We modeled effects of demographic stochasticity on extinction risk in a broad range of generalized life histories, using matrix models and branching processes. Extinction risks of life histories varied greatly in their sensitivity to demographic stochasticity. Comparing life histories, extinction risk generally increased with increasing fecundity and decreased with higher ages of maturation. Effects of adult survival depended on age of maturation. At lower ages of maturation, extinction risk peaked at intermediate levels of adult survival, but it increased along with adult survival at higher ages of maturation. These differences were largely explained by differences in sensitivities of population growth to perturbations of life-history traits. Juvenile survival rate contributed most to total demographic variance in the majority of life histories. Our general results confirmed earlier findings, suggesting that empirical patterns can be explained by a relatively simple model. Thus, basic life-history information can be used to assign life-history-specific sensitivity to demographic stochasticity. This is of great value when assessing the vulnerability of small populations.     

Combining capture–recapture data and known ages allows estimation of age‐dependent survival rates

In many animal populations, demographic parameters such as survival and recruitment vary markedly with age, as do parameters related to sampling, such as capture probability. Failing to account for such variation can result in biased estimates of population‐level rates. However, estimating age‐dependent survival rates can be challenging because ages of individuals are rarely known unless tagging is done at birth. For many species, it is possible to infer age based on size. In capture–recapture studies of such species, it is possible to use a growth model to infer the age at first capture of individuals. We show how to build estimates of age‐dependent survival into a capture–mark–recapture model based on data obtained in a capture–recapture study. We first show how estimates of age based on length increments closely match those based on definitive aging methods. In simulated analyses, we show that both individual ages and age‐dependent survival rates estimated from simulated data closely match true values. With our approach, we are able to estimate the age‐specific apparent survival rates of Murray and trout cod in the Murray River, Australia. Our model structure provides a flexible framework within which to investigate various aspects of how survival varies with age and will have extensions within a wide range of ecological studies of animals where age can be estimated based on size.     

Pattern of variation in avian population growth rates

A central question in population ecology is to understand why population growth rates differ over time. Here, we describe how the long-term growth of populations is not only influenced by parameters affecting the expected dynamics, for example form of density dependence and specific population growth rate, but is also affected by environmental and demographic stochasticity. Using long-term studies of fluctuations of bird populations, we show an interaction between the stochastic and the deterministic components of the population dynamics: high specific growth rates at small densities r(1) are typically positively correlated with the environmental variance sigma(e)(2). Furthermore, theta, a single parameter describing the form of the density regulation in the theta-logistic density-regulation model, is negatively correlated with r(1). These patterns are in turn correlated with interspecific differences in life-history characteristics. Higher specific growth rates, larger stochastic effects on the population dynamics and stronger density regulation at small densities are found in species with large clutch sizes or high adult mortality rates than in long-lived species. Unfortunately, large uncertainties in parameter estimates, as well as strong stochastic effects on the population dynamics, will often make even short-term population projections unreliable. We illustrate that the concept of population prediction interval can be useful in evaluating the consequences of these uncertainties in the population projections for the choice of management actions.     

Contrasting age-specific recruitment and survival at different spatial scales: a case study with the European storm petrel

Evolutionary studies on optimal decisions or conservation guidelines are often derived by generalising patterns from a single population, while inter‐population variability in life‐history traits is seldom considered. We investigated here how survival and recruitment probabilities changed with age at different geographical scales using the encounter histories of 5523 European storm petrels from three Mediterranean colonies, and also how our estimates of these parameters might be expected to affect population growth rates using population matrix models. We recorded similar patterns among colonies, but also important biological differences. Local survival, recruitment and breeding success increased with age at all colonies; the most distant of three colonies (Marettimo Is.) showed the largest differences. Strikingly, differences in recruitment were also found between two adjacent colonies (two caves from Benidorm Is.). Birds marked as adults from Marettimo and Benidorm colonies had a different survival, whereas we found no differences within Benidorm. Differences in survival were no longer apparent between the two islands at the end of the study following a reduction in predation by specialist gulls at Benidorm. Since birds marked as fledglings mostly recruited near the end of the study, their overall survival was high and in turn similar among colonies. Results from our population matrix models suggested that different age‐dependent patterns of demographic parameters can lead to similar population growth rates. Variability appeared to be greater for recruitment and the most sensitive parameter was adult survival. Thus conservation actions targeting this vulnerable species should focus on factors influencing adult survival. Differences in survival and recruitment among colonies could reflect the spatial heterogeneity in mortality due to predation and colony‐specific density dependent processes. Results highlight the importance of taking into account the potential spatio‐temporal heterogeneity among populations in vital rates, even in those traits that life‐history theory considers less important in driving population dynamics.     

Site‐specific dynamics in remnant populations of Northern Wheatears Oenanthe oenanthe in the Netherlands

Dynamics of populations may be synchronized at large spatial scales, indicating driving forces acting beyond local scales, but may also vary locally as a result of site‐specific conditions. Conservation measures for fragmented and declining populations may need to address such local effects to avoid local extinction before measures at large spatial scales become effective. To assess differences in local population dynamics, we aimed to determine the demographic drivers controlling population trends in three remaining populations of the Northern Wheatear Oenanthe oenanthe in the Netherlands, as a basis for conservation actions. An integrated population model (IPM) was fitted to field data collected in each site in 2007–2011 to estimate fecundity, survival and immigration. Sites were 40–120 km apart, yet first‐year recruits were observed to move between some of the sites, albeit rarely. All three populations were equally sensitive to changes in fecundity and first‐year survival. One population was less sensitive to adult survival but more sensitive to immigration. A life table response experiment suggested that differences in immigration were important determinants of differences in population growth between sites. Given the importance of immigration for local dynamics along with high philopatry, resulting in low exchange between sites, creating a metapopulation structure by improving connectivity and the protection of local populations are important for the conservation of these populations. Site‐specific conservation actions will therefore be efficient and, for the short term, we propose different site‐specific conservation actions.     

Small-scale spatial and temporal variation in the demographic processes underlying the large-scale decline of eiders in the Baltic Sea

The application of uniform conservation schemes often fails to account for small-scale spatial variation in the drivers of population decline. Demographic comparisons of imperilled populations across locations are therefore crucial for successful conservation, but progress is hampered by lack of long-term data from more than a single population. The recent large-scale decline of eider ducks (Somateria mollissima) in the Baltic Sea is ideal for determining to what extent mechanisms underlying population decline can be extrapolated over larger areas. We utilized stochastic demographic methods incorporating both environmental and sampling variation to assess small-scale spatial and temporal variation in the population dynamics of eiders at Söderskär (eastern range-margin) and Tvärminne (core breeding area), situated 130 km apart. The stochastic growth rate models accurately predicted the observed differences in the rate of decline between sites and time periods. At Söderskär, established breeder survival had by far the greatest elasticity, whereas elasticity was more evenly distributed among vital rates at Tvärminne. Although the study sites showed the single largest difference in fecundity, stochastic life table response experiment analyses revealed that reduced adult female survival at Tvärminne mainly determined the observed difference in growth rates between sites. In contrast, reduced fecundity primarily differentiated the past population increase from the present population decline at Söderskär. Our results demonstrate that different mechanisms may be associated with population decline across adjacent geographic locations, and indicate that dispersal of first-time breeders may be important for population dynamics. Safeguarding adult female survival and/or fecundity should be prioritized in management efforts.