Stage,age and individual stochasticity in demography

Demography is the study of the population consequences of the fates of individuals. Individuals are differentiated on the basis of age or, in general, life cycle stages. The movement of an individual through its life cycle is a random process, and although the eventual destination (death) is certain, the pathways taken to that destination are stochastic and will differ even between identical individuals; this is individual stochasticity. A stage‐classified demographic model contains implicit age‐specific information, which can be analyzed using Markov chain methods. The living stages in the life cycles are transient states in an absorbing Markov chain; death is an absorbing state. This paper presents Markov chain methods for computing the mean and variance of the lifetime number of visits to any transient state, the mean and variance of longevity, the net reproductive rate R₀, and the cohort generation time. It presents the matrix calculus methods needed to calculate the sensitivity and elasticity of all these indices to any life history parameters. These sensitivities have many uses, including calculation of selection gradients. It is shown that the use of R₀ as a measure of fitness or an invasion exponent gives erroneous results except when R₀=λ=1. The Markov chain approach is then generalized to variable environments (deterministic environmental sequences, periodic environments, iid random environments, Markovian environments). Variable environments are analyzed using the vec‐permutation method to create a model that classifies individuals jointly by the stage and environmental condition. Throughout, examples are presented using the North Atlantic right whale (Eubaleana glacialis) and an endangered prairie plant (Lomatium bradshawii) in a stochastic fire environment.     

Lifetime reproductive success in common buzzard, Buteo buteo: from individual variation to population demography

We studied annual and lifetime reproductive success (LRS) of both sexes of common buzzard Buteo buteo in eastern Westphalia, Germany. We followed a bottom‐up approach starting from individual breeding attempts, over lifetime reproductive success to derive population demography. Annual breeding performance and survival followed a quadratic relationship with breeding experience; individuals starting their breeding career were less likely to survive and breed successfully than birds of intermediate breeding experience. According to an analysis of selection gradients, both the opportunity and intensity of selection peaked in the early stages of the breeding career. The distribution of both LRS and another fitness measure, λ, was highly skewed, with ca 17% of adult birds producing 50% of fledglings in both sexes. Besides breeding life span and number of breeding attempts, habitat quality and plumage morph were significant predictors of LRS. There were strong differences in LRS and λ between the plumage morphs in both sexes: intermediate pigmented buzzards were much more successful than either dark or light ones. There was no significant difference between buzzard cohorts either in LRS or λ, nor did these fitness measures differ between individuals starting their breeding career at different conditions of food availability. Based on individual life histories, we formed a transition matrix and analysed its properties to study the population as a whole. This analysis showed that the population growth rate was close to unity (0.906, bootstrapped 95% confidence limits: 0.834 and 0.962). Analysis of reproductive values and elasticities further emphasised colour morph differences: the contribution of intermediate individuals to population growth greatly exceeded that of dark or light individuals. Thus most phenomena on all levels from individual breeding attempts over lifetime reproductive success to population demography can be explained by the fitness differences between the colour morphs with the intermediate morph maintaining the current population renewal potential.     

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.     

Lifetime reproductive success and density-dependent, multi-variable resource selection

Individuals are predicted to maximize lifetime reproductive success (LRS) through selective use of resources; however, a wide range of ecological and social processes may prevent individuals from always using the highest-quality resources available. Resource selection functions (RSFs) estimate the relative amount of time an individual spends using a resource as a function of the proportional availability of that resource. We quantified the association between LRS and coefficients of individual-based RSFs describing lifetime resource selection for 267 female red deer (Cervus elaphus) of the Isle of Rum, Scotland, from 1970 to 2001. LRS was significantly related to first- and second-order effects of selection for Agrostis/Festuca grassland and proximity to the sea coast (quality of forage within Agrostis/Festuca grassland was highest nearest the coast (ratio of short:long grassland)). The benefits of selecting for quality in Agrostis/Festuca grassland, however, traded-off with increases in LRS gained by avoiding conspecific density. LRS was inversely associated with local density, which was highest along the coast, and reproductive benefits of selecting Agrostis/Festuca grassland diminished with increasing density. We discuss the relevance of these results to our understanding of the spatial distribution of red deer abundance, and potential applications of our approach to evolutionary and applied ecology.     

Lifetime reproductive output in a hermaphrodite cestode when reproducing alone or in pairs: a time cost of pairing

The cestode Schistocephalus solidus is a facultatively self-fertilising simultaneous hermaphrodite. Here we test for differences in the starting point, the rate, and the magnitude of egg production between individuals allowed to reproduce alone (only self-fertilisation possible) or in pairs (both self- and cross-fertilisation possible). Specifically, we want to distinguish between alternative processes responsible for the lower egg production in paired individuals observed in an earlier study (Wedekind et al., 1998). We designed an improved in vitro system, replacing the bird final host that allows us to measure, with high temporal resolution, the timing and magnitude of lifetime egg production of worms in these two social situations. We found that the experimental groups did not differ significantly in the starting point of egg production. However, the temporal pattern in egg production differed between them, in that paired individuals had a lower rate of egg production. This, however, did not lead to a significant reduction in lifetime egg production, as pairs compensated for the lower rate by producing eggs longer than single individuals. We argue that the lower rate of egg production may nevertheless lead to a time cost of pairing in the study species, and that this cost is likely to represent a cost of outcrossing due to sexual selection. This revised version was published online in July 2006 with corrections to the Cover Date.     

Temporal partitioning and aggression among foragers: modeling the effects of stochasticity and individual state

In many natural systems, individuals compete with conspecificsand heterospecifics for food and in some cases, individualshave been observed to partition their foraging times or fightover food. In this study, I investigated when it is optimalfor a consumer to partition time and be aggressive. I formulatedan individual-based model of foraging and used game theoryto find evolutionarily stable strategies (ESSs) that maximizethe probability that consumers survive each day and acquiretheir daily food requirements. Consumers choose when to forageand when to behave aggressively during confrontations overfood. Consumers are each associated with a state variable,representing the amount of food eaten, and a dominance ranking, which describes how likely they are to forage and fight forfood. The ESS is sensitive to food abundance, consumer state,and the dominance ranking. When food is abundant, temporalpartitioning is often an ESS where the dominant consumer foragesfirst; however, partitioning is unlikely to be an ESS when food abundance is low. Fights over food are typically avoidedbut may be part of an ESS when food abundance is low, bothconsumers are hungry, or the time available for foraging eachday is drawing to a close. Because the ESS is sensitive toconsumer state, the stochastic nature of finding food often results in considerable variation in observed foraging dynamicsfrom one day to the next, even when consumers adopt the samestate-dependent strategy each day. Results are compared withempirical observations, and I discuss implications for consumercoexistence.     

Metapopulation moments: coupling, stochasticity and persistence

1.  Spatial heterogeneity has long been viewed as a reliable means of increasing persistence. Here, an analytical model is developed to consider the variation and, hence, the persistence of stochastic metapopulations. This model relies on a novel moment closure technique, which is equivalent to assuming log-normal distributions for the population sizes.
2.  Single-species models show the greatest persistence when the mixing between subpopulations is large, so spatial heterogeneity is of no benefit. This result is confirmed by stochastic simulation of the full metapopulation.
3.  In contrast, natural-enemy models exhibit the greatest persistence for intermediate levels of coupling. When the coupling is too low, there are insufficient rescue effects between the subpopulations to sustain the dynamics, whereas when the coupling is too high all spatial heterogeneity is lost.
4.  The difference in behaviour between the one- and two-species models can be attributed to the oscillatory nature of the natural-enemy system.     

Comparative sensitivity analysis of stable stage structures and reproductive values

New formulas for deriving the sensitivities of stable stage structures and reproductive values to changes in vital rates are presented. They enable comparison of the sensities to changes of different elements in the projection matrix; in other words, comparison of partial derivatives of the eigenvectors. These kinds of sensitivities can be used in applied problems such as an analysis of the effect of harvesting on the population structure. However, in this paper, we examine the application of the sensitivities in a more general ecological context. We investigate why the stable stage structure of the mustard aphid,Lipaphis erysimi, changes very little in the temperature interval 10–30°C. The sensitivities of the stable stage structure at 15°C and 25°C were derived. The character of the sensitivites were the same in both temperatures although the stage structure was more sensitive to changes at 15°C than at 25°C. The sensitivity analysis also revealed that the temperature variation results in changes in fecundity and developmental rate that have a counteractive effect on the population structure.     

Skewed distributions of lifetime reproductive success: beyond mean and variance

Lifetime reproductive performance is quantified here by the LRS (lifetime reproductive success), the random number of offspring an individual produces over its lifetime. Many field studies find that distributions of LRS among individuals are non‐normal, zero‐inflated and highly skewed. These results beg the question, what is the distribution of LRS predicted by demographic models when the only source of randomness is demographic stochasticity? Here we present the first exact analysis of the probability distribution of LRS for species described by age + stage models; our analysis starts with estimated vital rates. We illustrate with three examples: the Hadza, human hunter‐foragers (age‐only), the evergreen tree Tsuga canadensis (stage‐only) and Roe deer, Capreolus capreolus (age + stage). For each we obtain the exact distribution of LRS, but also calculate and discuss the first three moments. Our results point to important questions about how such LRS distributions affect natural selection, and life history evolution.     

On net reproductive rate and the timing of reproductive output

Abstract: Understanding the relationship between life-history patterns and population growth is central to demographic studies. Here we derive a new method for calculating the timing of reproductive output, from which the generation time and its variance can also be calculated. The method is based on the explicit computation of the net reproductive rate (R0) using a new graphical approach. Using nodding thistle, desert tortoise, creeping aven, and cat's ear as examples, we show how R0 and the timing of reproduction is calculated and interpreted, even in cases with complex life cycles. We show that the explicit R0 formula allows us to explore the effect of all reproductive pathways in the life cycle, something that cannot be done with traditional analysis of the population growth rate (lambda). Additionally, we compare a recently published method for determining population persistence conditions with the condition R0 > 1 and show how the latter is simpler and more easily interpreted biologically. Using our calculation of the timing of reproductive output, we illustrate how this demographic measure can be used to understand the effects of life-history traits on population growth and control.     

Life span,reproductive output,and reproductive opportunity in captive Goeldi's monkeys (Callimico goeldii)

In the absence of long‐term field studies, demographic and reproductive records from animals housed in zoos and research laboratories are a valuable tool for the study of life history variables relating to reproduction. In this study, we analyzed studbook records of more than 2,000 individuals born over a 40‐year period (1965–2004) to describe life history patterns of captive Goeldi's monkeys (Callimico goeldii) housed in North America and Europe. Using Kaplan–Meier survival analysis methods, we found the mean life span to be 5.5 years. The rate of infant mortality, defined as death before 30 days, was approximately 30%, with European animals being more likely to survive infancy than North American animals. When individuals surviving at least 1.5 years are considered, lifetime reproductive output averaged 3.5 offspring, yet more than one‐third of individuals did not produce any offspring. Using a smaller dataset of individuals with known pairing histories, we developed a measure of opportunity for reproduction (OFR), which represented the total time an individual was known to be housed with a potential mate. For both sexes, we found that the correlation between OFR and number of offspring produced was much higher than the correlation between life span and number of offspring produced. This result highlights the importance of taking into account an individual's OFR. As a whole, our findings help characterize the life histories of captive Goeldi's monkeys and emphasize the impact management practices may have on reproductive success. Zoo Biol 29:1–15, 2010. © 2009 Wiley‐Liss, Inc.     

Energetic composition,reproductive output,and resource allocation of Antarctic asteroids

Summary Thirteen species of antarctic asteroids were collected in McMurdo Sound during the austral spring and summer of 1984–1985 and 1985–1986. Variability in the energetic composition of the body wall, pyloric cecum, and gonad was directly related to differences in levels of protein, carbohydrate, lipid and ash in these tissue types. Reproductive output was generally similar between males and females when expressed in gravimetric units. When considered in energy units, females generally had greater reproductive output than males. Reproductive output values of antarctic asteroids were similar to published values for temperate, tropical, and subpolar species. In a given year, the majority of antarctic asteroids have the greatest energy resources in the body wall, a tissue representing a long-term energy investment. Lesser amounts of energy are devoted to annual nutrient storage (pyloric ceca) and reproduction (gonads).     

Additive genetic variance, heritability, and inbreeding depression in male extra-pair reproductive success

The hypothesis that female extra-pair reproduction in socially monogamous animals reflects indirect genetic benefits requires that there be additive and/or nonadditive genetic variance in fitness. However, the specific hypotheses that male extra-pair reproductive success (EPRS) shows additive genetic variance (V(A)), heritability (h2), or inbreeding depression, and hence that females could acquire indirect genetic benefits through increased EPRS of sons, have not been explicitly tested. We used comprehensive genetic pedigree data from song sparrows (Melospiza melodia) to estimate V(A), h2, and inbreeding depression in the number of extra-pair offspring a male sired per year and the probability that a male would sire any extra-pair offspring per year. Inbreeding depression was substantial: more inbred males sired fewer extra-pair offspring and were less likely to sire any extra-pair offspring. In contrast, estimates of V(A) and h2 were close to 0, although 95% credible intervals were relatively wide. These data suggest that females could accrue indirect genetic benefits, in terms of increased EPRS of outbred sons, by mating with unrelated social or extra-pair mates. In contrast, any indirect benefit of extra-pair reproduction in terms of producing sons with high additive genetic value for EPRS is most likely to be small.     

Sexual selection: harem size and the variance in male reproductive success

Sexual selection is potentially stronger than natural selection when the variance in male reproductive fitness exceeds all other components of fitness variance combined. However, measuring the variance in male reproductive fitness is difficult when nonmating males are absent, inconspicuous, or otherwise difficult to find. Omitting the nonmating males inflates estimates of average male reproductive success and diminishes the variance, leading to underestimates of the potential strength of sexual selection. We show that, in theory, the proportion of the total variance in male fitness owing to sexual selection is approximately equal to H, the mean harem size, as long as H is large and females are randomly distributed across mating males (i.e., Vharem=H). In this case, mean harem size not only provides an easy way to estimate the potential strength of sexual selection but also equals the opportunity for sexual selection, I(mates). In nature, however, females may be overdispersed with VharemH. We show that H+(k-1) is a good measure of the opportunity for sexual selection, where k is the ratio Vharem/H. A review of mating system data reveals that in nature the median ratio for Vharem/H is 1.04, but as H increases, females tend to become more aggregated across mating males with V(harem) two to three times larger than H.     

Armor morphology and reproductive output in threespine stickleback,Gasterosteus aculeatus

Synopsis The threespine stickleback, Gasterosteus aculeatus, is an extensively armored fish inhabiting both marine and fresh waters across its holarctic distribution. Marine fish nearly always possess a full complement of bony lateral plates running from just behind the head to the tail, and a robust pelvic girdle complex. These armor features appear to constrain lateral and ventral abdominal distention, and therefore clutch volume. Freshwater populations in many areas exhibit variable reduction in lateral plate number, and in some regions the pelvic girdle is also reduced or lost. Freshwater populations also vary in the degree of abdominal distention exhibited by gravid females. We tested whether reduction in armoring might be correlated with increased clutch volume using five populations from the Cook Inlet area of Alaska. The hypothesis that populations having reduced pelvic girdle complexes would have greater size-adjusted clutch volumes was not supported. In fact, our two full-pelvic populations as a group had larger volumes. Similarly, size-adjusted clutch volumes were not related to pelvic phenotype within either of our two pelvic-reduced populations, nor to lateral plate morph within a fifth population. Other factors that may explain the interpopulation differences in clutch volume in threespine stickleback include body shape, food quantity and quality, intensity of predation, and even behavior. Except for a preliminary analysis of body shape, these possibilities remain unexplored. The concept of phenotypic integration suggests that these factors should be analyzed as a suite rather than individually.     

Robustness of individual and marginal model-based estimates: A sensitivity analysis of flexible parametric models

BackgroundFlexible parametric survival models (FPMs) are commonly used in epidemiology. These are preferred as a wide range of hazard shapes can be captured using splines to model the log-cumulative hazard function and can include time-dependent effects for more flexibility. An important issue is the number of knots used for splines. The reliability of estimates are assessed using English data for 10 cancer types and the use of online interactive graphs to enable a more comprehensive sensitivity analysis at the control of the user is demonstrated.MethodsSixty FPMs were fitted to each cancer type with varying degrees of freedom to model the baseline excess hazard and the main and time-dependent effect of age. For each model, we obtained age-specific, age-group and internally age-standardised relative survival estimates. The Akaike Information Criterion and Bayesian Information Criterion were also calculated and comparative estimates were obtained using the Ederer II and Pohar Perme methods. Web-based interactive graphs were developed to present results.ResultsAge-standardised estimates were very insensitive to the exact number of knots for the splines. Age-group survival is also stable with negligible differences between models. Age-specific estimates are less stable especially for the youngest and oldest patients, of whom there are very few, but for most scenarios perform well.ConclusionAlthough estimates do not depend heavily on the number of knots, too few knots should be avoided, as they can result in a poor fit. Interactive graphs engage researchers in assessing model sensitivity to a wide range of scenarios and their use is highly encouraged.     

Temporal change in the genetic structure between and within cohorts of a marine fish,Diplodus sargus,induced by a large variance in individual reproductive success

Temporal changes at 16 allozyme loci in the Diplodus sargus population of Banyuls-sur-Mer (Mediterranean Sea, France) were monitored. Temporal genetic variation within a single population was examined over two temporal scales: (i) among three year-classes sampled at the same age, and (ii) within a single year-class sampled three times over a two-year period. We observed a significant change in the genotypic structure within the same cohort during the first two years following settlement and before recruitment into the adult population. In addition, comparison of year-classes showed that cohorts differed significantly one year after settlement, whereas they became similar later on before recruitment into the adult population. The observed changes in the genetic structure within and between year-classes may be the result of complex selective processes or genetic drift. Linkage disequilibrium and genetic relatedness data suggest that these changes are due to large variation in reproductive success, followed by homogenization through adult movement. Overall, these results demonstrated a rapid genetic change within a population.