Age × stage-classified demographic analysis: a comprehensive approach

Planting long-lived trees in a warming climate: Theory shows the importance of stage-dependent climatic tolerance

Climate change poses a particular threat to long-lived trees, which may not adapt or migrate fast enough to keep up with rising temperatures. Assisted gene flow could facilitate adaptation of populations to future climates by using managed translocation of seeds from a warmer location (provenance) within the current range of a species. Finding the provenance that will perform best in terms of survival or growth is complicated by a trade-off. Because trees face a rapidly changing climate during their long lives, the alleles that confer optimal performance may vary across their lifespan. For instance, trees from warmer provenances could be well adapted as adults but suffer from colder temperatures while juvenile. Here we use a stage-structured model, using both analytical predictions and numerical simulations, to determine which provenance would maximize the survival of a cohort of long-lived trees in a changing climate. We parameterize our simulations using empirically estimated demographic transition matrices for 20 long-lived tree species. Unable to find reliable quantitative estimates of how climatic tolerance changes across stages in these same species, we varied this parameter to study its effect. Both our mathematical model and simulations predict that the best provenance depends strongly on how fast the climate changes and also how climatic tolerance varies across the lifespan of a tree. We thus call for increased empirical efforts to measure how climate tolerance changes over life in long-lived species, as our model suggests that it should strongly influence the best provenance for assisted gene flow.

Flexible transition timing in discrete-time multistate life tables using Markov chains with rewards

Discrete-time multistate life tables are attractive because they are easier to understand and apply in comparison with their continuous-time counterparts. While such models are based on a discrete time grid, it is often useful to calculate derived magnitudes (e.g. state occupation times), under assumptions which posit that transitions take place at other times, such as mid-period. Unfortunately, currently available models allow very few choices about transition timing. We propose the use of Markov chains with rewards as a general way of incorporating information on the timing of transitions into the model. We illustrate the usefulness of rewards-based multistate life tables by estimating working life expectancies using different retirement transition timings. We also demonstrate that for the single-state case, the rewards approach matches traditional life-table methods exactly. Finally, we provide code to replicate all results from the paper plus R and Stata packages for general use of the method proposed.

Age-specific sensitivity analysis of stable, stochastic and transient growth for stage-classified populations

Sensitivity analysis in ecology and evolution is a valuable guide to rank demographic parameters depending on their relevance to population growth. Here, we propose a method to make the sensitivity analysis of population growth for matrix models solely classified by stage more fine-grained by considering the effect of age-specific parameters. The method applies to stable population growth, the stochastic growth rate, and transient growth. The method yields expressions for the sensitivity of stable population growth to age-specific survival and fecundity from which general properties are derived about the pattern of age-specific selective forces molding senescence in stage-classified populations.

Individual stochasticity in the life history strategies of animals and plants

The life histories of organisms are expressed as rates of development, reproduction, and survival. However, individuals may experience differential outcomes for the same set of rates. Such individual stochasticity generates variance around familiar mean measures of life history traits, such as life expectancy and the reproductive number R0. By writing life cycles as Markov chains, we calculate variance and other indices of variability for longevity, lifetime reproductive output (LRO), age at offspring production, and age at maturity for 83 animal and 332 plant populations from the Comadre and Compadre matrix databases. We find that the magnitude within and variability between populations in variance indices in LRO, especially, are surprisingly high. We furthermore use principal components analysis to assess how the inclusion of variance indices of different demographic outcomes affects life history constraints. We find that these indices, to a similar or greater degree than the mean, explain the variation in life history strategies among plants and animals.

Sensitivity analysis on the declining population in Japan: Effects of prefecture-specific fertility and interregional migration

Japan has been facing a population decline since 2010 due to low birth rates, interregional migration, and regional traits. In this study, we modeled the demographic dynamics of Japan using a transition matrix model. Then, from the mathematical structure of the model, we quantitatively evaluated the domestic factors of population decline. To achieve this, we constructed a multi-regional Leslie matrix model and developed a method for representing the reproductive value and stable age distribution using matrix entries. Our method enabled us to interpret the mathematical indices using the genealogies of the migration history of individuals and their ancestors. Furthermore, by combining our method with sensitivity analysis, we analyzed the effect of region-specific fertility rates and interregional migration rates on the population decline in Japan. We found that the sensitivity of the population growth rate to the migration rate from urban areas with large populations to prefectures with high fertility rates was greatest for people aged under 30. In addition, compared to other areas, the fertility rates of urban areas exhibited higher sensitivity for people aged over 30. Because this feature is robust in comparison with those in 2010 and 2015, it can be said to be a unique structure in Japan in recent years. We also established a method to represent the reproductive value and stable age distribution in an irreducible non-negative matrix population model by using the matrix entries. Furthermore, we show the effects of fertility and migration rates numerically in urban and non-urban areas on the population growth rates for each age group in a society with a declining population.

A dynamically structured matrix population model for insect life histories observed under variable environmental conditions

Various environmental drivers influence life processes of insect vectors that transmit human disease. Life histories observed under experimental conditions can reveal such complex links; however, designing informative experiments for insects is challenging. Furthermore, inferences obtained under controlled conditions often extrapolate poorly to field conditions. Here, we introduce a pseudo-stage-structured population dynamics model to describe insect development as a renewal process with variable rates. The model permits representing realistic life stage durations under constant and variable environmental conditions. Using the model, we demonstrate how random environmental variations result in fluctuating development rates and affect stage duration. We apply the model to infer environmental dependencies from the life history observations of two common disease vectors, the southern (Culex quinquefasciatus) and northern (Culex pipiens) house mosquito. We identify photoperiod, in addition to temperature, as pivotal in regulating larva stage duration, and find that carefully timed life history observations under semi-field conditions accurately predict insect development throughout the year. The approach we describe augments existing methods of life table design and analysis, and contributes to the development of large-scale climate- and environment-driven population dynamics models for important disease vectors.

The Contributions of Maternal Age Heterogeneity to Variance in Lifetime Reproductive Output

AbstractVariance among individuals in fitness components reflects both genuine heterogeneity between individuals and stochasticity in events experienced along the life cycle. Maternal age represents a form of heterogeneity that affects both the mean and the variance of lifetime reproductive output (LRO). Here, we quantify the relative contribution of maternal age heterogeneity to the variance in LRO using individual-level laboratory data on the rotifer Brachionus manjavacas to parameterize a multistate age × maternal age matrix model. In B. manjavacas, advanced maternal age has large negative effects on offspring survival and fertility. We used multistate Markov chains with rewards to quantify the contributions to variance in LRO of heterogeneity and of the stochasticity inherent in the outcomes of probabilistic transitions and reproductive events. Under laboratory conditions, maternal age heterogeneity contributes 26% of the variance in LRO. The contribution changes when mortality and fertility are reduced to mimic more ecologically relevant environments. Over the parameter space where populations are near stationarity, maternal age heterogeneity contributes an average of 3% of the variance. Thus, the contributions of maternal age heterogeneity and individual stochasticity can be expected to depend strongly on environmental conditions; over most of the parameter space, the variance in LRO is dominated by stochasticity.

Evolution of heterogeneity under constant and variable environments

Various definitions of fitness are essentially based on the number of descendants of an allele or a phenotype after a sufficiently long time. However, these different definitions do not explicate the continuous evolution of life histories. Herein, we focus on the eigenfunction of an age-structured population model as fitness. The function generates an equation, called the Hamilton-Jacobi-Bellman equation, that achieves adaptive control of life history in terms of both the presence and absence of the density effect. Further, we introduce a perturbation method that applies the solution of this equation to the long-term logarithmic growth rate of a stochastic structured population model. We adopt this method to realize the adaptive control of heterogeneity for an optimal foraging problem in a variable environment as the analyzable example. The result indicates that the eigenfunction is involved in adaptive strategies under all the environments listed herein. Thus, we aim to systematize adaptive life histories in the presence of density effects and variable environments using the proposed objective function as a universal fitness candidate.

A demographic and evolutionary analysis of maternal effect senescence

Maternal effect senescence, the decline in offspring quality with increasing maternal age, is common in animals despite its negative impact on fitness. To understand how maternal effect senescence might evolve, we built matrix population models to calculate selection gradients on survival and fertility as functions of maternal age. We estimated the model’s parameters with data from an aquatic invertebrate. The strength of selection eventually declines with age and maternal age, implying that maternal effect senescence could be favored by selection and evolve in the same way as senescence. Our framework can be applied to investigate maternal effect senescence in organisms with diverse life histories and unifies the demographic approaches to age-related and maternal effect senescence.

Maternal effect senescence—a decline in offspring survival or fertility with maternal age—has been demonstrated in many taxa, including humans. Despite decades of phenotypic studies, questions remain about how maternal effect senescence impacts evolutionary fitness. To understand the influence of maternal effect senescence on population dynamics, fitness, and selection, we developed matrix population models in which individuals are jointly classified by age and maternal age. We fit these models to data from individual-based culture experiments on the aquatic invertebrate, Brachionus manjavacas (Rotifera). By comparing models with and without maternal effects, we found that maternal effect senescence significantly reduces fitness for B. manjavacas and that this decrease arises primarily through reduced fertility, particularly at maternal ages corresponding to peak reproductive output. We also used the models to estimate selection gradients, which measure the strength of selection, in both high growth rate (laboratory) and two simulated low growth rate environments. In all environments, selection gradients on survival and fertility decrease with increasing age. They also decrease with increasing maternal age for late maternal ages, implying that maternal effect senescence can evolve through the same process as in Hamilton’s theory of the evolution of age-related senescence. The models we developed are widely applicable to evaluate the fitness consequences of maternal effect senescence across species with diverse aging and fertility schedule phenotypes.

Ontogenetic niche shifts as a driver of seasonal migration

Ontogenetic niche shifts have helped to understand population dynamics. Here we show that ontogenetic niche shifts also offer an explanation, complementary to traditional concepts, as to why certain species show seasonal migration. We describe how demographic processes (survival, reproduction and migration) and associated ecological requirements of species may change with ontogenetic stage (juvenile, adult) and across the migratory range (breeding, non-breeding). We apply this concept to widely different species (dark-bellied brent geese (Branta b. bernicla), humpback whales (Megaptera novaeangliae) and migratory Pacific salmon (Oncorhynchus gorbuscha) to check the generality of this hypothesis. Consistent with the idea that ontogenetic niche shifts are an important driver of seasonal migration, we find that growth and survival of juvenile life stages profit most from ecological conditions that are specific to breeding areas. We suggest that matrix population modelling techniques are promising to detect the importance of the ontogenetic niche shifts in maintaining migratory strategies. As a proof of concept, we applied a first analysis to resident, partial migratory and fully migratory populations of barnacle geese (Branta leucopsis). We argue that recognition of the costs and benefits of migration, and how these vary with life stages, is important to understand and conserve migration under global environmental change.

Variance as a life history outcome: Sensitivity analysis of the contributions of stochasticity and heterogeneity

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Individuals vary in traits or in luck; both cause variance in life history outcomes.

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The variance components are calculated from a multistate group-stage cohort model.

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Sensitivity analysis shows how variance components relate to demographic parameters.

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Both mortality and fertility affect the variance components.

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Effects depend on life history timing, and the nature and mixture of differences.

Variance in life history outcomes among individuals is a requirement for natural selection, and a determinant of the ecological dynamics of populations. Heterogeneity among individuals will cause such variance, but so will the inherently stochastic nature of their demography. The relative contributions of these variance components – stochasticity and heterogeneity – to life history outcomes are presented here in a general, demographic calculation. A general formulation of sensitivity analysis is provided for the relationship between the variance components and the demographic rates within the life cycle. We illustrate these novel methods with two examples; the variance in longevity within and between frailty groups in a laboratory population of fruit flies, and the variance in lifetime reproductive output within and between initial environment states in a perennial herb in a stochastic fire environment. In fruit flies, an increase in mortality would increase the variance due to stochasticity and reduce that due to heterogeneity. In the plant example, increasing mortality reduces, and increasing fertility increases both variance components. Sensitivity analyses such as these can provide a powerful tool in identifying patterns among life history stages and heterogeneity groups and their contributions to variance in life history outcomes.

Selection in two-sex stage-structured populations: Genetics, demography, and polymorphism

The outcome of natural selection depends on the demographic processes of birth, death, and development. Here, we derive conditions for protected polymorphism in a population characterized by age- or stage-dependent demography with two sexes. We do so using a novel two-sex matrix population model including basic Mendelian genetics (one locus, two alleles, random mating). Selection may operate on survival, growth, or fertility, any or all of which may differ between the sexes. The model can therefore incorporate genes with arbitrary pleiotropic and sex-specific effects. Conditions for protected polymorphism are expressed in terms of the eigenvalues of the linearization of the model at the homozygote boundary equilibria. We show that in the absence of sexual dimorphism, polymorphism requires heterozygote superiority in the genotypic population growth rate. In the presence of sexual dimorphism, however, heterozygote superiority is not required; an inferior heterozygote may invade, reducing the population growth rate and even leading to extinction (so-called evolutionary suicide). Our model makes no assumptions about separation of time scales between ecological and evolutionary processes, and can thus be used to project sex×stage×genotype dynamics of eco-evolutionary processes. Empirical evidence that sexual dimorphism affects extinction risk is growing, yet sex differences are often ignored in evolutionary demography and in eco-evolutionary models. Our analysis highlights the importance of sexual dimorphism and suggests mechanisms by which an allele can be favored by selection, yet drive a population to extinction, as a result of the structure and interdependence of sex- and stage-specific processes.

Changing contribution of area-level deprivation to total variance in age at death: a population-based decomposition analysis

OBJECTIVES

Two processes generate total variance in age at death: heterogeneity (between-group variance) and individual stochasticity (within-group variance). Limited research has evaluated how these two components have changed over time. We quantify the degree to which area-level deprivation contributed to total variance in age at death in Scotland between 1981 and 2011.

DESIGN

Full population and mortality data for Scotland were obtained and matched with the Carstairs score, a standardised z-score calculated for each part-postcode sector that measures relative area-level deprivation. A z-score above zero indicates that the part-postcode sector experienced higher deprivation than the national average. A z-score below zero indicates lower deprivation. From the aggregated data we constructed 40 lifetables, one for each deprivation quintile in 1981, 1991, 2001 and 2011 stratified by sex.

PRIMARY OUTCOME MEASURES

Total variance in age at death and the proportion explained by area-level deprivation heterogeneity (between-group variance).

RESULTS

The most deprived areas experienced stagnating or slightly increasing variance in age at death. The least deprived areas experienced decreasing variance. For males, the most deprived quintile life expectancy was between 7% and 11% lower and the SD is between 6% and 25% higher than the least deprived. This suggests that the effect of deprivation on the SD of longevity is comparable to its effect on life expectancy. Decomposition analysis revealed that contributions from between-group variance doubled between 1981 and 2011 but at most only explained 4% of total variance.

CONCLUSIONS

This study adds to the emerging body of literature demonstrating that socio-economic groups have experienced diverging trends in variance in age at death. The contribution from area-level deprivation to total variance in age at death, which we were able to capture, has doubled since 1981. Area-level deprivation may play an increasingly important role in mortality inequalities.

Stage-Structured Evolutionary Demography: Linking Life Histories, Population Genetics, and Ecological Dynamics

Demographic processes and ecological interactions are central to understanding evolution and vice versa. We present a novel framework that combines basic Mendelian genetics with the powerful demographic approach of matrix population models. The ecological components of the model may be stage classified or age classified, linear or nonlinear, time invariant or time varying, and deterministic or stochastic. Genotypes may affect, in fully pleiotropic fashion, any mixture of demographic traits (viability, fertility, development) at any points in the life cycle. The dynamics of the stage × genotype structure of the population are given by a nonlinear population projection matrix. We show how to construct this matrix and use it to derive sufficient conditions for a protected genetic polymorphism for the case of linear, time-independent demography. These conditions demonstrate that genotype-specific population growth rates (λ) do not determine the outcome of selection. Except in restrictive special cases, heterozygote superiority in λ is neither necessary nor sufficient for a genetic polymorphism. As a consequence, the population growth rate does not always increase, and populations can be driven to extinction due to evolutionary suicide. We demonstrate the construction and analysis of the model using data on a color polymorphism in the common buzzard (Buteo buteo). The model exhibits a stable genetic polymorphism and declining growth rate, consistent with field data and previous models.

Variance in animal longevity: contributions of heterogeneity and stochasticity

Variance in longevity among individuals may arise as an effect of heterogeneity (differences in mortality rates experienced at the same age or stage) or as an effect of individual stochasticity (the outcome of random demographic events during the life cycle). Decomposing the variance into components due to heterogeneity and stochasticity is crucial for evolutionary analyses.In this study, we analyze longevity from ten studies of invertebrates in the laboratory, and use the results to partition the variance in longevity into its components. To do so, we fit finite mixtures of Weibull survival functions to each data set by maximum likelihood, using the EM algorithm. We used the Bayesian Information Criterion to select the most well supported model. The results of the mixture analysis were used to construct an age × stage-classified matrix model, with heterogeneity groups as stages, from which we calculated the variance in longevity and its components. Almost all data sets revealed evidence of some degree of heterogeneity. The median contribution of unobserved heterogeneity to the total variance was 35%, with the remaining 65% due to stochasticity. The differences among groups in mean longevity were typically on the order of 30% of the overall life expectancy. There was considerable variation among data sets in both the magnitude of heterogeneity and the proportion of variance due to heterogeneity, but no clear patterns were apparent in relation to sex, taxon, or environmental conditions.