Perturbation analysis of nonlinear matrix population models

Parameter Sensitivity of Transient Community Dynamics

AbstractTransient dynamics have always intrigued ecologists, but current rapid environmental change (inducing transients even in previously undisturbed systems) has highlighted their importance more than ever. Here, I introduce a method for analyzing the sensitivity of transient ecological dynamics to parameter perturbations. The question the method answers is: how would the community dynamics have unfolded for some time horizon had the parameters been slightly different? I apply the method to three empirically parameterized models: competition between native forbs and exotic grasses in California, a host-parasitoid system, and an experimental chemostat predator-prey model. These applications showcase the ecological insights one can gain from models using transient sensitivity analysis. First, one can find parameters and their combinations whose perturbations disproportionately affect a system. Second, one can identify particular windows of time during which the predicted deviation from the unperturbed trajectories is especially large and utilize this information for management purposes. Third, there is an inverse relationship between transient and long-term sensitivities whenever the interacting populations are ecologically similar; paradoxically, the smaller the immediate response of the system, the more extreme its long-term response will be.

Reactive persistence, spatial management, and conservation of metapopulations: An application to seagrass restoration

Assessing the conditions for persistence of spatially structured populations, especially those that are exploited by humans or threatened by global change, is of critical importance to inform management and conservation efforts. Observations for entire metapopulations are usually incomplete and rarely, if ever, sufficiently long to deduce population persistence from spatial patterns of abundance. Instead, insights based on metapopulation theory are often used for interpreting the demographic trajectories of real populations and for informing management decisions. The classical theoretical tool used to assess conditions for metapopulation persistence is the "invasibility criterion," which characterizes the asymptotic, or long-term, stability of a small colonizing population. Essentially, when the linear operator governing the metapopulation dynamics of an invasion event has a positive eigenvalue, recovery and resistance to extinction (resilience) are implied. The converse, however, is not necessarily the case-an invasion may grow over multiple generations, even when the eigenvalues indicate that extinction will eventually occur, a situation referred to here as "reactive persistence." For the management, restoration, and conservation of real metapopulations subject to continual disturbance, this transient behavior is often more relevant than the asymptotic behavior over long time scales. We develop the theoretical tools for assessing reactive persistence, demonstrating how the conditions for asymptotic and reactive persistence differ in both the patch-occupancy models suited to many terrestrial populations and those where local patch extinctions can be disregarded in the dynamics, often suited to marine species. After presenting the mathematical basis for generalizing the invasibility criterion to include reactive persistence, we illustrate how these concepts and tools can be applied in practice, using as a case study the population ecology and restoration of the seagrass Zostera muelleri (Irmisch ex Ascherson, 1867) in the Port of Gladstone in the Great Barrier Reef World Heritage Area Australia. It is shown how the analysis of the transient dynamics of the Z. muelleri metapopulation can be used to guide restoration efforts. Moreover, it is demonstrated that these reactive persistence concepts provide a more appropriate basis for site prioritization for restoration interventions than traditional stability analysis.

Population level consequences of facultatively cooperative behaviour in a stochastic environment

The social environment in which individuals live affects their fitness and in turn population dynamics as a whole. Birds with facultative cooperative breeding can live in social groups with dominants, subordinate helpers that assist with the breeding of others, and subordinate non-helpers. Helping behaviour benefits dominants through increased reproductive rates and reduced extrinsic mortality, such that cooperative breeding might have evolved in response to unpredictable, harsh conditions affecting reproduction and/or survival of the dominants. Additionally, there may be different costs and benefits to both helpers and non-helpers, depending on the time-scale. For example, early-life costs might be compensated by later-life benefits. These differential effects are rarely analysed in the same study. We examined whether helping behaviour affects population persistence in a stochastic environment and whether there are direct fitness consequences of different life-history tactics adopted by helpers and non-helpers. We parameterised a matrix population model describing the population dynamics of female Seychelles warblers Acrocephalus sechellensis, birds that display facultative cooperative breeding. The stochastic density-dependent model is defined by a (st)age structure that includes life-history differences between helpers and non-helpers and thus can estimate the demographic mechanisms of direct benefits of helping behaviour. We found that population dynamics are strongly influenced by stochastic variation in the reproductive rates of the dominants, that helping behaviour promotes population persistence and that there are only early-life differences in the direct fitness of helpers and non-helpers. Through a matrix population model, we captured multiple demographic rates simultaneously and analysed their relative importance in determining population dynamics of these cooperative breeders. Disentangling early-life versus lifetime effects of individual tactics sheds new light on the costs and benefits of helping behaviour. For example, the finding that helpers and non-helpers have similar lifetime reproductive outputs and that differences in reproductive values between the two life-history tactics arise only in early life suggests that overall, helpers and non-helpers have a similar balance of costs and benefits when analysing direct benefits. We recommend analysing the consequence of different life-history tactics, during both early life and over the lifetime, as analyses of these different time frames may produce conflicting results.

The interplay between hunting rate, hunting selectivity, and reproductive strategies shapes population dynamics of a large carnivore

Harvest, through its intensity and regulation, often results in selection on female reproductive traits. Changes in female traits can have demographic consequences, as they are fundamental in shaping population dynamics. It is thus imperative to understand and quantify the demographic consequences of changes in female reproductive traits to better understand and anticipate population trajectories under different harvest intensities and regulations. Here, using a dynamic, frequency-dependent, population model of the intensively hunted brown bear (Ursus arctos) population in Sweden, we quantify and compare population responses to changes in four reproductive traits susceptible to harvest-induced selection: litter size, weaning age, age at first reproduction, and annual probability to reproduce. We did so for different hunting quotas and under four possible hunting regulations: (i) no individuals are protected, (ii) mothers but not dependent offspring are protected, (iii) mothers and dependent offspring of the year (cubs) are protected, and (iv) entire family groups are protected (i.e., mothers and dependent offspring of any age). We found that population growth rate declines sharply with increasing hunting quotas. Increases in litter size and the probability to reproduce have the greatest potential to affect population growth rate. Population growth rate increases the most when mothers are protected. Adding protection on offspring (of any age), however, reduces the availability of bears for hunting, which feeds back to increase hunting pressure on the nonprotected categories of individuals, leading to reduced population growth. Finally, we found that changes in reproductive traits can dampen population declines at very high hunting quotas, but only when protecting mothers. Our results illustrate that changes in female reproductive traits may have context-dependent consequences for demography. Thus, to predict population consequences of harvest-induced selection in wild populations, it is critical to integrate both hunting intensity and regulation, especially if hunting selectivity targets female reproductive strategies.

Population Trends and Urbanization: Simulating Density Effects Using a Local Regression Approach

Density-dependent population growth regulates long-term urban expansion and shapes distinctive socioeconomic trends. Despite a marked heterogeneity in the spatial distribution of the resident population, Mediterranean European countries are considered more homogeneous than countries in other European regions as far as settlement structure and processes of metropolitan growth are concerned. However, rising socioeconomic inequalities among Southern European regions reflect latent demographic and territorial transformations that require further investigation. An integrated assessment of the spatio-temporal distribution of resident populations in more than 1000 municipalities (1961–2011) was carried out in this study to characterize density-dependent processes of metropolitan growth in Greece. Using geographically weighted regressions, the results of our study identified distinctive local relationships between population density and growth rates over time. Our results demonstrate that demographic growth rates were non-linearly correlated with other variables, such as population density, with positive and negative impacts during the first (1961–1971) and the last (2001–2011) observation decade, respectively. These findings outline a progressive shift over time from density-dependent processes of population growth, reflecting a rapid development of large metropolitan regions (Athens, Thessaloniki) in the 1960s, to density-dependent processes more evident in medium-sized cities and accessible rural regions in the 2000s. Density-independent processes of population growth have been detected in the intermediate study period (1971–2001). This work finally discusses how a long-term analysis of demographic growth, testing for density-dependent mechanisms, may clarify the intrinsic role of population concentration and dispersion in different phases of the metropolitan cycle in Mediterranean Europe.

Individual Specialization and Multihost Epidemics: Disease Spread in Plant-Pollinator Networks

Many parasites infect multiple species and persist through a combination of within- and between-species transmission. Multispecies transmission networks are typically constructed at the species level, linking two species if any individuals of those species interact. However, generalist species often consist of specialized individuals that prefer different subsets of available resources, so individual- and species-level contact networks can differ systematically. To explore the epidemiological impacts of host specialization, we build and study a model for pollinator pathogens on plant-pollinator networks, in which individual pollinators have dynamic preferences for different flower species. We find that modeling and analysis that ignore individual host specialization can predict die-off of a disease that is actually strongly persistent and can badly over- or underpredict steady-state disease prevalence. Effects of individual preferences remain substantial whenever mean preference duration exceeds half of the mean time from infection to recovery or death. Similar results hold in a model where hosts foraging in different habitats have different frequencies of contact with an environmental reservoir for the pathogen. Thus, even if all hosts have the same long-run average behavior, dynamic individual differences can profoundly affect disease persistence and prevalence.

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.

Age × stage-classified demographic analysis: a comprehensive approach

This paper presents a comprehensive theory for the demographic analysis of populations in which individuals are classified by both age and stage. The earliest demographic models were age classified. Ecologists adopted methods developed by human demographers and used life tables to quantify survivorship and fertility of cohorts and the growth rates and structures of populations. Later, motivated by studies of plants and insects, matrix population models structured by size or stage were developed. The theory of these models has been extended to cover all the aspects of age-classified demography and more. It is a natural development to consider populations classified by both age and stage. A steady trickle of results has appeared since the 1960s, analyzing one or another aspect of age × stage-classified populations, in both ecology and human demography. Here, we use the vec-permutation formulation of multistate matrix population models to incorporate age- and stage-specific vital rates into demographic analysis. We present cohort results for the life table functions (survivorship, mortality, and fertility), the dynamics of intra-cohort selection, the statistics of longevity, the joint distribution of age and stage at death, and the statistics of life disparity. Combining transitions and fertility yields a complete set of population dynamic results, including population growth rates and structures, net reproductive rate, the statistics of lifetime reproduction, and measures of generation time. We present a complete analysis of a hypothetical model species, inspired by poecilogonous marine invertebrates that produce two kinds of larval offspring. Given the joint effects of age and stage, many familiar demographic results become multidimensional, so calculations of marginal and mixture distributions are an important tool. From an age-classified point of view, stage structure is a form of unobserved heterogeneity. From a stage-classified point of view, age structure is unobserved heterogeneity. In an age × stage-classified model, variance in demographic outcomes can be partitioned into contributions from both sources. Because these models are formulated as matrices, they are amenable to a complete sensitivity analysis. As more detailed and longer longitudinal studies are developed, age × stage-classified demography will become more common and more important.

Density dependence governs when population responses to multiple stressors are magnified or mitigated

Population endangerment typically arises from multiple, potentially interacting anthropogenic stressors. Extensive research has investigated the consequences of multiple stressors on organisms, frequently focusing on individual life stages. Less is known about population-level consequences of exposure to multiple stressors, especially when exposure varies through life. We provide the first theoretical basis for identifying species at risk of magnified effects from multiple stressors across life history. By applying a population modeling framework, we reveal conditions under which population responses from stressors applied to distinct life stages are either magnified (synergistic) or mitigated. We find that magnification or mitigation critically depends on the shape of density dependence, but not the life stage in which it occurs. Stressors are always magnified when density dependence is linear or concave, and magnified or mitigated when it is convex. Using Bayesian numerical methods, we estimated the shape of density dependence for eight species across diverse taxa, finding support for all three shapes.

Matrix methods in health demography: a new approach to the stochastic analysis of healthy longevity and DALYs

BACKGROUND

Increases in human longevity have made it critical to distinguish healthy longevity from longevity without regard to health. Current methods focus on expectations of healthy longevity, and are often limited to binary health outcomes (e.g., disabled vs. not disabled). We present a new matrix formulation for the statistics of healthy longevity, based on health prevalence data and Markov chain theory, applicable to any kind of health outcome and which provides variances and higher moments as well as expectations of healthy life.

METHOD

The model is based on a Markov chain description of the life course coupled with the moments of health outcomes ("rewards") at each age or stage. As an example, we apply the method to nine European countries using the SHARE survey data on the binary outcome of disability as measured by activities of daily living, and the continuous health outcome of hand grip strength.

RESULTS

We provide analytical formulas for the mean, variance, coefficient of variation, skewness and other statistical properties of healthy longevity. The analysis is applicable to binary, categorical, ordinal, or interval scale health outcomes. The results are easily evaluated in any matrix-oriented software. The SHARE results reveal familiar patterns for the expectation of life and of healthy life: women live longer than men but spend less time in a healthy condition. New results on the variance shows that the standard deviation of remaining healthy life declines with age, but the coefficient of variation is nearly constant. Remaining grip strength years decrease with age more dramatically than healthy years but their variability pattern is similar to the pattern of healthy years. Patterns are similar across nine European countries.

CONCLUSIONS

The method extends, in several directions, current calculations of health expectancy (HE) and disability-adjusted life years (DALYs). It applies to both categorical and continuous health outcomes, to combinations of multiple outcomes (e.g., death and disability in the formulation of DALYs) and to age- or stage-classified models. It reveals previously unreported patterns of variation among individuals in the outcomes of healthy longevity.

Demographic causes of adult sex ratio variation and their consequences for parental cooperation

The adult sex ratio (ASR) is a fundamental concept in population biology, sexual selection, and social evolution. However, it remains unclear which demographic processes generate ASR variation and how biases in ASR in turn affect social behaviour. Here, we evaluate the demographic mechanisms shaping ASR and their potential consequences for parental cooperation using detailed survival, fecundity, and behavioural data on 6119 individuals from six wild shorebird populations exhibiting flexible parental strategies. We show that these closely related populations express strikingly different ASRs, despite having similar ecologies and life histories, and that ASR variation is largely driven by sex differences in the apparent survival of juveniles. Furthermore, families in populations with biased ASRs were predominantly tended by a single parent, suggesting that parental cooperation breaks down with unbalanced sex ratios. Taken together, our results indicate that sex biases emerging during early life have profound consequences for social behaviour.

Estimating adult sex ratios in nature

Adult sex ratio (ASR, the proportion of males in the adult population) is a central concept in population and evolutionary biology, and is also emerging as a major factor influencing mate choice, pair bonding and parental cooperation in both human and non-human societies. However, estimating ASR is fraught with difficulties stemming from the effects of spatial and temporal variation in the numbers of males and females, and detection/capture probabilities that differ between the sexes. Here, we critically evaluate methods for estimating ASR in wild animal populations, reviewing how recent statistical advances can be applied to handle some of these challenges. We review methods that directly account for detection differences between the sexes using counts of unmarked individuals (observed, trapped or killed) and counts of marked individuals using mark-recapture models. We review a third class of methods that do not directly sample the number of males and females, but instead estimate the sex ratio indirectly using relationships that emerge from demographic measures, such as survival, age structure, reproduction and assumed dynamics. We recommend that detection-based methods be used for estimating ASR in most situations, and point out that studies are needed that compare different ASR estimation methods and control for sex differences in dispersal.This article is part of the themed issue 'Adult sex ratios and reproductive decisions: a critical re-examination of sex differences in human and animal societies'.

Age-structure density-dependent fertility and individuals dispersal in a population model

In this work, we analyze the interplay between general age structured density-dependent fertility functions and age classes dispersal in a patchy environment. As novelties, (i) the fertility function depends on age classes (instead of on the total population size) and (ii) dispersal patterns are also allowed to be different for individuals belonging to different age classes. Our results highlight the interplay between the shape of the age structured density-dependent fertility function and the age classes dispersal patterns. We analyze this interaction from an environmental management point of view by exploring the consequences of connecting patches that can sustain a population (source patch) or cannot (sink patch), as well as its relation to component Allee effects and strong Allee effects. In particular, we have found scenarios such that the metapopulation goes extinct when two isolated source patches are connect due to heterogeneous age classes distribution. On the contrary, there are settings such that heterogeneous age classes distribution enables two isolated sink patches to be sustainable when connected. Besides, we discuss what kind of local interventions are helpful to manage component Allee effect and its impact at the metopopulation level. The source code used to simulations is fully available. The code is presented as a knitr reproducible document in the open source R computing system. Thus, free access and usability of the code are granted.

Trait level analysis of multitrait population projection matrices

In most matrix population projection models, individuals are characterized according to, usually, one or two traits such as age, stage, size or location. A broad theory of multitrait population projection matrices (MPPMs) incorporating larger number of traits was long held back by time and space computational complexity issues. As a consequence, no study has yet focused on the influence of the structure of traits describing a life-cycle on population dynamics and life-history evolution. We present here a novel vector-based MPPM building methodology that allows to computationally-efficiently model populations characterized by numerous traits with large distributions, and extend sensitivity analyses for these models. We then present a new method, the trait level analysis consisting in folding an MPPM on any of its traits to create a matrix with alternative trait structure (the number of traits and their characteristics) but similar asymptotic properties. Adding or removing one or several traits to/from the MPPM and analyzing the resulting changes in spectral properties, allows investigating the influence of the trait structure on the evolution of traits. We illustrate this by modeling a 3-trait (age, parity and fecundity) population designed to investigate the implications of parity-fertilitytrade-offs in a context of fecundity heterogeneity in humans. The trait level analysis, comparing models of the same population differing in trait structures, demonstrates that fertility selection gradients differ between cases with or without parity-fertility trade-offs. Moreover it shows that age-specific fertility has seemingly very different evolutionary significance depending on whether heterogeneity is accounted for. This is because trade-offs can vary strongly in strength and even direction depending on the trait structure used to model the population.

Sex-specific early survival drives adult sex ratio bias in snowy plovers and impacts mating system and population growth

Adult sex ratio (ASR) is a central concept in population biology and a key factor in sexual selection, but why do most demographic models ignore sex biases? Vital rates often vary between the sexes and across life history, but their relative contributions to ASR variation remain poorly understood-an essential step to evaluate sex ratio theories in the wild and inform conservation. Here, we combine structured two-sex population models with individual-based mark-recapture data from an intensively monitored polygamous population of snowy plovers. We show that a strongly male-biased ASR (0.63) is primarily driven by sex-specific survival of juveniles rather than adults or dependent offspring. This finding provides empirical support for theories of unbiased sex allocation when sex differences in survival arise after the period of parental investment. Importantly, a conventional model ignoring sex biases significantly overestimated population viability. We suggest that sex-specific population models are essential to understand the population dynamics of sexual organisms: reproduction and population growth are most sensitive to perturbations in survival of the limiting sex. Overall, our study suggests that sex-biased early survival may contribute toward mating system evolution and population persistence, with implications for both sexual selection theory and biodiversity conservation.

Lifetime reproductive output: individual stochasticity, variance, and sensitivity analysis

Lifetime reproductive output (LRO) determines per-generation growth rates, establishes criteria for population growth or decline, and is an important component of fitness. Empirical measurements of LRO reveal high variance among individuals. This variance may result from genuine heterogeneity in individual properties, or from individual stochasticity, the outcome of probabilistic demographic events during the life cycle. To evaluate the extent of individual stochasticity requires the calculation of the statistics of LRO from a demographic model. Mean LRO is routinely calculated (as the net reproductive rate), but the calculation of variances has only recently received attention. Here, we present a complete, exact, analytical, closed-form solution for all the moments of LRO, for age- and stage-classified populations. Previous studies have relied on simulation, iterative solutions, or closed-form analytical solutions that capture only part of the sources of variance. We also present the sensitivity and elasticity of all of the statistics of LRO to parameters defining survival, stage transitions, and (st)age-specific fertility. Selection can operate on variance in LRO only if the variance results from genetic heterogeneity. The potential opportunity for selection is quantified by Crow’s index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {I}$\end{document}I, the ratio of the variance to the square of the mean. But variance due to individual stochasticity is only an apparent opportunity for selection. In a comparison of a range of age-classified models for human populations, we find that proportional increases in mortality have very small effects on the mean and variance of LRO, but large positive effects on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {I}$\end{document}I. Proportional increases in fertility increase both the mean and variance of LRO, but reduce \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {I}$\end{document}I. For a size-classified tree population, the elasticity of both mean and variance of LRO to stage-specific mortality are negative; the elasticities to stage-specific fertility are positive.

Evaluating mortality rates with a novel integrated framework for nonmonogamous species

The conservation of wildlife requires management based on quantitative evidence, and especially for large carnivores, unraveling cause-specific mortalities and understanding their impact on population dynamics is crucial. Acquiring this knowledge is challenging because it is difficult to obtain robust long-term data sets on endangered populations and, usually, data are collected through diverse sampling strategies. Integrated population models (IPMs) offer a way to integrate data generated through different processes. However, IPMs are female-based models that cannot account for mate availability, and this feature limits their applicability to monogamous species only. We extended classical IPMs to a two-sex framework that allows investigation of population dynamics and quantification of cause-specific mortality rates in nonmonogamous species. We illustrated our approach by simultaneously modeling different types of data from a reintroduced, unhunted brown bear (Ursus arctos) population living in an area with a dense human population. In a population mainly driven by adult survival, we estimated that on average 11% of cubs and 61% of adults died from human-related causes. Although the population is currently not at risk, adult survival and thus population dynamics are driven by anthropogenic mortality. Given the recent increase of human-bear conflicts in the area, removal of individuals for management purposes and through poaching may increase, reversing the positive population growth rate. Our approach can be generalized to other species affected by cause-specific mortality and will be useful to inform conservation decisions for other nonmonogamous species, such as most large carnivores, for which data are scarce and diverse and thus data integration is highly desirable.

Sensitivity and elasticity analysis of a Lur'e system used to model a population subject to density-dependent reproduction

Sensitivity and elasticity analyzes have become central to the analysis of models in population biology and ecology. While much work has been done applying sensitivity and elasticity analysis to study density-independent (linear) matrix and integral projection models, little work has been done to study the sensitivity and elasticity of density-dependent models, especially integral projection models. In this paper we derive sensitivity and elasticity formulas for the equilibrium population n^* of a structured population modeled by a Lur'e system, which consists of a linear system plus a nonlinearity modeling density-dependent fecundity. Sensitivity and elasticity formulas are easy to interpret ecologically, and we apply these formulas to published models for Chinook Salmon and Platte thistle (Cirsium canescens). In the C. canescens example we show that models with identical equilibrium populations can have sensitivities that are an order-of-magnitude apart, depending on the functional form for the nonlinearity.

Modelling effects of nonbreeders on population growth estimates

Adult individuals that do not breed in a given year occur in a wide range of natural populations. However, such nonbreeders are often ignored in theoretical and empirical population studies, limiting our knowledge of how nonbreeders affect realized and estimated population dynamics and potentially impeding projection of deterministic and stochastic population growth rates. We present and analyse a general modelling framework for systems where breeders and nonbreeders differ in key demographic rates, incorporating different forms of nonbreeding, different life histories and frequency-dependent effects of nonbreeders on demographic rates of breeders. Comparisons of estimates of deterministic population growth rate, λ, and demographic variance, σd2, from models with and without distinct nonbreeder classes show that models that do not explicitly incorporate nonbreeders give upwardly biased estimates of σd2, particularly when the equilibrium ratio of nonbreeders to breeders, Nnb∗/Nb∗, is high. Estimates of λ from empirical observations of breeders only are substantially inflated when individuals frequently re-enter the breeding population after periods of nonbreeding. Sensitivity analyses of diverse parameterizations of our model framework, with and without negative frequency-dependent effects of nonbreeders on breeder demographic rates, show how changes in demographic rates of breeders vs. nonbreeders differentially affect λ. In particular, λ is most sensitive to nonbreeder parameters in long-lived species, when Nnb∗/Nb∗>0, and when individuals are unlikely to breed at several consecutive time steps. Our results demonstrate that failing to account for nonbreeders in population studies can obscure low population growth rates that should cause management concern. Quantifying the size and demography of the nonbreeding section of populations and modelling appropriate demographic structuring is therefore essential to evaluate nonbreeders' influence on deterministic and stochastic population dynamics.

A demographic model for sex ratio evolution and the effects of sex-biased offspring costs

The evolution of the primary sex ratio, the proportion of male births in an individual's offspring production strategy, is a frequency-dependent process that selects against the more common sex. Because reproduction is shaped by the entire life cycle, sex ratio theory would benefit from explicitly two-sex models that include some form of life cycle structure. We present a demographic approach to sex ratio evolution that combines adaptive dynamics with nonlinear matrix population models. We also determine the evolutionary and convergence stability of singular strategies using matrix calculus. These methods allow the incorporation of any population structure, including multiple sexes and stages, into evolutionary projections. Using this framework, we compare how four different interpretations of sex-biased offspring costs affect sex ratio evolution. We find that demographic differences affect evolutionary outcomes and that, contrary to prior belief, sex-biased mortality after parental investment can bias the primary sex ratio (but not the corresponding reproductive value ratio). These results differ qualitatively from the widely held conclusions of previous models that neglect demographic structure.

Changes in seasonal climate outpace compensatory density-dependence in eastern brook trout

Understanding how multiple extrinsic (density-independent) factors and intrinsic (density-dependent) mechanisms influence population dynamics has become increasingly urgent in the face of rapidly changing climates. It is particularly unclear how multiple extrinsic factors with contrasting effects among seasons are related to declines in population numbers and changes in mean body size and whether there is a strong role for density-dependence. The primary goal of this study was to identify the roles of seasonal variation in climate driven environmental direct effects (mean stream flow and temperature) vs. density-dependence on population size and mean body size in eastern brook trout (Salvelinus fontinalis). We use data from a 10-year capture-mark-recapture study of eastern brook trout in four streams in Western Massachusetts, USA to parameterize a discrete-time population projection model. The model integrates matrix modeling techniques used to characterize discrete population structures (age, habitat type, and season) with integral projection models (IPMs) that characterize demographic rates as continuous functions of organismal traits (in this case body size). Using both stochastic and deterministic analyses we show that decreases in population size are due to changes in stream flow and temperature and that these changes are larger than what can be compensated for through density-dependent responses. We also show that the declines are due mostly to increasing mean stream temperatures decreasing the survival of the youngest age class. In contrast, increases in mean body size over the same period are the result of indirect changes in density with a lesser direct role of climate-driven environmental change.

COMADRE: a global data base of animal demography

The open‐data scientific philosophy is being widely adopted and proving to promote considerable progress in ecology and evolution. Open‐data global data bases now exist on animal migration, species distribution, conservation status, etc. However, a gap exists for data on population dynamics spanning the rich diversity of the animal kingdom world‐wide. This information is fundamental to our understanding of the conditions that have shaped variation in animal life histories and their relationships with the environment, as well as the determinants of invasion and extinction.

Matrix population models (MPMs) are among the most widely used demographic tools by animal ecologists. MPMs project population dynamics based on the reproduction, survival and development of individuals in a population over their life cycle. The outputs from MPMs have direct biological interpretations, facilitating comparisons among animal species as different as Caenorhabditis elegans, Loxodonta africana and Homo sapiens.

Thousands of animal demographic records exist in the form of MPMs, but they are dispersed throughout the literature, rendering comparative analyses difficult. Here, we introduce the COMADRE Animal Matrix Database, an open‐data online repository, which in its version 1.0.0 contains data on 345 species world‐wide, from 402 studies with a total of 1625 population projection matrices. COMADRE also contains ancillary information (e.g. ecoregion, taxonomy, biogeography, etc.) that facilitates interpretation of the numerous demographic metrics that can be derived from its MPMs. We provide R code to some of these examples.

Synthesis: We introduce the COMADRE Animal Matrix Database, a resource for animal demography. Its open‐data nature, together with its ancillary information, will facilitate comparative analysis, as will the growing availability of databases focusing on other aspects of the rich animal diversity, and tools to query and combine them. Through future frequent updates of COMADRE, and its integration with other online resources, we encourage animal ecologists to tackle global ecological and evolutionary questions with unprecedented sample size.

Calculating second derivatives of population growth rates for ecology and evolution

1. Second derivatives of the population growth rate measure the curvature of its response to demographic, physiological or environmental parameters. The second derivatives quantify the response of sensitivity results to perturbations, provide a classification of types of selection and provide one way to calculate sensitivities of the stochastic growth rate.

2. Using matrix calculus, we derive the second derivatives of three population growth rate measures: the discrete-time growth rate λ, the continuous-time growth rate r = log λ and the net reproductive rate R₀, which measures per-generation growth.

3. We present a suite of formulae for the second derivatives of each growth rate and show how to compute these derivatives with respect to projection matrix entries and to lower-level parameters affecting those matrix entries.

4. We also illustrate several ecological and evolutionary applications for these second derivative calculations with a case study for the tropical herb Calathea ovandensis.

Inherent demographic stability in mutualist-resource-exploiter interactions

Core principles of ecological theory predict that, in the absence of other factors, mutualisms should experience destabilizing positive feedback and should be vulnerable to extinction through competitive exclusion by exploiter species. Many effective stabilizing mechanisms address one issue or the other, and many turn upon additional features. Using an explicitly demographic approach, I show that indirect, demography-mediated interactions between mutualists and exploiters can enable mutualist-exploiter coexistence, which in turn can stabilize the abundances of mutualists, exploiters, and their shared resources. This occurs because of the distinct resource demographic responses that are inherent to interaction with mutualistic and exploitative partners and can occur in long-lasting, exclusive interactions, such as protection mutualisms, as well as in apparently very different, short-lived mutualistic interactions, such as pollination. The key necessary factor-demographic response to interspecific interaction-is common in nature. Some demographic structure is also necessary and is generated through interspecific interaction in long-lasting associations; it is also very common in natural populations. Thus, the explicitly demographic and multispecies approach taken here constitutes a potentially promising single explanation for the apparent stability of mutualism in a wide range of natural systems.

Frequency-dependent population dynamics: effect of sex ratio and mating system on the elasticity of population growth rate

When vital rates depend on population structure (e.g., relative frequencies of males or females), an important question is how the long-term population growth rate λ responds to changes in rates. For instance, availability of mates may depend on the sex ratio of the population and hence reproductive rates could be frequency-dependent. In such cases change in any vital rate alters the structure, which in turn, affect frequency-dependent rates. We show that the elasticity of λ to a rate is the sum of (i) the effect of the linear change in the rate and (ii) the effect of nonlinear changes in frequency-dependent rates. The first component is always positive and is the classical elasticity in density-independent models obtained directly from the population projection matrix. The second component can be positive or negative and is absent in density-independent models. We explicitly express each component of the elasticity as a function of vital rates, eigenvalues and eigenvectors of the population projection matrix. We apply this result to a two-sex model, where male and female fertilities depend on adult sex ratio α (ratio of females to males) and the mating system (e.g., polygyny) through a harmonic mating function. We show that the nonlinear component of elasticity to a survival rate is negligible only when the average number of mates (per male) is close to α. In a strictly monogamous species, elasticity to female survival is larger than elasticity to male survival when α<1 (less females). In a polygynous species, elasticity to female survival can be larger than that of male survival even when sex ratio is female biased. Our results show how demography and mating system together determine the response to selection on sex-specific vital rates.

Why do lifespan variability trends for the young and old diverge? A perturbation analysis

BACKGROUND

Variation in lifespan has followed strikingly different trends for the young and old: while total lifespan variability has decreased as life expectancy at birth has risen, the variability conditional on survival to older ages has increased. These diverging trends reflect changes in the underlying demographic parameters determining age-specific mortality.

OBJECTIVE

We ask why the variation in the ages at death after survival to adult ages has followed a different trend than the variation at younger ages, and aim to explain the divergence in terms of the age pattern of historical mortality changes.

METHODS

Using simulations, we show that the empirical trends in lifespan variation are well characterized using the Siler model, which describes the mortality trajectory using functions representing early-life, later-life, and background mortality. We then obtain maximum likelihood estimates of the Siler parameters for Swedish females from 1900 to 2010. We express mortality in terms of a Markov chain model, and apply matrix calculus to compute the sensitivity of age-specific variance trends to the changes in Siler model parameters.

RESULTS

Our analysis quantifies the influence of changing demographic parameters on lifespan variability at all ages, highlighting the influence of declining childhood mortality on the reduction of lifespan variability, and the influence of subsequent improvements in adult survival on the rising variability of lifespans at older ages.

CONCLUSIONS

These findings provide insight into the dynamic relationship between the age pattern of survival improvements and time trends in lifespan variability.

Individual fitness and phenotypic selection in age-structured populations with constant growth rates

Powerful multiple regression-based approaches are commonly used to measure the strength of phenotypic selection, which is the statistical association between individual fitness and trait values. Age structure and overlapping generations complicate determinations of individual fitness, contributing to the popularity of alternative methods for measuring natural selection that do not depend upon such measures. The application of regression-based techniques for measuring selection in these situations requires a demographically appropriate, conceptually sound, and observable measure of individual fitness. It has been suggested that Fisher's reproductive value applied to an individual at its birth is such a definition. Here I offer support for this assertion by showing that multiple regression applied to this measure and vital rates (age-specific survival and fertility rates) yields the same selection gradients for vital rates as those inferred from Hamilton's classical results. I discuss how multiple regressions, applied to individual reproductive value at birth, can be used efficiently to estimate measures of phenotypic selection that are problematic for sensitivity analyses. These include nonlinear selection, components of the opportunity for selection, and multilevel selection.

The effects of disease-induced juvenile mortality on the transient and asymptotic population dynamics of Chinook salmon (Oncorhynchus tshawytscha)

The effects of an increased disease mortality rate on the transient and asymptotic dynamics of Chinook salmon (Oncorhynchus tshawytscha) were investigated. Disease-induced mortality of juvenile salmon has become a serious concern in recent years. However, the overall effects of disease mortality on the asymptotic and transient dynamics of adult spawning abundance are still largely unknown. We explored various scenarios with regard to the density-dependent process, the distribution of survivorship over the juvenile phase, the disease mortality rate, and the infusion of stray hatchery fish. Our results suggest that the sensitivity to the disease mortality rate of the equilibrium adult spawning abundance and resilience (asymptotic return rate toward this equilibrium following a small perturbation) varied widely and differently depending on the scenario. The resilience and coefficient of variation of adult spawning abundance following a large perturbation were consistent with each other under the scenarios investigated. We conclude that the increase in disease mortality likely has an effect on fishery yield under a fluctuating environment, not only because the mean equilibrium adult spawning abundance has likely been reduced, but also because the resilience has likely decreased and the variance in adult spawning abundance has likely increased. We also infer the importance of incorporating finer-scale spatiotemporal information into population models and demonstrate a means for doing so within a matrix population modeling framework.

Perturbation analysis of indices of lifespan variability

A number of indices exist to calculate lifespan variation, each with different underlying properties. Here, we present new formulae for the response of seven of these indices to changes in the underlying mortality schedule (life disparity, Gini coefficient, standard deviation, variance, Theil's index, mean logarithmic deviation, and interquartile range). We derive each of these indices from an absorbing Markov chain formulation of the life table, and use matrix calculus to obtain the sensitivity and the elasticity (i.e., the proportional sensitivity) to changes in age-specific mortality. Using empirical French and Russian male data, we compare the underlying sensitivities to mortality change under different mortality regimes to determine the conditions under which the indices might differ in their conclusions about the magnitude of lifespan variation. Finally, we demonstrate how the sensitivities can be used to decompose temporal changes in the indices into contributions of age-specific mortality changes. The result is an easily computable method for calculating the properties of this important class of longevity indices.

A seasonal, density-dependent model for the management of an invasive weed

The population effects of harvest depend on complex interactions between density dependence, seasonality, stage structure, and management timing. Here we present a periodic nonlinear matrix population model that incorporates seasonal density dependence with stage-selective and seasonally selective harvest. To this model, we apply newly developed perturbation analyses to determine how population densities respond to changes in harvest and demographic parameters. We use the model to examine the effects of popular control strategies and demographic perturbations on the invasive weed garlic mustard (Alliaria petiolata). We find that seasonality is a major factor in harvest outcomes, because population dynamics may depend significantly on both the season of management and the season of observation. Strategies that reduce densities in one season can drive increases in another, with strategies giving positive sensitivities of density in the target seasons leading to compensatory effects that invasive species managers should avoid. Conversely, demographic parameters to which density is very elastic (e.g., seeding survival, second-year rosette spring survival, and the flowering to fruiting adult transition for maximum summer densities) may indicate promising management targets.

Transient population dynamics of mosquitoes during sterile male releases: modelling mating behaviour and perturbations of life history parameters

The release of genetically-modified or sterile male mosquitoes offers a promising form of mosquito-transmitted pathogen control, but the insights derived from our understanding of male mosquito behaviour have not fully been incorporated into the design of such genetic control or sterile-male release methods. The importance of aspects of male life history and mating behaviour for sterile-male release programmes were investigated by projecting a stage-structured matrix model over time. An elasticity analysis of transient dynamics during sterile-male releases was performed to provide insight on which vector control methods are likely to be most synergistic. The results suggest that high mating competitiveness and mortality costs of released males are required before the sterile-release method becomes ineffective. Additionally, if released males suffer a mortality cost, older males should be released due to their increased mating capacity. If released males are of a homogenous size and size-assortative mating occurs in nature, this can lead to an increase in the abundance of large females and reduce the efficacy of the population-suppression effort. At a high level of size-assortative mating, the disease transmission potential of the vector population increases due to male releases, arguing for the release of a heterogeneously-sized male population. The female population was most sensitive to perturbations of density-dependent components of larval mortality and female survivorship and fecundity. These findings suggest source reduction might be a particularly effective complement to mosquito control based on the sterile insect technique (SIT). In order for SIT to realize its potential as a key component of an integrated vector-management strategy to control mosquito-transmitted pathogens, programme design of sterile-male release programmes must account for the ecology, behaviour and life history of mosquitoes. The model used here takes a step in this direction and can easily be modified to investigate additional aspects of mosquito behaviour or species-specific ecology.

Impacts of climate change on avian populations

This review focuses on the impacts of climate change on population dynamics. I introduce the MUP (Measuring, Understanding, and Predicting) approach, which provides a general framework where an enhanced understanding of climate-population processes, along with improved long-term data, are merged into coherent projections of future population responses to climate change. This approach can be applied to any species, but this review illustrates its benefit using birds as examples. Birds are one of the best-studied groups and a large number of studies have detected climate impacts on vital rates (i.e., life history traits, such as survival, maturation, or breeding, affecting changes in population size and composition) and population abundance. These studies reveal multifaceted effects of climate with direct, indirect, time-lagged, and nonlinear effects. However, few studies integrate these effects into a climate-dependent population model to understand the respective role of climate variables and their components (mean state, variability, extreme) on population dynamics. To quantify how populations cope with climate change impacts, I introduce a new universal variable: the 'population robustness to climate change.' The comparison of such robustness, along with prospective and retrospective analysis may help to identify the major climate threats and characteristics of threatened avian species. Finally, studies projecting avian population responses to future climate change predicted by IPCC-class climate models are rare. Population projections hinge on selecting a multiclimate model ensemble at the appropriate temporal and spatial scales and integrating both radiative forcing and internal variability in climate with fully specified uncertainties in both demographic and climate processes.

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.