The many growth rates and elasticities of populations in random environments

Despite considerable interest in the dynamics of populations subject to temporally varying environments, alternate population growth rates and their sensitivities remain incompletely understood. For a Markovian environment, we compare and contrast the meanings of the stochastic growth rate (lambdaS), the growth rate of average population (lambdaM), the growth rate for average transition rates (lambdaA), and the growth rate of an aggregate represented by a megamatrix (shown here to equal lambdaM). We distinguish these growth rates by the averages that define them. We illustrate our results using data on an understory shrub in a hurricane-disturbed landscape, employing a range of hurricane frequencies. We demonstrate important differences among growth rates: lambdaS lambdaM. We show that stochastic elasticity, ESij, and megamatrix elasticity, EMij, describe a complex perturbation of both means and variances of rates by the same proportion. Megamatrix elasticities respond slightly and stochastic elasticities respond strongly to changing the frequency of disturbance in the habitat (in our example, the frequency of hurricanes). The elasticity EAij of lambdaA does not predict changes in the other elasticities. Because ES, although commonly utilized, is difficult to interpret, we introduce elasticities with a more direct interpretation: ESmu for perturbations of means and ESsigma for variances. We argue that a fundamental tool for studying selection pressures in varying environments is the response of growth rate to vital rates in all habitat states.     

Time, transients and elasticity

How does life history affects the short‐term elasticities of population growth rate? We decompose short‐term elasticity as a sum of (i) the effect of the perturbation in rates on the unperturbed population structure and (ii) the effect of the original vital rates on the difference in structure between the original and the perturbed population. We provide exact analytical formulas for these components. In a population at its stable stage distribution (SSD), short‐term elasticity is determined mainly by the SSD and reproductive value. In a non‐stable population, short‐term elasticity depends also on the projection of initial structure on the SSD, equal to population momentum. Non‐stable stage structures matter most to elasticity if stages are missing that take time to fill in. We show how the demographic damping rate of the original population determines the rate at which short‐term elasticity converges to its limiting values.     

Sensitivity of the population growth rate to demographic variability within and between phases of the disturbance cycle

For species in disturbance-prone ecosystems, vital rates (survival, growth and reproduction) often vary both between and within phases of the cycle of disturbance and recovery; some of this variation is imposed by the environment, but some may represent adaptation of the life history to disturbance. Anthropogenic changes may amplify or impede these patterns of variation, and may have positive or negative effects on population growth. Using stochastic population projection matrix models, we develop stochastic elasticities (proportional derivatives of the long-run population growth rate) to gauge the population effects of three types of change in demographic variability (changes in within- and between-disturbance-phase variability and phase-specific changes). Computing these elasticities for five species of disturbance-influenced perennial plants, we pinpoint demographic rates that may reveal adaptation to disturbance, and we demonstrate that species may differ in their responses to different types of changes in demographic variability driven by climate change.     

Buffering and plasticity in vital rates of oldfield rodents

1. Under the hypothesis of environmental buffering, populations are expected to minimize the variance of the most influential vital rates; however, this may not be a universal principle. Species with a life span <1 year may be less likely to exhibit buffering because of temporal or seasonal variability in vital rate sensitivities. Further, plasticity in vital rates may be adaptive for species in a variable environment with reliable cues. 2. We tested for environmental buffering and plasticity in vital rates using stage-structured matrix models from long-term data sets in four species of grassland rodents. We used periodic matrices to estimate stochastic elasticity for each vital rate and then tested for correlations with a standardized coefficient of variation for each rate. 3. We calculated stochastic elasticities for individual months to test for an association between increased reproduction and the influence of reproduction, relative to survival, on the population growth rate. 4. All species showed some evidence of buffering. The elasticity of vital rates of Peromyscus leucopus (Rafinesque, 1818), Sigmodon hispidus Say & Ord, 1825 and Microtus ochrogaster (Wagner, 1842) was negatively related to vital rate CV. Elasticity and vital rate CV were negatively related in Peromyscus maniculatus (Wagner, 1845), but the relationship was not statistically significant. Peromyscus leucopus and M. ochrogaster showed plasticity in vital rates; reproduction was higher following months where elasticity for reproduction exceeded that of survival. 5. Our results suggest that buffering is common in species with fast life histories; however, some populations that exhibit buffering are capable of responding to short-term variability in environmental conditions through reproductive plasticity.     

The influence of time since introduction on the population growth of introduced species and the consequences for management

Several processes likely act to change the demographic rates of introduced species over time, and this changing demography could influence the optimal management of invasive populations. Optimal management strategies should be derived based on the demography. However, we have a poor understanding of the degree to which the demography of introduced species changes following initial introduction. We used published matrix population models of introduced plant populations to test how population growth rate and elasticity change with time since introduction. We did not find a significant relationship between population growth rate and time since introduction. However, elasticity to stasis increased while elasticity to growth decreased with time since introduction. Broadly, as time since introduction progressed the elasticities of the introduced plant populations became more similar to those that have been reported for native species. These results suggest that the optimal management strategy should be derived incorporating elasticity through time, especially when the time scope of management is long or the available demographic data were obtained in the past.     

Recruitment rates exhibit high elasticity and high temporal variation in populations of a short-lived perennial herb

Empirical studies for different life histories have shown an inverse relationship between elasticity (i.e. the proportional contribution to population growth rate) and temporal variation in vital rates. It is accepted that this relationship indicates the effect of selective pressures in reducing variation in those life‐history traits with a major impact on fitness. In this paper, we sought to determine whether changes in environmental conditions affect the relationship between elasticity of vital rates and their temporal variation, and whether vital rates with simultaneously large elasticity and temporal variation might represent a characteristic life‐history strategy. We used demographic data on 13 populations of the short‐lived Hypericum cumulicola over 5–6 years, in three time‐since‐fire classes. For each population of each time‐since‐fire, we computed the mean matrix over years and its respective elasticity matrix, and the coefficients of variation in matrix entries over study years as an estimate of temporal variability. We found that mean elasticity negatively significantly correlated with temporal variation in vital rates in populations (overall eight out of 13) included in each time‐since‐fire. However, seedling recruitment exhibited both high elasticity and high temporal variation in almost all study populations. These results indicated that (1) the general relationship between elasticity and temporal variation in vital rates was not modified by environmental changes due to time‐since‐fire, and (2) high elasticity and high temporal variation in seedling recruitment in H. cumulicola is a particular trait of the species' life history. After seed survival in the soil seed bank, seedling recruitment represents the most important life‐history trait influencing H. cumulicola population growth rate (and fitness). The high temporal variability in seedling recruitment suggests that this trait is determined by environmental cues, leading to an increase in population size and subsequent replenishment of the seed bank in favorable years.     

IS SURVIVORSHIP A BETTER FITNESS SURROGATE THAN FECUNDITY?

Although fitness depends on both survivorship and fecundity, we tend to assume fecundity relates to fitness more directly than survivorship. In fact, several recent ecological studies suggest fitness depends more heavily on annual survivorship than annual fecundity for most taxa with lifespans longer than one year. These studies review elasticities of transition matrices for a broad range of taxa. Elasticities covary monotonically with selection gradients for demographic rates and are identical to selection gradients for traits rescaled to have mean values of zero and variance of one. For all taxa except semelparous perennial plants, adult survivorship has consistently higher elasticity than other suites of demographic rates. Fecundity only rarely has the highest elasticity. Thus, differences in yearly survival affect fitness disproportionately more than differences in yearly fecundity, even in many exponentially growing populations. This pattern reinforces the importance of interpreting the contribution of vital rates to fitness in the context of life history and population dynamics.     

Evolution in the real world: stochastic variation and the determinants of fitness in Carlina vulgaris

Empirical studies of life histories often ignore stochastic variation, despite theoretical demonstrations of its potential impact on life-history evolution. Here we use a novel approach to explore the effects of stochastic variation on life-history evolution and estimate the selection pressures operating on the monocarpic perennial Carlina vulgaris, in which flowering may be delayed by up to eight years. The approach is novel in that we use modern theoretical techniques to estimate selection pressures and the fitness landscape from a fully parameterised individual-based model. These approaches take into account temporal variation in demographic rates and density dependence. Analysis of 16 years' data revealed significant temporal variation in growth, mortality, and recruitment in our study population. Flowering was strongly size dependent and, unusually for such a species, also age dependent. Individual-based models of the flowering strategy, parameterized using field data, consistently underestimated the size at flowering, when temporal variation in demographic rates was ignored. In contrast, models that incorporated temporal variation in growth, mortality, and recruitment predicted sizes at flowering not significantly different from those observed in the field. Temporal variation in mortality, which had the largest effect on the flowering strategy, selected for increased size at flowering. An analytical approximation is presented to explain this result, extending the "1-year look-ahead criterion" presented in Rees et al. (2000). A fitness landscape generated by following the fate of rare mutant invaders with a broad range of alternative flowering strategies demonstrated that the observed parameters were adaptive. However, the fitness landscape reveals that approximately equal fitness is achieved by a broad range of strategies, providing a mechanism for the maintenance of genetic variation. To understand how the different parameters that defined our models determine the fitness of rare mutants, we numerically estimated the elasticities and sensitivities of mutant fitness. This demonstrated strong selection on a number of the parameters. Elasticities and sensitivities estimated in constant and random environments were significantly positively correlated, and both were negatively related to the standard error of the parameter. This last result is surprising and, we argue, reflects the genetic and phenotypic responses to selection.     

Comparative demography of two giant caulescent rosettes (Espeletia timotensis and E. spicata) from the high tropical Andes

Using field data from previous studies we built matrix models for two populations of giant rosettes, Espeletia timotensis Cuatrec. and E. spicata Sch. Bip. Wedd., from the Andes Cordillera in Mérida, Venezuela. We analysed the models and calculated population growth rate (λ), sensitivities, elasticities and the sensitivity of the elasticities to changes in the vital rates. The analysis showed that the two species behave alike in general demographic terms. In both models, population growth rate is positive and sensitivities of λ to changes in vital rates decrease markedly in this order: plant establishment, progression of juvenile–adult, germination and survival. The relative contributions of vital rates to λ (elasticities) are very similar to those of other woody plant species: a higher contribution of survival and a very low contribution of fecundity. Transition from seedling to juvenile is most important and the younger established stages (juveniles and young adults) play a predominant demographic role in both populations. Seed banks and older adults are playing a relatively minor role in the dynamics of both populations. However, they may be important in relation to unpredictable, favourable or detrimental events. Perturbation analysis of elasticities showed that increasing the rate of plant establishment will decrease the relative importance of stasis. We conclude that both species are demographically very close, and similar to other long‐lived woody plant species. However, the two species differ in the role of the seed bank, which seems more important in the demography of E. spicata than in E. timotensis.     

The generality of management recommendations across populations of an invasive perennial herb

Demographic models are widely used to produce management recommendations for different species. For invasive plants, current management recommendations to control local population growth are often based on data from a limited number of populations per species, and the assumption of stable population structure (asymptotic dynamics). However, spatial variation in population dynamics and deviation from a stable structure may affect these recommendations, calling into question their generality across populations of an invasive species. Here, I focused on intraspecific variation in population dynamics and investigated management recommendations generated by demographic models across 37 populations of a short-lived, invasive perennial herb (Lupinus polyphyllus). Models that relied on the proportional perturbations of vital rates (asymptotic elasticities) indicated an essential role for plant survival in long-term population dynamics. The rank order of elasticities for different vital rates (survival, growth, retrogression, fecundity) varied little among the 37 study populations regardless of population status (increasing or declining asymptotically). Summed elasticities for fecundity increased, while summed elasticities for survival decreased with increasing long-term population growth rate. Transient dynamics differed from asymptotic dynamics, but were qualitatively similar among populations, that is, depending on the initial size structure, populations tended to either increase or decline in density more rapidly than predicted by asymptotic growth rate. These findings indicate that although populations are likely to exhibit transient dynamics, management recommendations based on asymptotic elasticities for vital rates might be to some extent generalised across established populations of a given short-lived invasive plant species.     

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

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

Population Dynamics of the Andean Lizard Anolis heterodermus: Fast‐slow Demographic Strategies in Fragmented Scrubland Landscapes

Habitat fragmentation and loss affect population stability and demographic processes, increasing the extinction risk of species. We studied Anolis heterodermus populations inhabiting large and small Andean scrubland patches in three fragmented landscapes in the Sabana de Bogotá (Colombia) to determine the effect of habitat fragmentation and loss on population dynamics. We used the capture‐mark‐recapture method and multistate models to estimate vital rates for each population. We estimated growth population rate and the most important processes that affect λ by elasticity analysis of vital rates. We tested the effects of habitat fragmentation and loss on vital rates of lizard populations. All six isolated populations showed a positive or an equilibrium growth rate (λ = 1), and the most important demographic process affecting λ was the growth to first reproduction. Populations from landscapes with less scrubland natural cover showed higher stasis of young adults. Populations in highly fragmented landscapes showed highest juvenile survival and growth population rates. Independent of the landscape's habitat configuration and connectivity, populations from larger scrubland patches showed low adult survivorship, but high transition rates. Populations varied from a slow strategy with low growth and delayed maturation in smaller patches to a fast strategy with high growth and early maturation in large patches. This variation was congruent with the fast‐slow continuum hypothesis and has serious implications for Andean lizard conservation and management strategies. We suggest that more stable lizard populations will be maintained if different management strategies are adopted according to patch area and habitat structure.     

A cautionary note on elasticity analyses in a ternary plot using randomly generated population matrices

Matrix population models are one of the most common mathematical models in ecology, which describe the dynamics of stage-structured populations and provide us many population statistics. One of the statistics, elasticity onto population growth rate, is frequently used and represents the degree of the relative impact of life history parameters to the population growth rate. Due to the utility of elasticities for cross-taxonomic comparisons, Silvertown and his coauthors have published multiple papers and reported the relationship between elasticities and life forms (or life history) in multiple plant species, using a triangle map (called “ternary plot”). To understand why their elasticities are located in specific regions of the ternary plot, we constructed four archetypes of population matrices, from which we simulated 24,000 randomly generated population matrices and obtained the consequent elasticities. We found a large discrepancy when comparing our results to those in Silvertown et al.'s study (Conserv Biol 10:591–597, 1996): for our simulated matrices where rapid transitions were not allowed (e.g., trees), the elasticity distribution resulted in a line across the ternary plot. We provided the mathematical proof for this result, and found that its slope depends on matrix dimension. We also used 1230 matrices from the COMPADRE Plant Matrix Database and calculated the elasticities. Our simulated results were validated with field data from COMPADRE: two straight lines appeared in the ternary plot. Furthermore, we answered several addressed questions, such as, “Is there any special elasticity distribution in matrices with high population growth rates?” and “Why are the elasticities of natural populations concentrated in the upper half of the ternary plot?”.     

Buffering of life histories against environmental stochasticity: accounting for a spurious correlation between the variabilities of vital rates and their contributions to fitness

Life-history theory predicts vital rates that on average make large contributions to the annual multiplication rate of a lineage should be highly buffered against environmental variability. This prediction has been tested by looking for a negative correlation between the sensitivities (or elasticities) of the elements in a projection matrix and their variances (or coefficients of variation). Here, we show by constructing random matrices that a spurious negative correlation exists between the sensitivities and variances, and between the elasticities and coefficients of variation, of matrix elements. This spurious correlation arises in part because size transition probabilities, which are bounded by 0 and 1, have a limit to their variability that often does not apply to matrix elements representing reproduction. We advocate an alternative analysis based on the underlying vital rates (not the matrix elements) that accounts for the inherent limit to the variability of zero-to-one vital rates, corrects for sampling variation, and tests for a declining upper limit to variability as a vital rate's fitness contribution increases. Applying this analysis to demographic data from five populations of the alpine cushion plant Silene acaulis, we provide evidence of stronger buffering in the vital rates that most influence fitness.     

Demography when history matters: construction and analysis of second-order matrix population models

History matters when individual prior conditions contain important information about the fate of individuals. We present a general framework for demographic models which incorporates the effects of history on population dynamics. The framework incorporates prior condition into the i-state variable and includes an algorithm for constructing the population projection matrix from information on current state dynamics as a function of prior condition. Three biologically motivated classes of prior condition are included: prior stages, linear functions of current and prior stages, and equivalence classes of prior stages. Taking advantage of the matrix formulation of the model, we show how to calculate sensitivity and elasticity of any demographic outcome. Prior condition effects are a source of inter-individual variation in vital rates, i.e., individual heterogeneity. As an example, we construct and analyze a second-order model of Lathyrus vernus, a long-lived herb. We present population growth rate, the stable population distribution, the reproductive value vector, and the elasticity of λ to changes in the second-order transition rates. We quantify the contribution of prior conditions to the total heterogeneity in the stable population of Lathyrus using the entropy of the stable distribution.     

Elasticities in variable environments: properties and implications

Elasticities in stochastic matrix models are used to understand both population and evolutionary dynamics. We examine three such elasticities: stochastic elasticity E(ij)(S) with respect to the (i, j) matrix element, the elasticity E(ij)(S mu) with respect to the mean mu(ij) of the matrix element, and the elasticity E(ij)(S sigma) with respect to the variability sigma(ij) of the matrix element. We show that the stochastic elasticity E(S) does not accurately describe the effect of variability; one should use E(S sigma) and E(S mu). We establish two general properties of these elasticities: a sum rule that connects them and a limit on the sum of the E(S sigma). We discuss the implications of these properties for the analysis of buffering and selection on the average rates versus the variability of rates.     

Managing multiple vital rates to maximize greater sage-grouse population growth

Despite decades of field research on greater sage-grouse, range-wide demographic data have yet to be synthesized into a sensitivity analysis to guide management actions. We reviewed range-wide demographic rates for greater sage-grouse from 1938 to 2011 and used data from 50 studies to parameterize a 2-stage, female-based population matrix model. We conducted life-stage simulation analyses to determine the proportion of variation in population growth rate (λ) accounted for by each vital rate, and we calculated analytical sensitivity, elasticity, and variance-stabilized sensitivity to identify the contribution of each vital rate to λ. As expected for an upland game bird, greater sage-grouse showed marked annual and geographic variation in several vital rates. Three rates were demonstrably important for population growth: female survival, chick survival, and nest success. Female survival and chick survival, in that order, had the most influence on λ per unit change in vital rates. However, nest success explained more of the variation in λ than did the survival rates. In lieu of quantitative data on specific mortality factors driving local populations, we recommend that management efforts for greater sage-grouse first focus on increasing female survival by restoring large, intact sagebrush-steppe landscapes, reducing persistent sources of human-caused mortality, and eliminating anthropogenic habitat features that subsidize species that prey on juvenile, yearling, and adult females. Our analysis also supports efforts to increase chick survival and nest success by eliminating anthropogenic habitat features that subsidize chick and nest predators, and by managing shrub, forb, and grass cover, height, and composition to meet local brood-rearing and nesting habitat guidelines. We caution that habitat management to increase chick survival and nest success should not reduce the cover or height of sagebrush below that required for female survival in other seasons (e.g., fall, winter). The success or failure of management actions for sage-grouse should be assessed by measuring changes in vital rates over long time periods to avoid confounding with natural, annual variation. © 2011 The Wildlife Society.     

Bayesian Estimates of Transition Probabilities in Seven Small Lithophytic Orchid Populations: Maximizing Data Availability from Many Small Samples

Predicting population dynamics for rare species is of paramount importance in order to evaluate the likelihood of extinction and planning conservation strategies. However, evaluating and predicting population viability can be hindered from a lack of data. Rare species frequently have small populations, so estimates of vital rates are often very uncertain due to lack of data. We evaluated the vital rates of seven small populations from two watersheds with varying light environment of a common epiphytic orchid using Bayesian methods of parameter estimation. From the Lefkovitch matrices we predicted the deterministic population growth rates, elasticities, stable stage distributions and the credible intervals of the statistics. Populations were surveyed on a monthly basis between 18–34 months. In some of the populations few or no transitions in some of the vital rates were observed throughout the sampling period, however, we were able to predict the most likely vital rates using a Bayesian model that incorporated the transitions rates from the other populations. Asymptotic population growth rate varied among the seven orchid populations. There was little difference in population growth rate among watersheds even though it was expected because of physical differences as a result of differing canopy cover and watershed width. Elasticity analyses of Lepanthes rupestris suggest that growth rate is more sensitive to survival followed by growth, shrinking and the reproductive rates. The Bayesian approach helped to estimate transition probabilities that were uncommon or variable in some populations. Moreover, it increased the precision of the parameter estimates as compared to traditional approaches.     

Evaluating the demographic buffering hypothesis with vital rates estimated for Weddell seals from 30 years of mark-recapture data

1. Life-history theory predicts that those vital rates that make larger contributions to population growth rate ought to be more strongly buffered against environmental variability than are those that are less important. Despite the importance of the theory for predicting demographic responses to changes in the environment, it is not yet known how pervasive demographic buffering is in animal populations because the validity of most existing studies has been called into question because of methodological deficiencies. 2. We tested for demographic buffering in the southern-most breeding mammal population in the world using data collected from 5558 known-age female Weddell seals over 30 years. We first estimated all vital rates simultaneously with mark-recapture analysis and then estimated process variance and covariance in those rates using a hierarchical Bayesian approach. We next calculated the population growth rate's sensitivity to changes in each of the vital rates and tested for evidence of demographic buffering by comparing properly scaled values of sensitivity and process variance in vital rates. 3. We found evidence of positive process covariance between vital rates, which indicates that all vital rates are affected in the same direction by changes in annual environment. Despite the positive correlations, we found strong evidence that demographic buffering occurred through reductions in variation in the vital rates to which population growth rate was most sensitive. Process variation in vital rates was inversely related to sensitivity measures such that variation was greatest in breeding probabilities, intermediate for survival rates of young animals and lowest for survival rates of older animals. 4. Our work contributes to a small but growing set of studies that have used rigorous methods on long-term, detailed data to investigate demographic responses to environmental variation. The information from these studies improves our understanding of life-history evolution in stochastic environments and provides useful information for predicting population responses to future environmental change. Our results for an Antarctic apex predator also provide useful baselines from a marine ecosystem when its top- and middle-trophic levels were not substantially impacted by human activity.     

Effective size of fluctuating populations with two sexes and overlapping generations

We derive formulas that can be applied to estimate the effective population size N(e) for organisms with two sexes reproducing once a year and having constant adult mean vital rates independent of age. Temporal fluctuations in population size are generated by demographic and environmental stochasticity. For populations with even sex ratio at birth, no deterministic population growth and identical mean vital rates for both sexes, the key parameter determining N(e) is simply the mean value of the demographic variance for males and females considered separately. In this case Crow and Kimura's generalization of Wright's formula for N(e) with two sexes, in terms of the effective population sizes for each sex, is applicable even for fluctuating populations with different stochasticity in vital rates for males and females. If the mean vital rates are different for the sexes then a simple linear combination of the demographic variances determines N(e), further extending Wright's formula. For long-lived species an expression is derived for N(e) involving the generation times for both sexes. In the general case with nonzero population growth and uneven sex ratio of newborns, we use the model to investigate numerically the effects of different population parameters on N(e). We also estimate the ratio of effective to actual population size in six populations of house sparrows on islands off the coast of northern Norway. This ratio showed large interisland variation because of demographic differences among the populations. Finally, we calculate how N(e) in a growing house sparrow population will change over time.