Variance In Population Growth Rate Research Articles

An exact version of Life Table Response Experiment analysis, and the R package exactLTRE

Abstract Matrix population models are frequently built and used by ecologists to analyse demography and elucidate the processes driving population growth or decline. Life Table Response Experiments (LTREs) are comparative analyses that decompose the realized difference or variance in population growth rate () into contributions from the differences or variances in the vital rates (i.e. the matrix elements). Since their introduction, LTREs have been based on approximations and have not included biologically relevant interaction terms. We used the functional analysis of variance framework to derive an exact LTRE method, which calculates the exact response of to the difference or variance in a given vital rate, for all interactions among vital rates—including higher‐order interactions neglected by the classical methods. We used the publicly available COMADRE and COMPADRE databases to perform a meta‐analysis comparing the results of exact and classical LTRE methods. We analysed 186 and 1487 LTREs for animal and plant matrix population models, respectively. We found that the classical methods often had small errors, but that very high errors were possible. Overall error was related to the difference or variance in the matrices being analysed, consistent with the Taylor series basis of the classical method. Neglected interaction terms accounted for most of the errors in fixed design LTRE, highlighting the importance of two‐way interaction terms. For random design LTRE, errors in the contribution terms present in both classical and exact methods were comparable to errors due to neglected interaction terms. In most examples we analysed, evaluating exact contributions up to three‐way interaction terms was sufficient for interpreting 90% or more of the difference or variance in . Relative error, previously used to evaluate the accuracy of classical LTREs, is not a reliable metric of how closely the classical and exact methods agree. Error compensation between estimated contribution terms and neglected contribution terms can lead to low relative error despite faulty biological interpretation. Trade‐offs or negative covariances among matrix elements can lead to high relative error despite accurate biological interpretation. Exact LTRE provides reliable and accurate biological interpretation, and the R package exactLTRE makes the exact method accessible to ecologists.

Robustness of life histories to environmental variability in complex versus simple life cycles

Most animal species have a complex life cycle (CLC) with metamorphosis. It is thus of interest to examine possible benefits of such life histories. The prevailing view is that CLC represents an adaptation for genetic decoupling of juvenile and adult traits, thereby allowing life stages to respond independently to different selective forces. Here I propose an additional potential advantage of CLCs that is, decreased variance in population growth rate due to habitat separation of life stages. Habitat separation of pre- and post-metamorphic stages means that the stages will experience different regimes of environmental variability. This is in contrast to species with simple life cycles (SLC) whose life stages often occupy one and the same habitat. The correlation in the fluctuations of the vital rates of life stages is therefore likely to be weaker in complex than in simple life cycles. By a theoretical framework using an analytical approach, I have (1) derived the relative advantage, in terms of long-run growth rate, of CLC over SLC phenotypes for a broad spectrum of life histories, and (2) explored which life histories that benefit most by a CLC, that is avoid correlation in vital rates between life stages. The direction and magnitude of gain depended on life history type and fluctuating vital rate. One implication of our study is that species with CLCs should, on average, be more robust to increased environmental variability caused by global warming than species with SLCs.

Sea ice predicts long-term trends in Adélie penguin population growth, but not annual fluctuations: Results from a range-wide multiscale analysis.

Understanding the scales at which environmental variability affects populations is critical for projecting population dynamics and species distributions in rapidly changing environments. Here we used a multilevel Bayesian analysis of range-wide survey data for Adélie penguins to characterize multidecadal and annual effects of sea ice on population growth. We found that mean sea ice concentration at breeding colonies (i.e., "prevailing" environmental conditions) had robust nonlinear effects on multidecadal population trends and explained over 85% of the variance in mean population growth rates among sites. In contrast, despite considerable year-to-year fluctuations in abundance at most breeding colonies, annual sea ice fluctuations often explained less than 10% of the temporal variance in population growth rates. Our study provides an understanding of the spatially and temporally dynamic environmental factors that define the range limits of Adélie penguins, further establishing this iconic marine predator as a true sea ice obligate and providing a firm basis for projection under scenarios of future climate change. Yet, given the weak effects of annual sea ice relative to the large unexplained variance in year-to-year growth rates, the ability to generate useful short-term forecasts of Adélie penguin breeding abundance will be extremely limited. Our approach provides a powerful framework for linking short- and longer term population processes to environmental conditions that can be applied to any species, facilitating a richer understanding of ecological predictability and sensitivity to global change.

Relating plant height to demographic rates and extinction vulnerability

To prioritize conservation efforts, it is important to know which plant species are most vulnerable to extinction. Intrinsic extinction vulnerabilities depend on demographic parameters, but for many species these demographic parameters are lacking. Body size has been successfully used as proxy of such parameters to estimate extinction vulnerability of birds and mammals. For plants, not all necessary demographic parameters have been related to size yet.Here, we derived allometric relationships with maximum plant height for the intrinsic population growth rate and the carrying capacity. Furthermore, for the first time, we derived a relationship between the variance in population growth rate due to environmental stochasticity and plant height. These relationships were used to relate extinction vulnerability to maximum plant height.Extinction vulnerability was found to be most sensitive to fluctuations in the population growth rate due to environmental stochasticity. Large plant species were less susceptible to environmental stochasticity, resulting in a lower vulnerability to extinction than small plant species. This negative relationship between plant size and extinction vulnerabilities is in contrast to previous results for mammals and birds.These results increase our theoretical understanding of the relationship between plant functional traits and extinction vulnerabilities and may aid in assessments of data deficient species. The uncertainty in the allometric relationships is, however, too large to quantify true extinction vulnerabilities. Further investigation in the relationship between demographic parameters and plant traits other than height is needed to further enhance our understanding of plant species extinction vulnerabilities.

Evolution of stochastic demography with life history tradeoffs in density-dependent age-structured populations

We analyze the stochastic demography and evolution of a density-dependent age- (or stage-) structured population in a fluctuating environment. A positive linear combination of age classes (e.g., weighted by body mass) is assumed to act as the single variable of population size, [Formula: see text], exerting density dependence on age-specific vital rates through an increasing function of population size. The environment fluctuates in a stationary distribution with no autocorrelation. We show by analysis and simulation of age structure, under assumptions often met by vertebrate populations, that the stochastic dynamics of population size can be accurately approximated by a univariate model governed by three key demographic parameters: the intrinsic rate of increase and carrying capacity in the average environment, [Formula: see text] and [Formula: see text], and the environmental variance in population growth rate, [Formula: see text] Allowing these parameters to be genetically variable and to evolve, but assuming that a fourth parameter, [Formula: see text], measuring the nonlinearity of density dependence, remains constant, the expected evolution maximizes [Formula: see text] This shows that the magnitude of environmental stochasticity governs the classical trade-off between selection for higher [Formula: see text] versus higher [Formula: see text] However, selection also acts to decrease [Formula: see text], so the simple life-history trade-off between [Formula: see text]- and [Formula: see text]-selection may be obscured by additional trade-offs between them and [Formula: see text] Under the classical logistic model of population growth with linear density dependence ([Formula: see text]), life-history evolution in a fluctuating environment tends to maximize the average population size.

Demographic response of mule deer to experimental reduction of coyotes and mountain lions in southeastern Idaho

AbstractManipulating predator populations is often posed as a solution to depressed ungulate populations. However, predator–prey dynamics are complex and the effect on prey populations is often an interaction of predator life history, climate, prey density, and habitat quality. The effect of predator removal on ungulate and, more specifically, mule deer (Odocoileus hemionus) populations has not been adequately investigated at a management scale. We tested the efficacy of removing coyotes (Canis latrans) and mountain lions (Puma concolor) for increasing survival and population growth rate of mule deer in southeastern Idaho, USA, during 1997–2003. We assigned 8 game management units (GMUs) to treatments under a 2 × 2 factorial design (treatments of coyote removal and lion removal) with 2 replicates of each treatment or reference area combination. We used methods typically available to wildlife managers to achieve predator removals and a combination of extensive and intensive monitoring in these 8 GMUs to test the hypothesis that predator removal increased vital rates and population growth rate of mule deer. We determined effects of predator removal on survival and causes of mortality in 2 intensive study sites, one with coyote and mountain lion removal and one without. We also considered the effects of other variables on survival including lagomorph abundance and climatic conditions. In these 2 intensive study areas, we monitored with radiotelemetry 250 neonates, 284 6‐month‐old fawns, and 521 adult females. At the extensive scale, we monitored mule deer population trend and December fawn ratios with helicopter surveys. Coyote removal decreased neonate mortality only when deer were apparently needed as alternate prey, thus removal was more effective when lagomorph populations were reduced. The best mortality model of mule deer captured at 6 months of age included summer precipitation, winter precipitation, fawn mass, and mountain lion removal. Over‐winter mortality of adult female mule deer decreased with removal of mountain lions. Precipitation variables were included in most competing mortality models for all age classes of mule deer. Mountain lion removal increased fawn ratios and our models predicted fawn ratios would increase 6% at average removal rates (3.53/1,000 km2) and 27% at maximum removal rates (14.18/1,000 km2). Across our extensive set of 8 GMUs, coyote removal had no effect on December fawn ratios. We also detected no strong effect of coyote or mountain lion removal alone on mule deer population trend; the best population‐growth‐rate model included previous year's mountain lion removal and winter severity, yet explained only 27% of the variance in population growth rate. Winter severity in the current and previous winter was the most important influence on mule deer population growth. The lack of response in fawn ratio or mule deer abundance to coyote reduction at this extensive (landscape) scale suggests that decreased neonate mortality due to coyote removal is partially compensatory. Annual removal of coyotes was not an effective method to increase mule deer populations in Idaho because coyote removal increased radiocollared neonate fawn survival only under particular combinations of prey densities and weather conditions, and the increase did not result in population growth. Coyote‐removal programs targeted in areas where mortality of mule deer fawns is known to be additive and coyote‐removal conditions are successful may influence mule deer population vital rates but likely will not change direction of population trend. Although mountain lion removal increased mule‐deer survival and fawn ratios, we were unable to demonstrate significant changes in population trend with mountain lion removal. In conclusion, benefits of predator removal appear to be marginal and short term in southeastern Idaho and likely will not appreciably change long‐term dynamics of mule deer populations in the intermountain west. © 2011 The Wildlife Society.

An evolutionary maximum principle for density-dependent population dynamics in a fluctuating environment

The evolution of population dynamics in a stochastic environment is analysed under a general form of density-dependence with genetic variation in r and K, the intrinsic rate of increase and carrying capacity in the average environment, and in sigma(e)(2), the environmental variance of population growth rate. The continuous-time model assumes a large population size and a stationary distribution of environments with no autocorrelation. For a given population density, N, and genotype frequency, p, the expected selection gradient is always towards an increased population growth rate, and the expected fitness of a genotype is its Malthusian fitness in the average environment minus the covariance of its growth rate with that of the population. Long-term evolution maximizes the expected value of the density-dependence function, averaged over the stationary distribution of N. In the theta-logistic model, where density dependence of population growth is a function of N(theta), long-term evolution maximizes E[N(theta)]=[1-sigma(e)(2)/(2r)]K(theta). While sigma(e)(2) is always selected to decrease, r and K are always selected to increase, implying a genetic trade-off among them. By contrast, given the other parameters, theta has an intermediate optimum between 1.781 and 2 corresponding to the limits of high or low stochasticity.

The effects of spatial and temporal heterogeneity on the population dynamics of four animal species in a Danish landscape

BackgroundVariation in carrying capacity and population return rates is generally ignored in traditional studies of population dynamics. Variation is hard to study in the field because of difficulties controlling the environment in order to obtain statistical replicates, and because of the scale and expense of experimenting on populations. There may also be ethical issues. To circumvent these problems we used detailed simulations of the simultaneous behaviours of interacting animals in an accurate facsimile of a real Danish landscape. The models incorporate as much as possible of the behaviour and ecology of skylarks Alauda arvensis, voles Microtus agrestis, a ground beetle Bembidion lampros and a linyphiid spider Erigone atra. This allows us to quantify and evaluate the importance of spatial and temporal heterogeneity on the population dynamics of the four species.ResultsBoth spatial and temporal heterogeneity affected the relationship between population growth rate and population density in all four species. Spatial heterogeneity accounted for 23–30% of the variance in population growth rate after accounting for the effects of density, reflecting big differences in local carrying capacity associated with the landscape features important to individual species. Temporal heterogeneity accounted for 3–13% of the variance in vole, skylark and spider, but 43% in beetles. The associated temporal variation in carrying capacity would be problematic in traditional analyses of density dependence. Return rates were less than one in all species and essentially invariant in skylarks, spiders and beetles. Return rates varied over the landscape in voles, being slower where there were larger fluctuations in local population sizes.ConclusionOur analyses estimated the traditional parameters of carrying capacities and return rates, but these are now seen as varying continuously over the landscape depending on habitat quality and the mechanisms of density dependence. The importance of our results lies in our demonstration that the effects of spatial and temporal heterogeneity must be accounted for if we are to have accurate predictive models for use in management and conservation. This is an area which until now has lacked an adequate theoretical framework and methodology.

Adaptive topography of fluctuating selection in a Mendelian population

An adaptive topography is derived for a large randomly mating diploid population under weak density-independent selection in a fluctuating environment. Assuming a stationary distribution of environmental states with no temporal autocorrelation, a diffusion approximation for population size and allele frequency, p, reveals that the expected change in p involves the gradient with respect to p of the stochastic intrinsic rate of increase (the density-independent long-run growth rate), r = r - sigma 2 e/2, where r is the mean Malthusian fitness in the average environment and is the environmental variance in population growth rate. The expected relative fitness of a genotype is its Malthusian fitness in the average environment minus the covariance of its fitness with population growth rate. The influence of fitness correlation between genotypes is illustrated by an analysis of the Haldane-Jayakar model of fluctuating selection on a single diallelic locus, and on two loci with additive effects on a quantitative character.

Individual Size Variation and Population Stability in a Seasonal Environment: A Discrete‐Time Model and Its Calibration Using Grasshoppers

Much recent literature is concerned with how variation among individuals (e.g., variability in their traits and fates) translates into higher-level (i.e., population and community) dynamics. Although several theoretical frameworks have been devised to deal with the effects of individual variation on population dynamics, there are very few reports of empirically based estimates of the sign and magnitude of these effects. Here we describe an analytical model for size-dependent, seasonal life cycles and evaluate the effect of individual size variation on population dynamics and stability. We demonstrate that the effect of size variation on the population net reproductive rate varies in both magnitude and sign, depending on season length. We calibrate our model with field data on size- and density-dependent growth and survival of the generalist grasshopper Melanoplus femurrubrum. Under deterministic dynamics (fixed season length), size variation impairs population stability, given naturally occurring densities. However, in the stochastic case, where season length exhibits yearly fluctuations, size variation reduces the variance in population growth rates, thus enhancing stability. This occurs because the effect of size variation on net reproductive rate is dependent on season length. We discuss several limitations of the current model and outline possible routes for future model development.

EXPECTED RELATIVE FITNESS AND THE ADAPTIVE TOPOGRAPHY OF FLUCTUATING SELECTION

Wright's adaptive topography describes gene frequency evolution as a maximization of mean fitness in a constant environment. I extended this to a fluctuating environment by unifying theories of stochastic demography and fluctuating selection, assuming small or moderate fluctuations in demographic rates with a stationary distribution, and weak selection among the types. The demography of a large population, composed of haploid genotypes at a single locus or normally distributed phenotypes, can then be approximated as a diffusion process and transformed to produce the dynamics of population size, N, and gene frequency, p, or mean phenotype, . The expected evolution of p or is a product of genetic variability and the gradient of the long-run growth rate of the population, , with respect to p or . This shows that the expected evolution maximizes , the mean Malthusian fitness in the average environment minus half the environmental variance in population growth rate. Thus, as a function of p or represents an adaptive topography that, despite environmental fluctuations, does not change with time. The haploid model is dominated by environmental stochasticity, so the expected maximization is not realized. Different constraints on quantitative genetic variability, and stabilizing selection in the average environment, allow evolution of the mean phenotype to undergo a stochastic maximization of . Although the expected evolution maximizes the long-run growth rate of the population, for a genotype or phenotype the long-run growth rate is not a valid measure of fitness in a fluctuating environment. The haploid and quantitative character models both reveal that the expected relative fitness of a type is its Malthusian fitness in the average environment minus the environmental covariance between its growth rate and that of the population.

The contributions of age and sex to variation in common tern population growth rate

1. The decomposition of population growth rate into contributions from different demographic rates has many applications, ranging from evolutionary biology to conservation and management. Demographic rates with low variance may be pivotal for population persistence, but variable rates can have a dramatic influence on population growth rate. 2. In this study, the mean and variance in population growth rate (lambda) is decomposed into contributions from different ages and demographic rates using prospective and retrospective matrix analyses for male and female components of an increasing common tern (Sterna hirundo) population. 3. Three main results emerged: (1) subadult return was highly influential in prospective and retrospective analyses; (2) different age-classes made different contributions to variation in lambda: older age classes consistently produced offspring whereas young adults performed well only in high quality years; and (3) demographic rate covariation explained a significant proportion of variation in both sexes. A large contribution to lambda did not imply a large contribution to its variation. 4. This decomposition strengthens the argument that the relationship between variation in demographic rates and variation in lambda is complex. Understanding this relationship and its consequences for population persistence and evolutionary change demands closer examination of the lives, and deaths, of the individuals within populations within species.

LONG-TERM CONTRASTED RESPONSES TO CLIMATE OF TWO ANTARCTIC SEABIRD SPECIES

We examined the population dynamics of two Antarctic seabirds and the influence of environmental variability over a 40-year period by coupling the estimation of demographic parameters, based on capture–recapture data, and modeling, using Leslie matrix population models. We demonstrated that the demographic parameters showing the greatest contribution to the variance of population growth rate were adult survival for both species. Breeding success showed the same contribution as adult survival for Emperor Penguins, whereas the proportion of breeders had the next stronger contribution for Snow Petrels. The sensitivity of population growth rate to adult survival was very high and the adult survival variability was weak for both species. Snow Petrel males survived better than females, whereas Emperor Penguin males had lower survival than females. These differences may be explained by the different investment in breeding. Emperor Penguin adult survival was negatively affected by air temperature during summer and winter for both sexes; male survival was negatively affected by sea ice concentration during summer, autumn, and winter. On the other hand, there was no effect of environmental covariates on Snow Petrel adult survival. The Emperor Penguin population has declined by 50% because of a decrease in adult survival related to a warming event during a regime shift in the late 1970s, whereas Snow Petrels showed their lowest numbers in 1976, but were able to skip reproduction. Indeed, the retrospective analysis of projection population matrix entries indicated that breeding abstention played a critical role in the population dynamics of Snow Petrels but not Emperor Penguins. Snow Petrels did not breed either when air temperature decreased during spring (probably reducing nest attendance and laying) or when sea ice decreased during autumn (reducing food availability). Emperor Penguin and Snow Petrel breeding population sizes were positively influenced by sea ice through its effect on adult survival for Emperor Penguins and on the proportion of breeders for Snow Petrels. Therefore, we hypothesize that the population sizes of the two species could be negatively affected by reduced sea ice in the context of global warming.

SENSITIVITY ANALYSIS OF THE STOCHASTIC GROWTH RATE: THREE EXTENSIONS†

SummaryThe perturbation analysis of population growth rate plays an important role in population biology. The sensitivity and/or elasticity (proportional sensitivity) of population growth rate to changes in the vital rates are regularly used (i) to predict the effects of environmental perturbations, (ii) to characterize selection gradients on life‐history traits, (iii) to evaluate management tactics, (iv) to analyse life table response experiments, and (v) to calculate the sampling variance in population growth rate. In a stochastic environment, population growth is described by the stochastic growth rate, which gives, with probability 1, the asymptotic time‐averaged growth rate of any realization. Tuljapurkar derived the sensitivity and elasticity of the stochastic growth rate to changes in the entries of the stochastic matrices. This paper extends his result to cover three cases, each of which has arisen recently in applications. The first gives the response of the stochastic growth rate to environment‐specific perturbations, applied only in a specified subset of the possible environments. The second gives the sensitivity and elasticity of the stochastic growth rate to changes in lower‐level parameters. The third applies to stochastic seasonal models, in which the projection matrix for each year is a periodic product of matrices describing seasonal transitions. In this case interest focuses on the sensitivity of the stochastic growth rate to changes in the entries of the seasonal matrices, not entries in the annual matrices. The paper describes examples of problems where each of these extensions is needed, and the algorithms for each of the new calculations.

Demography and population dynamics of Drosera anglica and D. rotundifolia

Summary 1 We studied demography and population dynamics of the sympatric perennial herbs Drosera anglica and D. rotundifolia on a boreal bog in SE Norway. Dry mass of 2872 D. anglica plants and 2467 D. rotundifolia plants (estimated from field morphological measurements) was used to classify plants into five species-specific size classes. Demographic changes within these two populations were followed from 1995 to 1999, and within segments (quartiles) along the water table gradient and the peat productivity gradient. 2 Mortality was strongly size dependent, and varied between years, for both species; it was high for seedlings, low for the smallest mature rosettes and increased again for the largest mature rosettes. The proportion of fertile rosettes increased with increasing rosette size. Fecundity varied considerably between years, but little relative to gradient position. 3 Growth rate (λ) was > 1, except in the second year, when it fell to 0.572 for D. anglica and 0.627 for D. rotundifolia . For D. anglica small, but significant, differences were found between the two extremities of the water table gradient, and for D. rotundifolia between the second and the uppermost quartile. There was a tendency for D. anglica populations to have a lower growth rate in the most productive sites, whereas D. rotundifolia grew less on both low and high peat productivity. Elasticity analysis showed that stasis and size increase (primarily within mature stages) made major contributions to λ for D. anglica in all years. 4 The variance in population growth rate (var λ) was high between years, and higher for D. anglica than for D. rotundifolia , while the variance between quartiles along the two main gradients was low. Life-table response experiment (LTRE) analyses revealed that for both species, differences in probabilities of transitions within mature stages, and in growth to larger stages, contributed most to var λ. 5 The effects of global warming are uncertain: drier growing seasons would affect Drosera populations negatively, while initially positive responses to a wetter climate may be balanced by competition from increased Sphagnum growth.

Modelling the role of social behavior in the persistence of the alpine marmotMarmota marmota

A general rule of thumb for biological conservation obtained from simple models of hypothetical species is that for populations with strong environmental noise moderate increases in habitat size or quality do not substantially reduce extinction risk. However, whether this rule also holds for real species with complex behavior, such as social species with breeding units and reproductive suppression, is uncertain. Here we present a population viability analysis of the alpine marmotMarmota marmota, which displays marked social behavior, i.e. it lives in social groups of up to twenty individuals. Our analysis is based on a long‐term field study carried out in the Bavarian Alps since 1982. During the first fifteen years of this study, 687 marmots were individually marked and the movements and fate of 98 dispersing marmots were recorded with radio‐telemetry. Thus, in contrast to most other viability analyses of spatially structured populations, good data about dispersal exist. A model was constructed which is individual‐based, spatially explicit at the scale of clusters of neighbouring territories, and spatially implicit at larger scales. The decisive aspect of marmot life history, winter mortality, is described by logistic regression where mortality is increased by age and the severity of winter, and decreased by the number of subdominant individuals present in a group. Model predictions of group size distribution are in good agreement with the results of the field study. The model shows that the effect of sociality on winter mortality is very effective in buffering environmental harshness and fluctuations. This underpins theoretical results stating that the appropriate measure of the strength of environmental noise is the ratio between the variance of population growth rate and the intrinsic rate of increase. The lessons from our study for biological conservation are that simple, unstructured models may not be sufficient to assess the viability of species with complex behavioral traits, and that even moderate increases in habitat capacity may substantially reduce extinction risk even if environmental fluctuations seem high.

AN EXTENDED FLOWERING AND FRUITING SEASON HAS FEW DEMOGRAPHIC EFFECTS IN A MEDITERRANEAN PERENNIAL HERB

The Mediterranean perennial herb Lobularia maritima shows an exceptionally extended flowering and fruiting that lasts for 10 months, from early September to late June. We hypothesized that such an extended phenology may be a flexible mechanism that enhances population persistence in variable Mediterranean environments, as fecundity in one part of the season could compensate for reproductive failure in another part of the season. We predicted that (1) fecundities throughout the year would significantly contribute to mean population growth rate λ, (2) negative covariances among fecundities in different periods of the year would reduce variance in population growth rate, and (3) extended flowering and fruiting would enhance long-term stochastic population growth rates. We used a periodic matrix model to describe the demography of a L. maritima population over five years, in which each year was divided into six periods of two months each. Population growth rates varied from a low of 0.25 to a high of 4.81 ove...

Model complexity and population predictions. The alpine marmot as a case study

SummaryDuring the past 15 years, models have been used increasingly in predictive population ecology. Matrix models used for predicting the fates of populations are often extremely basic, ignoring density dependence, spatial scale and behaviour, and often based on one sex only. We tested the importance of some of these omissions for model realism, by comparing the performance of a variety of population models of varying levels of complexity.Detailed data from more than 13 years of behavioural and demographic research on a population of alpine marmotsMarmota marmotain Berchtesgaden National Park, southern Germany, were used to parameterize four different population models. The models ranged from a simple population‐based matrix model, to a spatially explicit behaviour‐based model.The performance of the models was judged by their ability to predict basic population dynamics under equilibrium conditions. Only a spatially explicit individual‐based model ignoring optimal behaviour predicted dynamics significantly different to those observed in the field, highlighting the importance of considering realistic patterns of behaviour in spatially explicit models.Using realistic levels of environmental and demographic stochasticity, variance in population growth rates predicted by all models was high, even within the range of population densities experienced in the field. This emphasizes the difficulty of using population‐level field data to determine overall patterns of density dependence for use in population models.All models were also used to predict potential density‐dependent effects on alpine marmot population growth. In this regard, the models differed greatly. It was concluded that the simplest matrix model was adequate for making predictions regarding population sizes or densities under equilibrium conditions, but that for predictions requiring an understanding of transient dynamics only the behavioural model would be adequate.An emergent feature of this study of alpine marmot population dynamics was the prediction of a demographic Allee effect with a profound influence on population dynamics across a very broad range of population sizes. Three mechanisms were identified as underlying this Allee effect: stochastic skews in sex ratio and demographic composition at low population sizes; less efficient social thermoregulation during hibernation in small groups; and difficulties with mate finding during dispersal, even at relatively high population sizes.

Interactions between Operophtera brumata and Tortrix viridana on oak: new evidence from time‐series analysis

1. Yearly population estimates of the green oak leaf roller moth, Tortrix viridana (Lepidoptera: Tortricidae), and the winter moth, Operophtera brumata (Lepidoptera: Geometridae), were taken from the pedunculate oak, Quercus robur, from 1951 to 1966 in Wytham Woods, Oxfordshire, U.K. Larval mortality from parasitoids was recorded for both species over the same period.2. Operophtera brumata population density fluctuated around a constant mean, whereas T. viridana population density exhibited a linear decline over the sampling period. Population counts were subjected to time‐series analysis after the linear decline was removed from the T. viridana data by detrending. Multiple regression models were built so that variation in the per capita rate of increase of each population could be partitioned among (a) current and previous population density, (b) current and previous population density of the second moth species, and (c) rates of larval parasitism.3. Multiple regression analysis suggested that variation in the per capita rate of increase of O. brumata could be explained by negative feedback from O. brumata density at T–2 (32%), a negative relationship with T. viridana density at T–1 (18%), and a positive relationship with parasitism suffered by T. viridana at T–1 (24%).4. The T. viridana population time‐series was dominated by a rapid feedback process such that per capita rate of increase at time T was negatively related to population density at time T–1, explaining 53% of variance in population growth rate. Per capita rate of increase was unrelated to larval parasitism or densities of O. brumata.5. In light of previous life‐table and experimental studies on O. brumata and T. viridana, the current analyses suggest that pupal predation (data not presented) and interspecific competition are significant determinants of O. brumata population growth rates in Wytham Woods. In contrast, T. viridana population growth rates appear to be dominated by rapid negative feedback consistent with intraspecific competition.

Patterns of variance in stage-structured populations: evolutionary predictions and ecological implications.

Variability in population growth rate is thought to have negative consequences for organism fitness. Theory for matrix population models predicts that variance in population growth rate should be the sum of the variance in each matrix entry times the squared sensitivity term for that matrix entry. I analyzed the stage-specific demography of 30 field populations from 17 published studies for pattern between the variance of a demographic term and its contribution to population growth. There were no instances in which a matrix entry both was highly variable and had a large effect on population growth rate; instead, correlations between estimates of temporal variance in a term and contribution to population growth (sensitivity or elasticity) were overwhelmingly negative. In addition, survivorship or growth sensitivities or elasticities always exceeded those of fecundity, implying that the former two terms always contributed more to population growth rate. These results suggest that variable life history stages tend to contribute relatively little to population growth rates because natural selection may alter life histories to minimize stages with both high sensitivity and high variation.