Integral projection models perform better for small demographic data sets than matrix population models: a case study of two perennial herbs

1. Matrix population models are widely used to describe population dynamics, conduct population viability analyses and derive management recommendations for plant populations. For endangered or invasive species, management decisions are often based on small demographic data sets. Hence, there is a need for population models which accurately assess population performance from such small data sets.
2. We used demographic data on two perennial herbs with different life histories to compare the accuracy and precision of the traditional matrix population model and the recently developed integral projection model (IPM) in relation to the amount of data.
3. For large data sets both matrix models and IPMs produced identical estimates of population growth rate (λ). However, for small data sets containing fewer than 300 individuals, IPMs often produced smaller bias and variance for λ than matrix models despite different matrix structures and sampling techniques used to construct the matrix population models.
4. Synthesis and applications . Our results suggest that the smaller bias and variance of λ estimates make IPMs preferable to matrix population models for small demographic data sets with a few hundred individuals. These results are likely to be applicable to a wide range of herbaceous, perennial plant species where demographic fate can be modelled as a function of a continuous state variable such as size. We recommend the use of IPMs to assess population performance and management strategies particularly for endangered or invasive perennial herbs where little demographic data are available.     

Stochastic matrix models for conservation and management: a comparative review of methods

Stochastic matrix models are frequently used by conservation biologists to measure the viability of species and to explore various management actions. Models are typically parameterized using two or more sets of estimated transition rates between age/size/stage classes. While standard methods exist for analyzing a single set of transition rates, a variety of methods have been employed to analyze multiple sets of transition rates. We review applications of stochastic matrix models to problems in conservation and use simulation studies to compare the performance of different analytic methods currently in use. We find that model conclusions are likely to be robust to the choice of parametric distribution used to model vital rate fluctuations over time. However, conclusions can be highly sensitive to the within-year correlation structure among vital rates, and therefore we suggest using analytical methods that provide a means of conducting a sensitivity analysis with respect to correlation parameters. Our simulation results also suggest that the precision of population viability estimates can be improved by using matrix models that incorporate environmental covariates in conjunction with experiments to estimate transition rates under a range of environmental conditions.     

Importance of correlations among matrix entries in stochastic models in relation to number of transition matrices

Stochastic matrix models are used to predict population viability and the risk of extinction. Different stochastic methods require different amounts of estimation effort and may lead to divergent estimates. We used 16 transition matrices collected from ten populations of the perennial herb Primula veris to compare population estimates produced by different stochastic methods, such as selection of matrices, selection of vital rates, selection of matrix elements, and Tuljapurkar's approximation. Specifically, we tested the reliability of the methods using different numbers of transition matrices, and examined the importance of correlations among matrix entries. When correlations among matrix entries were included in the models, selection of vital rates produced the lowest and Tuljapurkar's approximation produced the highest estimates of mean population growth rates. Selection of matrices and matrix elements often produced nearly similar population estimates. Simulations based on incompletely estimated correlations among matrix entries considerably differed from those based on all correlations estimated, particularly when correlations were strong. The magnitude of correlations among matrix entries depended on the number of matrices, which made it difficult to generalize correlations within a species. Given that selection of vital rates or matrix elements is used, correlations among matrix entries should usually be included in the model, and they should preferably be estimated from the present data rather than according to other information of the species.     

Effects of sample size on estimates of population growth rates calculated with matrix models

Background

Matrix models are widely used to study the dynamics and demography of populations. An important but overlooked issue is how the number of individuals sampled influences estimates of the population growth rate (λ) calculated with matrix models. Even unbiased estimates of vital rates do not ensure unbiased estimates of λ–Jensen''s Inequality implies that even when the estimates of the vital rates are accurate, small sample sizes lead to biased estimates of λ due to increased sampling variance. We investigated if sampling variability and the distribution of sampling effort among size classes lead to biases in estimates of λ.

Methodology/Principal Findings

Using data from a long-term field study of plant demography, we simulated the effects of sampling variance by drawing vital rates and calculating λ for increasingly larger populations drawn from a total population of 3842 plants. We then compared these estimates of λ with those based on the entire population and calculated the resulting bias. Finally, we conducted a review of the literature to determine the sample sizes typically used when parameterizing matrix models used to study plant demography.

Conclusions/Significance

We found significant bias at small sample sizes when survival was low (survival = 0.5), and that sampling with a more-realistic inverse J-shaped population structure exacerbated this bias. However our simulations also demonstrate that these biases rapidly become negligible with increasing sample sizes or as survival increases. For many of the sample sizes used in demographic studies, matrix models are probably robust to the biases resulting from sampling variance of vital rates. However, this conclusion may depend on the structure of populations or the distribution of sampling effort in ways that are unexplored. We suggest more intensive sampling of populations when individual survival is low and greater sampling of stages with high elasticities.     

Dynamics of age- and size-structured populations in fluctuating environments: Applications of stochastic matrix models to natural populations

Recent developments of the theory of stochastic matrix modeling have made it possible to estimate general properties of age- and size-structured populations in fluctuating environments. However, applications of the theory to natural populations are still few. The empirical studies which have used stochastic matrix models are reviewed here to examine whether predictions made by the theory can be generally found in wild populations. The organisms studied include terrestrial grasses and herbs, a seaweed, a fish, a reptile, a deer and some marine invertebrates. In all the studies, the stochastic population growth rate (ln λ _s ) was no greater than the deterministic population growth rate determined using average vital rates, suggesting that the model based only on average vital rates may overestimate growth rates of populations in fluctuating environments. Factors affecting ln λ _s include the magnitude of variation in vital rates, probability distribution of random environments, fluctuation in different types of vital rates, covariances between vital rates, and autocorrelation between successive environments. However, comprehensive rules were hardly found through the comparisons of the empirical studies. Based on shortcomings of previous studies, I address some important subjects which should be examined in future studies.     

Stochastic population dynamics of clonal plants: Numerical experiments with ramet and genet models

Dynamics of ramer and genet populations were analyzed by use of stochastic matrix models. Based on field data, population development and extinction rates during 50 simulated years were estimated for ramet populations of three speciesPotentilla anserina, Rubus saxatilis andLinnaea borealis. Only small initial populations (below 125–250 ramets), experienced a detectable risk of extinction within this time interval. ForP. anserina andR. saxatilis, population increase occurred in some simulations despite negative average growth rates. A model for stochastic genet dynamics was constructed by combining field data and hypothesized parameter values. Growth rate and population structure were insensitive to variation in disturbance intensity and frequency, whereas variation in recruitment affected population structure but only to a minor extent growth rate. Decreasing recruitment causes extinction of genet populations, but the time-scale for the decline is in the magnitude of centuries for initial genet populations of about 1000 individuals. Dynamics of genets in clonal plants thus incorporate processes occurring on widely different scales. Some implications of the results for models of population dynamics in long-lived clonal plants are discussed.     

Census data aggregation decisions can affect population‐level inference in heterogeneous populations

1. Conservation and population management decisions often rely on population models parameterized using census data. However, the sampling regime, precision, sample size, and methods used to collect census data are usually heterogeneous in time and space. Decisions about how to derive population‐wide estimates from this patchwork of data are complicated and may bias estimated population dynamics, with important implications for subsequent management decisions.
2. Here, we explore the impact of site selection and data aggregation decisions on pup survival estimates, and downstream estimates derived from parameterized matrix population models (MPMs), using a long‐term dataset on grey seal (Halichoerus grypus) pup survival from southwestern Wales. The spatiotemporal and methodological heterogeneity of the data are fairly typical for ecological census data and it is, therefore, a good model to address this topic.
3. Data were collected from 46 sampling locations (sites) over 25 years, and we explore the impact of data handling decisions by varying how years and sampling locations are combined to parameterize pup survival in population‐level MPMs. We focus on pup survival because abundant high‐quality data are available on this developmental stage.
4. We found that survival probability was highly variable with most variation being at the site level, and poorly correlated among sampling sites. This variation could generate marked differences in predicted population dynamics depending on sampling strategy. The sample size required for a confident survival estimate also varied markedly geographically.
5. We conclude that for populations with highly variable vital rates among sub‐populations, site selection and data aggregation methods are important. In particular, including peripheral or less frequently used areas can introduce substantial variation into population estimates. This is likely to be context‐dependent, but these choices, including the use of appropriate weights when summarizing across sampling areas, should be explored to ensure that management actions are successful.

     

Demography of the Pacific walrus (Odobenus rosmarus divergens): 1974–2006

Global climate change may fundamentally alter population dynamics of many species for which baseline population parameter estimates are imprecise or lacking. Historically, the Pacific walrus is thought to have been limited by harvest, but it may become limited by global warming‐induced reductions in sea ice. Loss of sea ice, on which walruses rest between foraging bouts, may reduce access to food, thus lowering vital rates. Rigorous walrus survival rate estimates do not exist, and other population parameter estimates are out of date or have well‐documented bias and imprecision. To provide useful population parameter estimates we developed a Bayesian, hidden process demographic model of walrus population dynamics from 1974 through 2006 that combined annual age‐specific harvest estimates with five population size estimates, six standing age structure estimates, and two reproductive rate estimates. Median density independent natural survival was high for juveniles (0.97) and adults (0.99), and annual density dependent vital rates rose from 0.06 to 0.11 for reproduction, 0.31 to 0.59 for survival of neonatal calves, and 0.39 to 0.85 for survival of older calves, concomitant with a population decline. This integrated population model provides a baseline for estimating changing population dynamics resulting from changing harvests or sea ice.     

Consequences of past and present harvest management in a declining flyway population of common eiders Somateria mollissima

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Incorporating harvest rates into the sex-age-kill model for white-tailed deer

Although monitoring population trends is an essential component of game species management, wildlife managers rarely have complete counts of abundance. Often, they rely on population models to monitor population trends. As imperfect representations of real-world populations, models must be rigorously evaluated to be applied appropriately. Previous research has evaluated population models for white-tailed deer (Odocoileus virginianus); however, the precision and reliability of these models when tested against empirical measures of variability and bias largely is untested. We were able to statistically evaluate the Pennsylvania sex-age-kill (PASAK) population model using realistic error measured using data from 1,131 radiocollared white-tailed deer in Pennsylvania from 2002 to 2008. We used these data and harvest data (number killed, age-sex structure, etc.) to estimate precision of abundance estimates, identify the most efficient harvest data collection with respect to precision of parameter estimates, and evaluate PASAK model robustness to violation of assumptions. Median coefficient of variation (CV) estimates by Wildlife Management Unit, 13.2% in the most recent year, were slightly above benchmarks recommended for managing game species populations. Doubling reporting rates by hunters or doubling the number of deer checked by personnel in the field reduced median CVs to recommended levels. The PASAK model was robust to errors in estimates for adult male harvest rates but was sensitive to errors in subadult male harvest rates, especially in populations with lower harvest rates. In particular, an error in subadult (1.5-yr-old) male harvest rates resulted in the opposite error in subadult male, adult female, and juvenile population estimates. Also, evidence of a greater harvest probability for subadult female deer when compared with adult (≥2.5-yr-old) female deer resulted in a 9.5% underestimate of the population using the PASAK model. Because obtaining appropriate sample sizes, by management unit, to estimate harvest rate parameters each year may be too expensive, assumptions of constant annual harvest rates may be necessary. However, if changes in harvest regulations or hunter behavior influence subadult male harvest rates, the PASAK model could provide an unreliable index to population changes. © 2012 The Wildlife Society.     

Demography_Lab,an educational application to evaluate population growth: Unstructured and matrix models

Training in Population Ecology asks for scalable applications capable of embarking students on a trip from basic concepts to the projection of populations under the various effects of density dependence and stochasticity. Demography_Lab is an educational tool for teaching Population Ecology aspiring to cover such a wide range of objectives. The application uses stochastic models to evaluate the future of populations. Demography_Lab may accommodate a wide range of life cycles and can construct models for populations with and without an age or stage structure. Difference equations are used for unstructured populations and matrix models for structured populations. Both types of models operate in discrete time. Models can be very simple, constructed with very limited demographic information or parameter‐rich, with a complex density‐dependence structure and detailed effects of the different sources of stochasticity. Demography_Lab allows for deterministic projections, asymptotic analysis, the extraction of confidence intervals for demographic parameters, and stochastic projections. Stochastic population growth is evaluated using up to three sources of stochasticity: environmental and demographic stochasticity and sampling error in obtaining the projection matrix. The user has full control on the effect of stochasticity on vital rates. The effect of the three sources of stochasticity may be evaluated independently for each vital rate. The user has also full control on density dependence. It may be included as a ceiling population size controlling the number of individuals in the population or it may be evaluated independently for each vital rate. Sensitivity analysis can be done for the asymptotic population growth rate or for the probability of extinction. Elasticity of the probability of extinction may be evaluated in response to changes in vital rates, and in response to changes in the intensity of density dependence and environmental stochasticity.     

The importance of within-year repeated counts and the influence of scale on long-term monitoring of sage-grouse

Long-term population monitoring is the cornerstone of animal conservation and management. The accuracy and precision of models developed using monitoring data can be influenced by the protocols guiding data collection. The greater sage-grouse (Centrocercus urophasianus) is a species of concern that has been monitored over decades, primarily, by counting the number of males that attend lek (breeding) sites. These lek count data have been used to assess long-term population trends and for multiple mechanistic studies. However, some studies have questioned the efficacy of lek counts to accurately identify population trends. In response, monitoring protocols were changed to have a goal of counting lek sites multiple times within a season. We assessed the influence of this change in monitoring protocols on model accuracy and precision applying generalized additive models to describe trends over time. We found that at large spatial scales including >50 leks, the absence of repeated counts within a year did not significantly alter population trend estimates or interpretation. Increasing sample size decreased the model confidence intervals. We developed a population trend model for Wyoming greater sage-grouse from 1965 to 2008, identifying significant changes in the population indices and capturing the cyclic nature of this species. Most sage-grouse declines in Wyoming occurred between 1965 and the 1990s and lek count numbers generally increased from the mid-1990s to 2008. Our results validate the combination of monitoring data collected under different protocols in past and future studies—provided those studies are addressing large-scale questions. We suggest that a larger sample of individual leks is preferable to multiple counts of a smaller sample of leks. © 2011 The Wildlife Society.     

Alternative spatial sampling in studies of plant demography: consequences for estimates of population growth rate

Ecologists commonly use matrix models to study the population dynamics of plants. Most studies of plant demography use plot-based methods to collect data, in part, because mapped individuals are easier to relocate in subsequent surveys and survey methods can be standardized among sites. However, there is tremendous variation among studies, both in terms of plot arrangement and the total area sampled. In addition, there has been little discussion of how alternative sampling arrangements influence estimates of population growth rates (λ) calculated with matrix models. We surveyed the literature to determine what sampling designs are most used in studies of plant demography using matrix models. We then used simulations of three common sampling techniques—using a single randomly placed plot, multiple randomly placed plots, and systematically distributed plots—to evaluate how these alternative strategies influenced the precision of estimates of λ. These simulations were based on long-term demographic data collected on 13 populations of the Amazonian understory herb Heliconia acuminate (Heliconiaceae). We found that the method used to collect data did not affect the bias or precision of estimates in our system—a surprising result, since the advantage in efficiency that is gained from systematic sampling is a well-known result from sampling theory. Because the statistical advantage of systematic sampling is most evident when there is spatial structure in demographic vital rates, we attribute this result to the lack of spatially structured vital rates in our focal populations. Given the likelihood of spatial autocorrelation in most ecological systems, we advocate sampling with a systematic grid of plots in each study site, as well as that researchers ensure that enough area is sampled—both within and across sites—to encompass the range of spatial variation in plant survival, growth, and reproduction.     

Accounting for uncertainty in dormant life stages in stochastic demographic models

Dormant life stages are often critical for population viability in stochastic environments, but accurate field data characterizing them are difficult to collect. Such limitations may translate into uncertainties in demographic parameters describing these stages, which then may propagate errors in the examination of population‐level responses to environmental variation. Expanding on current methods, we 1) apply data‐driven approaches to estimate parameter uncertainty in vital rates of dormant life stages and 2) test whether such estimates provide more robust inferences about population dynamics. We built integral projection models (IPMs) for a fire‐adapted, carnivorous plant species using a Bayesian framework to estimate uncertainty in parameters of three vital rates of dormant seeds – seed‐bank ingression, stasis and egression. We used stochastic population projections and elasticity analyses to quantify the relative sensitivity of the stochastic population growth rate (log λs) to changes in these vital rates at different fire return intervals. We then ran stochastic projections of log λs for 1000 posterior samples of the three seed‐bank vital rates and assessed how strongly their parameter uncertainty propagated into uncertainty in estimates of log λs and the probability of quasi‐extinction, P_q(t). Elasticity analyses indicated that changes in seed‐bank stasis and egression had large effects on log λs across fire return intervals. In turn, uncertainty in the estimates of these two vital rates explained > 50% of the variation in log λs estimates at several fire‐return intervals. Inferences about population viability became less certain as the time between fires widened, with estimates of P_q(t) potentially > 20% higher when considering parameter uncertainty. Our results suggest that, for species with dormant stages, where data is often limited, failing to account for parameter uncertainty in population models may result in incorrect interpretations of population viability.     

Open removal models with temporary emigration and population dynamics to inform invasive animal management

Removal sampling data are the primary source of monitoring information for many populations (e.g., invasive species, fisheries). Population dynamics, temporary emigration, and imperfect detection are common sources of variation in monitoring data and are key parameters for informing management. We developed two open robust‐design removal models for simultaneously modeling population dynamics, temporary emigration, and imperfect detection: a random walk linear trend model (estimable without ancillary information), and a 2‐age class informed population model (InfoPM, closely related to integrated population models) that incorporated prior information for age‐structured vital rates and relative juvenile availability. We applied both models to multiyear, removal trapping time‐series of a large invasive lizard (Argentine black and white tegu, Salvator merianae) in three management areas of South Florida to evaluate the effectiveness of management programs. Although estimates of the two models were similar, the InfoPMs generally returned more precise estimates, partitioned dynamics into births, deaths, net migration, and provided a decision support tool to predict population dynamics under different effort scenarios while accounting for uncertainty. Trends in tegu superpopulation abundance estimates were increasing in two management areas despite generally high removal rates. However, tegu abundance appeared to decline in the Core management area, where trapping density was the highest and immigration the lowest. Finally, comparing abundance predictions of no‐removal scenarios to those estimated in each management area suggested significant population reductions due to management. These results suggest that local tegu population control via systematic trapping may be feasible with high enough trap density and limited immigration; and highlights the value of these trapping programs. We provided the first estimates of tegu abundance, capture probabilities, and population dynamics, which is critical for effective management. Furthermore, our models are applicable to a wide range of monitoring programs (e.g., carcass recovery or removal point‐counts).     

Matrix dimensionality in demographic analyses of plants: when to use smaller matrices?

Large data requirements may restrict the use of matrix population models for analysis of population dynamics. Less data are required for a small population matrix than for a large matrix because the smaller matrix contains fewer vital rates that need to be estimated. Smaller matrices, however, tend to have a lower precision. Based on 37 plant species, we studied the effects of matrix dimensionality on the long-term population growth rate (λ) and the elasticity of λ in herbaceous and woody species. We found that when matrix dimensionality was reduced, changes in λ were significantly larger for herbaceous than for woody species. In many cases, λ of woody species remained virtually the same after a substantial decrease in matrix dimensionality, suggesting that woody species are less susceptible to matrix dimensionality. We demonstrated that when adjacent stages of a transition matrix are combined, the magnitude of a change in λ depends on the distance of the population structure from a stable stage distribution, and the difference in the combined vital rates weighted by their reproductive values. Elasticity of λ to survival and fecundity usually increased, whereas elasticity to growth decreased both in herbaceous and in woody species with reduced matrix dimensionality. Changes in elasticity values tended to be larger for herbaceous than for woody species. Our results show that by reducing matrix dimensionality, the amount of demographic data can be decreased to save time, money, and field effort. We recommend the use of a small matrix dimensionality especially when a limited amount of data is available, and for slow-growing species having a simple matrix structure that mainly consists of stasis and growth to the next stage.     

Finding confidence limits on population growth rates: Bootstrap and analytic methods

When predicting population dynamics, the value of the prediction is not enough and should be accompanied by a confidence interval that integrates the whole chain of errors, from observations to predictions via the estimates of the parameters of the model. Matrix models are often used to predict the dynamics of age- or size-structured populations. Their parameters are vital rates. This study aims (1) at assessing the impact of the variability of observations on vital rates, and then on model’s predictions, and (2) at comparing three methods for computing confidence intervals for values predicted from the models. The first method is the bootstrap. The second method is analytic and approximates the standard error of predictions by their asymptotic variance as the sample size tends to infinity. The third method combines use of the bootstrap to estimate the standard errors of vital rates with the analytical method to then estimate the errors of predictions from the model. Computations are done for an Usher matrix models that predicts the asymptotic (as time goes to infinity) stock recovery rate for three timber species in French Guiana. Little difference is found between the hybrid and the analytic method. Their estimates of bias and standard error converge towards the bootstrap estimates when the error on vital rates becomes small enough, which corresponds in the present case to a number of observations greater than 5000 trees.     

An Evaluation of Sex-Age-Kill (SAK) Model Performance

ABSTRACT The sex-age-kill (SAK) model is widely used to estimate abundance of harvested large mammals, including white-tailed deer (Odocoileus virginianus). Despite a long history of use, few formal evaluations of SAK performance exist. We investigated how violations of the stable age distribution and stationary population assumption, changes to male or female harvest, stochastic effects (i.e., random fluctuations in recruitment and survival), and sampling efforts influenced SAK estimation. When the simulated population had a stable age distribution and λ > 1, the SAK model underestimated abundance. Conversely, when λ < 1, the SAK overestimated abundance. When changes to male harvest were introduced, SAK estimates were opposite the true population trend. In contrast, SAK estimates were robust to changes in female harvest rates. Stochastic effects caused SAK estimates to fluctuate about their equilibrium abundance, but the effect dampened as the size of the surveyed population increased. When we considered both stochastic effects and sampling error at a deer management unit scale the resultant abundance estimates were within ±121.9% of the true population level 95% of the time. These combined results demonstrate extreme sensitivity to model violations and scale of analysis. Without changes to model formulation, the SAK model will be biased when λ ≠ 1. Furthermore, any factor that alters the male harvest rate, such as changes to regulations or changes in hunter attitudes, will bias population estimates. Sex-age-kill estimates may be precise at large spatial scales, such as the state level, but less so at the individual management unit level. Alternative models, such as statistical age-at-harvest models, which require similar data types, might allow for more robust, broad-scale demographic assessments.     

Modeling vital rates improves estimation of population projection matrices

Population projection matrices are commonly used by ecologists and managers to analyze the dynamics of stage-structured populations. Building projection matrices from data requires estimating transition rates among stages, a task that often entails estimating many parameters with few data. Consequently, large sampling variability in the estimated transition rates increases the uncertainty in the estimated matrix and quantities derived from it, such as the population multiplication rate and sensitivities of matrix elements. Here, we propose a strategy to avoid overparameterized matrix models. This strategy involves fitting models to the vital rates that determine matrix elements, evaluating both these models and ones that estimate matrix elements individually with model selection via information criteria, and averaging competing models with multimodel averaging. We illustrate this idea with data from a population of Silene acaulis (Caryophyllaceae), and conduct a simulation to investigate the statistical properties of the matrices estimated in this way. The simulation shows that compared with estimating matrix elements individually, building population projection matrices by fitting and averaging models of vital-rate estimates can reduce the statistical error in the population projection matrix and quantities derived from it.     

Population dynamics of three songbird species in a nestbox population in Central Europe show effects of density,climate and competitive interactions

Unravelling the contributions of density‐dependent and density‐independent factors in determining species population dynamics is a challenge, especially if the two factors interact. One approach is to apply stochastic population models to long‐term data, yet few studies have included interactions between density‐dependent and density‐independent factors, or explored more than one type of stochastic population model. However, both are important because model choice critically affects inference on population dynamics and stability. Here, we used a multiple models approach and applied log‐linear and non‐linear stochastic population models to time series (spanning 29 years) on the population growth rates of Blue Tits Cyanistes caeruleus, Great Tits Parus major and Pied Flycatchers Ficedula hypoleuca breeding in two nestbox populations in southern Germany. We focused on the roles of climate conditions and intra‐ and interspecific competition in determining population growth rates. Density dependence was evident in all populations. For Blue Tits in one population and for Great Tits in both populations, addition of a density‐independent factor improved model fit. At one location, Blue Tit population growth rate increased following warmer winters, whereas Great Tit population growth rates decreased following warmer springs. Importantly, Great Tit population growth rate also decreased following years of high Blue Tit abundance, but not vice versa. This finding is consistent with asymmetric interspecific competition and implies that competition could carry over to influence population dynamics. At the other location, Great Tit population growth rate decreased following years of high Pied Flycatcher abundance but only when Great Tit population numbers were low, illustrating that the roles of density‐dependent and density‐independent factors are not necessarily mutually exclusive. The dynamics of this Great Tit population, in contrast to the other populations, were unstable and chaotic, raising the question of whether interactions between density‐dependent and density‐independent factors play a role in determining the (in) stability of the dynamics of species populations.