Estimating density dependence from population time series using demographic theory and life-history data

For populations with a density-dependent life history reproducing at discrete annual intervals, we analyze small or moderate fluctuations in population size around a stable equilibrium, which is applicable to many vertebrate populations. Using a life history having age at maturity alpha, with stochasticity and density dependence in adult recruitment and mortality, we derive a linearized autoregressive equation with time lags from 1 to alpha yr. Contrary to current interpretations, the coefficients corresponding to different time lags in the autoregressive dynamics are not simply measures of delayed density dependence but also depend on life-history parameters. The theory indicates that the total density dependence in a life history, D, should be defined as the negative elasticity of population growth rate per generation with respect to change in population size, [Formula: see text], where lambda is the asymptotic multiplicative growth rate per year, T is the generation time, and N is adult population size. The total density dependence in the life history, D, can be estimated from the sum of the autoregression coefficients. We estimate D in populations of seven vertebrate species for which life-history studies and unusually long time series of complete population censuses are available. Estimates of D were statistically significant and large, on the order of 1 or higher, indicating strong density dependence in five of the seven species. We also show that life history can explain the qualitative features of population autocorrelation functions and power spectra and observations of increasing empirical variance in population size with increasing length of time series.     

Pattern of variation in avian population growth rates

A central question in population ecology is to understand why population growth rates differ over time. Here, we describe how the long-term growth of populations is not only influenced by parameters affecting the expected dynamics, for example form of density dependence and specific population growth rate, but is also affected by environmental and demographic stochasticity. Using long-term studies of fluctuations of bird populations, we show an interaction between the stochastic and the deterministic components of the population dynamics: high specific growth rates at small densities r(1) are typically positively correlated with the environmental variance sigma(e)(2). Furthermore, theta, a single parameter describing the form of the density regulation in the theta-logistic density-regulation model, is negatively correlated with r(1). These patterns are in turn correlated with interspecific differences in life-history characteristics. Higher specific growth rates, larger stochastic effects on the population dynamics and stronger density regulation at small densities are found in species with large clutch sizes or high adult mortality rates than in long-lived species. Unfortunately, large uncertainties in parameter estimates, as well as strong stochastic effects on the population dynamics, will often make even short-term population projections unreliable. We illustrate that the concept of population prediction interval can be useful in evaluating the consequences of these uncertainties in the population projections for the choice of management actions.     

Estimating density dependence in time-series of age-structured populations

For a life history with age at maturity alpha, and stochasticity and density dependence in adult recruitment and mortality, we derive a linearized autoregressive equation with time-lags of from 1 to alpha years. Contrary to current interpretations, the coefficients for different time-lags in the autoregressive dynamics do not simply measure delayed density dependence, but also depend on life-history parameters. We define a new measure of total density dependence in a life history, D, as the negative elasticity of population growth rate per generation with respect to change in population size, D = - partial differential lnlambda(T)/partial differential lnN, where lambda is the asymptotic multiplicative growth rate per year, T is the generation time and N is adult population size. We show that D can be estimated from the sum of the autoregression coefficients. We estimated D in populations of six avian species for which life-history data and unusually long time-series of complete population censuses were available. Estimates of D were in the order of 1 or higher, indicating strong, statistically significant density dependence in four of the six species.     

Overcompensatory population dynamic responses to environmental stochasticity

1. To quantify the interactions between density-dependent, population regulation and density-independent limitation, we studied the time-series dynamics of an experimental laboratory insect microcosm system in which both environmental noise and resource limitation were manipulated. 2. A hierarchical Bayesian state-space approach is presented through which it is feasible to capture all sources of uncertainty, including observation error to accurately quantify the density dependence operating on the dynamics. 3. The regulatory processes underpinning the dynamics of two different bruchid beetles (Callosobruchus maculatus and Callosobruchus chinensis) are principally determined by environmental conditions, with fluctuations in abundance explained in terms of changes in overcompensatory dynamics and stochastic processes. 4. A general, stochastic population model is developed to explore the link between abundance fluctuations and the interaction between density dependence and noise. Taking account of time-lags in population regulation can substantially increase predicted population fluctuations resulting from underlying noise processes.     

Spatial scales in the study of reef fishes: A theoretical perspective

Abstract Theoretical models imply that spatial scale derives its greatest importance through interactions between density-dependent processes and spatial variation in population densities and environmental variables. Such interactions cause population dynamics on large spatial scales to differ in important ways from predictions based on measurements of population dynamics at smaller scales, a phenomenon called the scale transition. These differences can account for large-scale population stability and species coexistence. The interactions between density dependence and spatial variation that lead to the scale transition can be understood by the process of non-linear averaging, which shows how variance originating on various spatial scales contributes to large-scale population dynamics. Variance originating below the scale of density dependence contributes less to the scale transition as the spatial scale of the variation declines, while variation originating on or above the scale of density dependence contributes independently of the spatial scale of the variation.     

Non-linear density dependence in time series is not evidence of non-logistic growth

Time series of population density are often used to seek deviations from logistic regulation by testing for a non-linear decline in per capita growth rate with density. Here I show that this method fails when the interval between observations is not matched by the timing of density impacts on growth. Time series overestimate instantaneous density impacts at low density and underestimate them at high density. More generally, logistic growth produces a deterministically decelerating decline in per capita growth with density if the interval between measures of population size exceeds any lag in density response. Deceleration arises independently out of stochastic density fluctuations, and under-compensating regulation. These multiple influences lead to the conclusion that sequential density estimates provide insufficient information on their own to reveal the identity of non-logistic growth processes. They can yield estimates of density compensation, however, which may suggest time lags in density dependence. Analysis of an empirical time series illustrates the issues.     

Genetic drift in range expansions is very sensitive to density dependence in dispersal and growth

Theory predicts rapid genetic drift during invasions, yet many expanding populations maintain high genetic diversity. We find that genetic drift is dramatically suppressed when dispersal rates increase with the population density because many more migrants from the diverse, high‐density regions arrive at the expansion edge. When density dependence is weak or negative, the effective population size of the front scales only logarithmically with the carrying capacity. The dependence, however, switches to a sublinear power law and then to a linear increase as the density dependence becomes strongly positive. We develop a unified framework revealing that the transitions between different regimes of diversity loss are controlled by a single, universal quantity: the ratio of the expansion velocity to the geometric mean of dispersal and growth rates at expansion edge. Our results suggest that positive density dependence could dramatically alter evolution in expanding populations even when its contribution to the expansion velocity is small.     

On a stochastic version of a modified Nicholson-Baily model

‘Deterministic’ models in population dynamics often are really approximations to stochastic models, justified by an appeal to ‘the law of large numbers’. It is proposed to call such models ‘pseudodeterministic’. Four questions are discussed in this article: (1) What errors may be made by equating deterministically predicted values to expectations? (2) When, and in what sense, may numbers be assumed to be large? (3) How large are the variances, coefficients of variations, etc., as assigned to the variables in the stochastic versions of the models? (4) What role may pseudodeterministic models play in empirical research, where problems of statistical reliability arise? As an example, a modified Nicholson-Bailey model of the interaction between insect parasitoids and their hosts is discussed; the modification consists of assigning a random (density-independent) mortality to the parasitoid population. A stochastic version of this model is discussed. The expectation of the final host density is compared with the value computed from the deterministic model. The latter value is systematically lower than the former. The magnitude of the difference depends on parameter values. The variability to be expected with the stochastic model is characterized by the coefficient of variation of the final host density; its dependence on parameter values and initial conditions is discussed. It is concluded that it is worthwhile in practical applications to estimate parasitoid mortality, and that the coefficient of variation in real situations may be far from negligible.     

Interspecific differences in stochastic population dynamics explains variation in Taylor's temporal power law

Taylor’s power law, i.e. that the slope for the increase in variance with mean population size is between 1 and 2 at a logarithmic scale, provides one of the few quantitative relationships in population ecology, yet the underlying ecological mechanisms are only poorly understood. Stochastic theory of population dynamics predicts that demographic and environmental stochasticity will affect the slope differently. In a stable environment under the influence of demographic stochasticity alone the slope will be equal to 1. In large populations in which demographic variance will have a negligible effect on the dynamics the slope will approach 2. In addition, the slope will also be influenced by how the strength of density dependence is related to mean population size. To disentangle the relative contribution of these processes we estimate the mean‐variance relationship for a large number of populations of British birds. The variance in population size of most species decreased with the mean due to decreased influence of demographic stochasticity at larger population sizes. Interspecific differences in demographic stochasticity was the main factor influencing variation in slopes of Taylor’s power law among species through a significant negative relationship between the slope and demographic variance. In addition, slopes were influenced by interspecific variation in life history parameters such as adult survival and clutch size. These analyses show that Taylor’s power law is generated from an interplay between stochastic and density dependent factors, modulated by life history.     

Climate and spatio-temporal variation in the population dynamics of a long distance migrant, the white stork

1. A central question in ecology is to separate the relative contribution of density dependence and stochastic influences to annual fluctuations in population size. Here we estimate the deterministic and stochastic components of the dynamics of different European populations of white stork Ciconia ciconia. We then examined whether annual changes in population size was related to the climate during the breeding period (the 'tap hypothesis' sensu Saether, Sutherland & Engen (2004, Advances in Ecological Research, 35, 185 209) or during the nonbreeding period, especially in the winter areas in Africa (the 'tube hypothesis'). 2. A general characteristic of the population dynamics of this long-distance migrant is small environmental stochasticity and strong density regulation around the carrying capacity with short return times to equilibrium. 3. Annual changes in the size of the eastern European populations were correlated by rainfall in the wintering areas in Africa as well as local weather in the breeding areas just before arrival and in the later part of the breeding season and regional climate variation (North Atlantic Oscillation). This indicates that weather influences the population fluctuations of white storks through losses of sexually mature individuals as well as through an effect on the number of individuals that manages to establish themselves in the breeding population. Thus, both the tap and tube hypothesis explains climate influences on white stork population dynamics. 4. The spatial scale of environmental noise after accounting for the local dynamics was 67 km, suggesting that the strong density dependence reduces the synchronizing effects of climate variation on the population dynamics of white stork. 5. Several climate variables reduced the synchrony of the residual variation in population size after accounting for density dependence and demographic stochasticity, indicating that these climate variables had a synchronizing effect on the population fluctuations. In contrast, other climatic variables acted as desynchronizing agents. 6. Our results illustrate that evaluating the effects of common environmental variables on the spatio-temporal variation in population dynamics require estimates and modelling of their influence on the local dynamics.     

Landscape structure and population density affect intraspecific aggression in beavers

Intraspecific competition plays an important role for territory acquisition and occupancy, in turn affecting individual fitness. Thus, understanding the drivers of intraspecific aggression can increase our understanding of population dynamics. Here, we investigated intraspecific aggression in Eurasian (Castor fiber) and North American (Castor canadensis) beavers that are both monogamous, territorial mammals. Combined, we examined tail scars from >1,000 beavers (>2,000 capture events) as part of two long‐term studies in Norway and the USA. We investigated the influence of landscape structure, population density, sex, age, and (for Eurasian beavers only) social status and group size on the number of tail scars caused by conspecifics. The number of tail scars was affected by population density in well‐connected landscape types (large lakes and rivers), but not in more isolated areas (ponds), where individuals generally had fewer tail scars. Further, the relationship of population density was not linear. In the North American beaver population occurring in large lakes, intraspecific aggression increased with population density. Conversely, in the saturated Eurasian beaver population, intraspecific aggression was in a negative relationship with population density (except at the highest densities), likely due to inverse density‐dependent intruder pressure via dispersers. Our findings emphasize that population density can affect intraspecific aggression depending on landscape structure, which might have important consequences for local patterns of dispersal, mate change, and territory occupancy, all of which can affect population dynamics.     

Modeling density dependence and climatic disturbances in caribou: a case study from the Bathurst Island complex, Canadian High Arctic

Peary caribou Rangifer tarandus pearyi is the northernmost subspecies of Rangifer in North America and endemic to the Canadian High Arctic. Because of severe population declines following years of unfavorable winter weather with ice coating on the ground or thicker snow cover, it is believed that density-independent disturbance events are the primary driver for Peary caribou population dynamics. However, it is unclear to what extent density dependence may affect population dynamics of this species. Here, we test for different levels of density dependence in a stochastic, single-stage population model, based on available empirical information for the Bathurst Island complex (BIC) population in the Canadian High Arctic. We compare predicted densities with observed densities during 1961–2001 under various assumptions of the strength of density dependence. On the basis of our model, we found that scenarios with no or very low density dependence led to population densities far above observed densities. For average observed disturbance regimes, a carrying capacity of 0.1 caribou km⁻² generated an average caribou density similar to that estimated for the BIC population over the past four decades. With our model we also tested the potential effects of climate change-related increases in the probability and severity of disturbance years, that is unusually poor winter conditions. On the basis of our simulation results, we found that, in particular, potential increases in disturbance severity (as opposed to disturbance frequency) may pose a considerable threat to the persistence of this species.     

Seven forms of rarity in mammals

Conservation biologists have identified threats to the survival of about a quarter of the mammalian species; to identify patterns of rarity and commonness of mammals, we studied a global sample of 1212 species (about 28% of the mammals) using the ‘7 forms of rarity’ model (in which species are roughly divided into above and below the median for local population density, species’ range area, and number of habitat types). From a niche‐based hypothesis of abundance and distribution, we predicted that mammals would exhibit a bimodal pattern of rarity and commonness, with an overabundance of species in the relatively rarest and most common categories; and just such a significant bimodal pattern emerged, with over a quarter of the species classified as exceedingly rare and a further quarter very common, supporting the niche‐based hypothesis. Orders that include large mammals, including perissodactyls, primates, diprotodonts, and carnivores, exhibited significantly high proportions of relatively rare species; and tropical zoogeographic regions, especially Indomalaya, had relatively high proportions of species in the rarest category. Significant biases in the available data on mammals included under‐sampling of small species like rodents and bats, and a relative paucity of data on zoogeographic regions outside of North America and Australia. Mammalian species listed as of conservation concern by the IUCN occurred in all cells of the model, indicating that even relatively common species can be listed as threatened under some conditions; but we also found that sixty‐three species were relatively rare in all three criteria of the 7‐forms model but were not listed as threatened, indicating potential candidates for further study. Mammals may be a group of animals where rarity or commonness is a natural aspect of species biology, both confirming and perhaps partly explaining the large proportion of mammals assigned threatened status.     

The taxonomic distribution of rare and common species among families in the vascular plant flora of Fennoscandia

Many plant traits are not randomly distributed among families. The question considered here is ‘are rarity and commonness of vascular plants in Fennoscandia randomly distributed among families?’ If more rare or more common species are found within a family, this may give some initial indications about which traits may predict rarity and commonness of species. A species was defined as rare or common based on its abundance and on the number of grid squares it occupies. 1521 naturally occurring species in 229 75×75 km grid squares were used. Permutation tests were performed to assess statistically if rarity and commonness are randomly distributed among families. Several families can be identified as having more rare or more common species than would be expected under a random allocation model. However, there are little deviations from what would be expected if rarity and commonness were randomly distributed among families in the whole Fennoscandian flora. It is proposed that the arbitrary geographical limits of the study area may account for the lack of any clear patterns of rarity and commonness among and between families.     

Rarity facets of biodiversity: Integrating Zeta diversity and Dark diversity to understand the nature of commonness and rarity

Measuring commonness and rarity is pivotal to ecology and conservation. Zeta diversity, the average number of species shared by multiple sets of assemblages, and Dark diversity, the number of species that could occur in an assemblage but are missing, have been recently proposed to capture two aspects of the commonness‐rarity spectrum. Despite a shared focus on commonness and rarity, thus far, Zeta and Dark diversities have been assessed separately. Here, we review these two frameworks and suggest their integration into a unified paradigm of the “rarity facets of biodiversity.” This can be achieved by partitioning Alpha and Beta diversities into five components (the Zeta, Eta, Theta, Iota, and Kappa rarity facets) defined based on the commonness and rarity of species. Each facet is assessed in traditional and multiassemblage fashions to bridge conceptual differences between Dark diversity and Zeta diversity. We discuss applications of the rarity facets including comparing the taxonomic, functional, and phylogenetic diversity of rare and common species, or measuring species'' prevalence in different facets as a metric of species rarity. The rarity facets integrate two emergent paradigms in biodiversity science to better understand the ecology of commonness and rarity, an important endeavor in a time of widespread changes in biodiversity across the Earth.     

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.     

Population dynamics of coral-reef fishes: Controversial concepts and hypotheses

Abstract Knowledge of processes that drive the local population dynamics of coral-reef fishes is important for managing reef fisheries, and for using these species as models for understanding the ecology of demersal marine fishes in general. However, the reef-fish literature is replete with poorly defined concepts and vague hypotheses regarding the issue of population dynamics. Dichotomous arguments, such as whether or not recruitment drives population dynamics, are misdirected because they fail to incorporate several important concepts. First, changes in local population size are driven by four demographic rates (birth, death, immigration and emigration), all of which must be studied to understand population dynamics. Second, all populations that persist do so because at least one of these demographic rates operates in a density-dependent way that is both sufficiently strong and appropriately time-lagged. Therefore, identifying the source(s) of direct density dependence is critical for understanding the limits to variation in population size (i.e. population regulation). Third, regulation does not imply a simple point equilibrium in population size; density dependence in populations of reef fishes is bound to lie within a field of stochastic variation, and thus be difficult to detect. Since its formal origin in 1981, the ‘recruitment limitation’ hypothesis for explaining local population dynamics in reef fishes has undergone ambiguous changes in definition that threaten its usefulness. ‘Recruitment, ‘originally defined as the appearance of newly settled fish on a reef, more recently is often measured months after settlement, thus confounding pre- and post-settlement processes. ‘Limitation, ‘ which originally referred to recruitment being so low as to preclude local populations from reaching densities where resources were limiting, is more recently defined as an absence of any form of density dependence after settlement. The most effective means of testing whether post-settlement mortality is in fact density-independent is to examine patterns of mortality directly, rather than indirectly by interpreting the shape of the relationship between initial recruit density and subsequent adult density within a cohort (the recruit-adult function). Understanding the population dynamics of coral-reef fishes will require a more equitable focus on all four demographic rates, be they density dependent or not, as well as greater attention to identifying sources of density dependence. Such a pluralistic focus necessitates integrated studies of both pre- and post-settlement processes conducted at multiple spatial and temporal scales. For example, recent evidence suggests that density-dependent pre-dation on new recruits that have settled among reefs at different densities may prove to be an important source of local population regulation, especially via the aggregative response of transient piscivores.     

Direct negative density‐dependence in a pond‐breeding frog population

Understanding population dynamics is critical for the management of animal populations. Comparatively little is known about the relative importance of endogenous (i.e. density‐dependent) and exogenous (i.e. density‐independent) factors on the population dynamics of amphibians with complex life cycles. We examined the potential effects of density‐dependent and ‐independent (i.e. climatic) factors on population dynamics by analyzing a 15‐yr time series data of the agile frog Rana dalmatina population from Târnava Mare Valley, Romania. We used two statistical models: 1) the partial rate correlation function to identify the feedback structure and the potential time lags in the time series data and 2) a Gompertz state‐space model to simultaneously investigate direct and delayed density dependence as well as climatic effects on population growth rate. We found evidence for direct negative density dependence, whereas delayed density dependence and climate did not show a strong influence on population growth rate. Here we demonstrated that direct density dependence rather than delayed density dependence or climate determined the dynamics of our study population. Our results confirm the findings of many experimental studies and suggest that density dependence may buffer amphibian populations against environmental stress. Consequently, it may not be easy to scale up from individual‐level effects to population‐level effects.     

Predictions of Taylor's power law, density dependence and pink noise from a neutrally modeled time series

There has recently been increasing interest in neutral models of biodiversity and their ability to reproduce the patterns observed in nature, such as species abundance distributions. Here we investigate the ability of a neutral model to predict phenomena observed in single-population time series, a study complementary to most existing work that concentrates on snapshots in time of the whole community. We consider tests for density dependence, the dominant frequencies of population fluctuation (spectral density) and a relationship between the mean and variance of a fluctuating population (Taylor's power law). We simulated an archipelago model of a set of interconnected local communities with variable mortality rate, migration rate, speciation rate, size of local community and number of local communities. Our spectral analysis showed ‘pink noise’: a departure from a standard random walk dynamics in favor of the higher frequency fluctuations which is partly consistent with empirical data. We detected density dependence in local community time series but not in metacommunity time series. The slope of the Taylor's power law in the model was similar to the slopes observed in natural populations, but the fit to the power law was worse. Our observations of pink noise and density dependence can be attributed to the presence of an upper limit to community sizes and to the effect of migration which distorts temporal autocorrelation in local time series. We conclude that some of the phenomena observed in natural time series can emerge from neutral processes, as a result of random zero-sum birth, death and migration. This suggests the neutral model would be a parsimonious null model for future studies of time series data.     

Invasion theory and biological control

Recent advances in the mathematical theory of invasion dynamics have much to offer to biological control. Here we synthesize several results concerning the spatiotemporal dynamics that occur when a biocontrol agent spreads into a population of an invading pest species. We outline conditions under which specialist and generalist predators can influence the density and rate of spatial spread of the pest, including the rather stringent conditions under which a specialist predator can successfully reverse a pest invasion. We next discuss the connections between long distance dispersal and invasive spread, emphasizing the different consequences of fast spreading pests and predators. Recent theory has considered the effects of population stage-structure on invasion dynamics, and we discuss how population demography affects the biological control of invading pests. Because low population densities generally characterize early stages of an invasion, we discuss the lessons invasion theory teaches concerning the detectability of invasions. Stochasticity and density-dependent dynamics are common features of many real invasions, influencing both the spatial character (e.g. patchiness) of pest invasions and the success of biocontrol agents. We conclude by outlining theoretical results delineating how stochastic effects and complex dynamics generated by density dependence can facilitate or impede biological pest control.