Process and measurement errors of population size: their mutual effects on precision and bias of estimates for demographic parameters

Knowing the parameters of population growth and regulation is fundamental for answering many ecological questions and the successful implementation of conservation strategies. Moreover, detecting a population trend is often a legal obligation. Yet, inherent process and measurement errors aggravate the ability to estimate these parameters from population time-series. We use numerical simulations to explore how the lengths of the time-series, process and measurement error influence estimates of demographic parameters. We first generate time-series of population sizes with given demographic parameters for density-dependent stochastic population growth, but assume that these population sizes are estimated with measurement errors. We then fit parameters for population growth, habitat capacity, total error and long-term trends to the ‘measured’ time-series data using non-linear regression. The length of the time-series and measurement error introduce a substantial bias in the estimates for population growth rate and to a lesser degree on estimates for habitat capacity, while process error has little effect on parameter bias. The total error term of the statistical model is dominated by process error as long as the latter is larger than the measurement error. A decline in population size is difficult to document as soon as either error becomes moderate, trends are not very pronounced, and time-series are short (<10–15 seasons). Detecting an annual decline of 1% within 6-year reporting periods, as required for the European Union for the species of Community Interest, appears unachievable.     

Density dependence, regulation and open-closed populations: insights from the wigeon, Anas penelope

We investigated whether the degree of exchange with other populations affects the occurrence of density-dependent regulation. We contrasted data from an Icelandic and a Finnish population of breeding wigeons ( Anas penelope ), the former population being more closed than the later. We looked for density dependence in time-series data and investigated whether breeding success is density dependent and plays a role in population dynamics and regulation. Time-series analysis did not reveal density-dependent regulation in either population. Nor did we find evidence of density-dependent breeding success in either population. However, population growth rate appeared to be strongly dependent on the breeding success in the previous year in the closed population but not in the open population. Our findings underline how important it is to link time-series analysis to the study of potential stabilizing mechanisms in order to understand population dynamics and regulation. We also suggest that it may be a difficult task to achieve sustainability in waterfowl harvesting, the theoretical basis of which is density-dependent population regulation.     

Estimation of a Nonlinear Density-Dependence Parameter for Wild Turkey

Abstract: Although previous research and theory has suggested that wild turkey (Meleagris gallopavo) populations may be subject to some form of density dependence, there has been no effort to estimate and incorporate a density-dependence parameter into wild turkey population models. To estimate a functional relationship for density dependence in wild turkey, we analyzed a set of harvest-index time series from 11 state wildlife agencies. We tested for lagged correlations between annual harvest indices using partial autocorrelation analysis. We assessed the ability of the density-dependent theta-Ricker model to explain harvest indices over time relative to exponential or random walk growth models. We tested the homogeneity of the density-dependence parameter estimates (θ) from 3 different harvest indices (spring harvest no. reported harvest/effort, survey harvest/effort) and calculated a weighted average based on each estimate's variance and its estimated covariance with the other indices. To estimate the potential bias in parameter estimates from measurement error, we conducted a simulation study using the theta-Ricker with known values and lognormally distributed measurement error. Partial autocorrelation function analysis indicated that harvest indices were significantly correlated only with their value at the previous time step. The theta-Ricker model performed better than the exponential growth or random walk models for all 3 indices. Simulation of known parameters and measurement error indicated a strong positive upward bias in the density-dependent parameter estimate, with increasing measurement error. The average density-dependence estimate, corrected for measurement error ranged 0.25 ≤ θC ≤ 0.49, depending on the amount of measurement error and assumed spring harvest rate. We infer that density dependence is nonlinear in wild turkey, where growth rates are maximized at 39-42% of carrying capacity. The annual yield produced by density-dependent population growth will tend to be less than that caused by extrinsic environmental factors. This study indicates that both density-dependent and density-independent processes are important to wild turkey population growth, and we make initial suggestions on incorporating both into harvest management strategies.     

Testing for density-dependent effects in sequential censuses

Summary In response to Gaston and Lawton (1987), we evaluated the ability of four statistical procedures to detect density dependence. We used data from the same 16 populations as Gaston and Lawton (1987). In each population, density dependence had been previously established with techniques that use more extensive data. The major axis test (Slade 1977) was rarely (3 populations of 16) capable of detecting density dependence. The autocorrelation test (Bulmer 1975) detected density dependence in 5 of 16 species (14 of 59 tests overall). The randomization procedure (Pollard et al. 1987) detected density dependence in 7 of the 16 data sets (10 of 59 tests overall). The simulation procedure (Vickery and Nudds 1984) detected density dependence in 5 of the 16 data sets (11 of 59 tests overall). We suggest that not all annual census data taken from populations subject to density-dependent effects will actually show evidence of such effects. We conclude that Pollard et al. 's (1987) randomization procedure is the best test for detecting density dependence in sequential census data but it is not as powerful as more elaborate techniques (k-factor analysis, experimentation, etc.), nor is it meant to replace more extensive analyses.     

Multiple sources of evidence for density dependence in the endangered Hawaiian stilt (Himantopus mexicanus knudseni)

Hawaiian stilts (Himantopus mexicanus knudseni) are an endangered subspecies of the Black-necked stilt endemic to the Hawaiian Islands. Despite long-term study, the main drivers of Hawaiian stilt population dynamics are poorly understood. We tested for density dependence using two sources of evidence: a 30-year time series of annual estimated range-wide abundance, and two 15+ year time series of reproductive success. Using separate methods with independent data, sources allowed us to make up for the potentially positive bias of one approach with the more conservative nature of the second. We compared nonlinear density-dependent and density-independent population model fits to our time-series data, using both frequentist and Bayesian state-space approaches. Across both approaches, density-dependent models best fit observed population dynamics, with lower AICc and cross-validation statistics compared to density-independent models. Among density-dependent models, a conditional model in which density-independent dynamics occur below a population size threshold (~850–1,000 birds), and then density-dependent dynamics occur above that threshold, performed best across Bayesian and frequentist model comparisons, with the Ricker model ranked next or equivalently. Our analysis of reproduction data revealed a strong negative effect of local adult density on nest success (proportion of nests hatching at least one chick) at Kealia National Wildlife Refuge on Maui, where few alternative breeding habitats are available, but no such effect at another site where many nearby alternative wetlands are available. These congruent results across independent datasets and analytical approaches support the hypothesis that Hawaiian stilts exhibit density dependence across their range.     

Visibility of the impact of environmental noise: a response to Kaitala and Ranta

Based on simulation modelling, Kaitala and Ranta (2001 Proc. R. Soc. Lond. B 268, 1769-1774) have argued that detecting the statistical relationships between environmental variability and population fluctuations will be difficult. However, their study was limited in that only one pattern of density dependence and one detection method were used. Here, we show that their conclusion is in part a consequence of their choice of population model and in part a consequence of using relatively weak or inappropriate statistical methods. Other patterns of density dependence respond differently to environmental fluctuations, and the impact of the disturbance on these is clearly visible using their methods. For some patterns of population dynamics, environmental impacts are more readily detectable by correlating running-average environmental conditions with the population time-series or by correlating the first differences of the population time-series with environmental noise. When more appropriate statistical methods are used, environmental forcing is detectable in the majority of cases used by Kaitala and Ranta. The interplay between environmental stochasticity and density-dependent population growth means that there is no single best method to detect the influence of environmental forcing, even when population dynamics are approximately linear. But environmental forcing will often be detectable, contrary to Kaitala and Ranta's assertions.     

Density dependence and the spread of anthelmintic resistance

Variation in the strength of selection pressures acting upon different subpopulations may cause density-dependent regulatory processes to act differentially on particular genotypes and may influence the rate of selection of adaptive traits. Using host-helminth parasite systems as examples, we investigate the impact of different positive and negative density dependence on the potential spread of anthelmintic resistance. Following chemotherapy, the negative density-dependent processes restricting parasite population growth will be relaxed, increasing the genetic contribution of resistant parasites to the next generation. Simple deterministic models of directly transmitted nematodes that merge population dynamics and genetics show that the frequency of drug-resistant alleles may increase faster in species whose population size is down-regulated by density-dependent parasite fecundity than in species with density-dependent establishment or parasite mortality. A genetically structured population dynamics model of an indirectly transmitted nematode is used to highlight how population regulation will influence the resistance allele frequency in different parasite lifestages. Results indicate that surveys aimed at monitoring the evolution of drug resistance should consider carefully which life stage to sample, and the time following treatment samples should be collected. Anthelmintic resistance offers a good opportunity to apply fundamental evolutionary and ecological principles to the management of a potentially crucial public health problem.     

The relative roles of density and climatic variation on population dynamics and fecundity rates in three contrasting ungulate species

The relative influences of density-dependent and -independent processes on vital rates and population dynamics have been debated in ecology for over half a century, yet it is only recently that both processes have been shown to operate within the same population. However, generalizations on the role of each process across species are rare. Using a process-orientated generalized linear modelling approach we show that variations in fecundity rates in populations of three species of ungulates with contrasting life histories are associated with density and winter weather in a remarkably similar manner. However, there are differences and we speculate that they are a result of differences in size between the species. Much previous research exploring the association between vital rates, population dynamics and density-dependent and -independent processes has used pattern-orientated approaches to decompose time-series into contributions from density-dependent and -independent processes. Results from these analyses are sometimes used to infer associations between vital rates, density and climatic variables. We compare results from pattern-orientated analyses of time-series with process-orientated analyses and report that the two approaches give different results. The approach of analysing relationships between vital rates, density and climatic variables may detect important processes influencing population dynamics that time-series methodologies may overlook.     

Simulating density-dependent relationships in south Texas northern bobwhite populations

Density dependence influences northern bobwhite (Colinus virginianus) reproduction and overwinter mortality. However, the functional forms of these density-dependent relationships or the factors that influence them during the annual life cycle events of this bird are not clear. We used a systems analysis approach with a compartment model based on difference equations (Δt = 3 months) for bobwhites in South Texas to simulate population behavior using 16 different functional forms of density-dependent production and overwinter mortality. During the reproductive season, a weak linear density-dependent relationship resulted in the longest population persistence (up to 100.0 yr), whereas a reverse-sigmoid density-dependent relationship had the worst population persistence (2.5–3.5 yr). Regarding overwinter mortality, a sigmoid or weak linear density-dependent relationship and a weak linear or no density-dependent reproduction relationship had the longest population persistence (87.5–100.0 yr). Weak linear density-dependent reproduction with either sigmoid or weak linear overwinter mortality produced stable fall population trends. Our results indicated that density dependence may have a greater influence on overwinter survival of bobwhites than previously thought. Inclusion of density-dependent functional relationships that represent both density-dependent reproduction and overwinter mortality, were critical for our simulation model to function properly. Therefore, integrating density-dependent relationships for both reproductive and overwinter periods of the annual cycle of bobwhite life history events is essential for conducting realistic bobwhite population simulation analyses that can be used to test different management scenarios in an integrated and interdisciplinary manner. © 2012 The Wildlife Society.     

Recruitment limitation: A theoretical perspective

Abstract A theoretical analysis of the concept of recruitment limitation leads to the conclusion that most populations should he regarded as jointly limited by recruitment and interactions between individuals after recruitment. The open nature of local marine systems does not permit avoidance of density-dependent interactions; it simply may make them more difficult to detect. Local populations consisting of settled organisms may not experience density-dependent interactions under some circumstances, but the entire species population consisting of the collection of local populations and their planktonic larvae must have density-dependent dynamics. Any local population of settled individuals can escape density dependence if sufficient density dependence occurs among planktonic larvae or within other local populations. Common conceptions of density dependence are too narrow, leading too often to the conclusion that it is absent from a system. It is equally wrong to expect that density-dependent interactions after settlement determine local population densities independently of recruitment. Special circumstances allowing density dependence to act strongly and quickly are needed before density dependence can neutralize the effects of recruitment. Recruitment limitation and density-dependent interactions therefore should not be regarded as alternatives but as jointly acting to determine the densities of marine benthic populations. Moreover, the interaction between fluctuating recruitment and density dependence is potentially the most interesting feature of recruitment limitation. For example, this interaction may be an important diversity-maintaining mechanism for marine systems.     

Population fluctuations and regulation in great snipe: a time-series analysis

1. During the last centuries, the breeding range of the great snipe Gallinago media has declined dramatically in the western part of its distribution. To examine present population dynamics in the Scandinavian mountains, we collected and analysed a 19-year time series of counts of great snipe males at leks in central Norway, 1987-2005. 2. The population showed large annual fluctuations in the number of males displaying at lek sites (range 45-90 males at the peak of the mating season), but no overall trend. 3. We detected presence of direct density-dependent mechanisms regulating this population. Inclusion of the density-dependent term in a Ricker-type model significantly improved the fit with observed data (evaluated with Parametric Bootstrap Likelihood Ratio tests and Akaike's Information Criterion for small sample size). 4. An analysis of (a number of a priori likely) environmental covariates suggests that the population dynamics were affected by conditions influencing reproduction and survival of offspring during the summer, but not by conditions influencing survival at the wintering grounds in Africa. This is in contrast to many altricial birds breeding in the northern hemisphere, and supports the idea that population dynamics of migratory nidifugous birds are more influenced by conditions during reproduction. 5. Inclusion of these external factors into our model improved the detectability of density dependence. This illustrates that allowing for external effects may increase statistical power of density dependence tests and thus be of particular importance in relatively short time series. 6. In our best model of the population dynamics, two likely density-independent offspring survival covariates explained 47.3% of the variance in great snipe numbers (predation pressure estimated by willow grouse reproductive success and food availability estimated by the amount of precipitation in June), whereas density dependence explained 35.5%. Demographic stochasticity and unidentified environmental stochasticity may account for the remaining 17.2%.     

Regulation of numbers in the Great tit (Aves: Passeriformes)

The census data of the Great tit collected by Perrins (1965) and others in Marley Wood near Oxford are analysed for density-dependence. Clutch size and hatching success are density-dependent and sufficiently so to regulate the population at the observed level (assuming that there is in addition a fairly large density-independent mortality). There may also be some weak density-dependent mortality outside the breeding season. The density-dependent variations in clutch size are probably in the main due to shortage of available food and density-dependent hatching failure is caused by predation. Territorial behaviour has been shown experimentally to determine breeding density, and may produce a density-dependent effect outside the breeding season. These three factors are responsible for regulation of the Great tit population in Marley Wood.     

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.     

Quantile regression estimates of animal population trends

Ecologists often estimate population trends of animals in time series of counts using linear regression to estimate parameters in a linear transformation of multiplicative growth models, where logarithms of rates of change in counts in time intervals are used as response variables. We present quantile regression estimates for the median (0.50) and interquartile (0.25, 0.75) relationships as an alternative to mean regression estimates for common density-dependent and density-independent population growth models. We demonstrate that the quantile regression estimates are more robust to outliers and require fewer distributional assumptions than conventional mean regression estimates and can provide information on heterogeneous rates of change ignored by mean regression. We provide quantile regression trend estimates for 2 populations of greater sage-grouse (Centrocercus urophasianus) in Wyoming, USA, and for the Crawford population of Gunnison sage-grouse (Centrocercus minimus) in southwestern Colorado, USA. Our selected Gompertz models of density dependence for both populations of greater sage-grouse had smaller negative estimates of density-dependence terms and less variation in corresponding predicted growth rates (λ) for quantile than mean regression models. In contrast, our selected Gompertz models of density dependence with piecewise linear effects of years for the Crawford population of Gunnison sage-grouse had predicted changes in λ across years from quantile regressions that varied more than those from mean regression because of heterogeneity in estimated λs that were both less and greater than mean estimates. Our results add to literature establishing that quantile regression provides better behaved estimates than mean regression when there are outlying growth rates, including those induced by adjustments for zeros in the time series of counts. The 0.25 and 0.75 quantiles bracketing the median provide robust estimates of population changes (λ) for the central 50% of time series data and provide a 50% prediction interval for a single new prediction without making parametric distributional assumptions or assuming homogeneous λs. Compared to mean estimates, our quantile regression trend estimates for greater sage-grouse indicated less variation in density-dependent λs by minimizing sensitivity to outlying values, and for Gunnison sage-grouse indicated greater variation in density-dependent λs associated with heterogeneity among quantiles.     

Estimation of Density-Dependent Mortality of Juvenile Bivalves in the Wadden Sea

We investigated density-dependent mortality within the early months of life of the bivalves Macoma balthica (Baltic tellin) and Cerastoderma edule (common cockle) in the Wadden Sea. Mortality is thought to be density-dependent in juvenile bivalves, because there is no proportional relationship between the size of the reproductive adult stocks and the numbers of recruits for both species. It is not known however, when exactly density dependence in the pre-recruitment phase occurs and how prevalent it is. The magnitude of recruitment determines year class strength in bivalves. Thus, understanding pre-recruit mortality will improve the understanding of population dynamics. We analyzed count data from three years of temporal sampling during the first months after bivalve settlement at ten transects in the Sylt-Rømø-Bay in the northern German Wadden Sea. Analyses of density dependence are sensitive to bias through measurement error. Measurement error was estimated by bootstrapping, and residual deviances were adjusted by adding process error. With simulations the effect of these two types of error on the estimate of the density-dependent mortality coefficient was investigated. In three out of eight time intervals density dependence was detected for M. balthica, and in zero out of six time intervals for C. edule. Biological or environmental stochastic processes dominated over density dependence at the investigated scale.     

A technique for analysis of density dependence in population models

The concept of density-dependent population growth is fundamental to our understanding of how populations persist. While it is generally agreed that negative density dependence must occur at high densities, the direction of density dependence may be negative (pure negative density dependence) or positive (demographic Allee effect) at low densities. In this article, we present a technique to link the direction of density dependence to generic ecological factors. This technique involves exploiting the presence of a particular bifurcation, known as a saddle-node-transcritical interaction. We first provide a method to detect this bifurcation in a given model and then demonstrate its ecological relevance using several existing mechanistic models. With a mathematical framework in place, we are able to identify scenarios in which neither a weak Allee effect nor pure negative density dependence are possible. More generally, we find conditions on parameter values that are necessary for transitions between pure negative density dependence and demographic Allee effects to occur.     

Population regulation in adult, but not juvenile, resident trout (Salmo trutta) in a Lake District stream

1. Unlike a neighbouring sea-trout population that showed strong density-dependent survival, a resident trout population ( Salmo trutta L.) showed simple proportionate survival in the early life-stages. However, this persistent population fluctuated within narrow limits. Mature adults, especially during spawning, were the only possible life-stage left in which regulation might occur. 2. An October census, just prior to spawning, was made at five sites (total area 300 m ² ) from 1965 to 1983. Gravel nests (redds) associated with females of known size were excavated outside these sites to obtain a power-function relationship between egg density per redd and female length (range 181–280 mm, n = 26). This relationship and the census data for females (range 186–284 mm) were used to estimate egg densities in each year-class. 3. The census data for the early life-stages (0+, 1+, 2+ trout) confirmed proportionate survival with no evidence for density-dependent regulation. In contrast, the number of spawning females produced in each year-class was strongly density dependent on the initial number of females that laid eggs at the start of the year-class. Similarly, total egg production in each year-class was density dependent on initial egg density. 4 Both relationships were well described by the Ricker and Beverton-Holt stock-recruitment models (P < 0.001) and the goodness-of-fit was similar for both models. This study is probably the first to provide clear evidence for fish population regulation in the adult, rather than the juvenile, stage.     

Impact of density and environmental factors on population fluctuations in a migratory passerine

1. Populations of plants and animals typically fluctuate because of the combined effects of density-dependent and density-independent processes. The study of these processes is complicated by the fact that population sizes are typically not known exactly, because population counts are subject to sampling variance. Although the existence of sampling variance is broadly acknowledged, relatively few studies on time-series data have accounted for it, which can result in wrong inferences about population processes. 2. To increase our understanding of population dynamics, we analysed time series from six Central European populations of the migratory red-backed shrike Lanius collurio by simultaneously assessing the strength of density dependence, process and sampling variance. In addition, we evaluated hypotheses predicting effects of factors presumed to operate on the breeding grounds, at stopover sites in eastern Africa during fall and spring migration and in the wintering grounds in southern Africa. We used both simple and state-space formulations of the Gompertz equation to model population size. 3. Across populations and modelling approaches, we found consistent evidence for negative density-dependent population regulation. Further, process variance contributed substantially to variation in population size, while sampling variance did not. Environmental conditions in eastern and southern Africa appear to influence breeding population size, as rainfall in the Sahel during fall migration and in the south African wintering areas were positively related to population size in the following spring in four of six populations. In contrast, environmental conditions in the breeding grounds were not related to population size. 4. Our findings suggest negative density-dependent regulation of red-backed shrike breeding populations and are consistent with the long-standing hypothesis that conditions in the African staging and wintering areas influence population numbers of species breeding in Europe. 5. This study highlights the importance of jointly investigating density-dependent and density-independent processes to improve our understanding of factors influencing population fluctuations in space and time.     

Interactions between density-dependent processes, population dynamics and control of an invasive plant species, Tripleurospermum perforatum (scentless chamomile)

Tripleurospermum perforatum is an invasive weedy species which exhibits strong over-compensating density dependence. Interactions between density-dependent survival, probability of flowering and fecundity were modelled and their impact on the population dynamics were examined. When only fecundity was density-dependent, the dynamics were similar to those observed in the model containing all three density-dependent terms. Density-dependent survival was a stabilizing process when acting in combination with density-dependent fecundity and probability of flowering; removing density-dependent survival from the model produced two-point cycles. The addition of a seed bank was also stabilizing. Simulations of control strategies at different life-history stages indicated that full control would be difficult due to the strong over-compensating density dependence, with severe reductions in fecundity and late season survival necessary in order to reduce equilibrium seed density and biomass.