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

Incorporating capture heterogeneity in the estimation of autoregressive coefficients of animal population dynamics using capture-recapture data

Population dynamic models combine density dependence and environmental effects. Ignoring sampling uncertainty might lead to biased estimation of the strength of density dependence. This is typically addressed using state-space model approaches, which integrate sampling error and population process estimates. Such models seldom include an explicit link between the sampling procedures and the true abundance, which is common in capture-recapture settings. However, many of the models proposed to estimate abundance in the presence of capture heterogeneity lead to incomplete likelihood functions and cannot be straightforwardly included in state-space models. We assessed the importance of estimating sampling error explicitly by taking an intermediate approach between ignoring uncertainty in abundance estimates and fully specified state-space models for density-dependence estimation based on autoregressive processes. First, we estimated individual capture probabilities based on a heterogeneity model for a closed population, using a conditional multinomial likelihood, followed by a Horvitz-Thompson estimate for abundance. Second, we estimated coefficients of autoregressive models for the log abundance. Inference was performed using the methodology of integrated nested Laplace approximation (INLA). We performed an extensive simulation study to compare our approach with estimates disregarding capture history information, and using R-package VGAM, for different parameter specifications. The methods were then applied to a real data set of gray-sided voles Myodes rufocanus from Northern Norway. We found that density-dependence estimation was improved when explicitly modeling sampling error in scenarios with low process variances, in which differences in coverage reached up to 8% in estimating the coefficients of the autoregressive processes. In this case, the bias also increased assuming a Poisson distribution in the observational model. For high process variances, the differences between methods were small and it appeared less important to model heterogeneity.

What Have Long-Term Field Studies Taught Us About Population Dynamics?

Long-term studies have been crucial to the advancement of population biology, especially our understanding of population dynamics. We argue that this progress arises from three key characteristics of long-term research. First, long-term data are necessary to observe the heterogeneity that drives most population processes. Second, long-term studies often inherently lead to novel insights. Finally, long-term field studies can serve as model systems for population biology, allowing for theory and methods to be tested under well-characterized conditions. We illustrate these ideas in three long-term field systems that have made outsized contributions to our understanding of population ecology, evolution, and conservation biology. We then highlight three emerging areas to which long-term field studies are well positioned to contribute in the future: ecological forecasting, genomics, and macrosystems ecology. Overcoming the obstacles associated with maintaining long-term studies requires continued emphasis on recognizing the benefits of such studies to ensure that long-term research continues to have a substantial impact on elucidating population biology.

Advances in population ecology and species interactions in mammals

The study of mammals has promoted the development and testing of many ideas in contemporary ecology. Here we address recent developments in foraging and habitat selection, source–sink dynamics, competition (both within and between species), population cycles, predation (including apparent competition), mutualism, and biological invasions. Because mammals are appealing to the public, ecological insight gleaned from the study of mammals has disproportionate potential in educating the public about ecological principles and their application to wise management. Mammals have been central to many computational and statistical developments in recent years, including refinements to traditional approaches and metrics (e.g., capture-recapture) as well as advancements of novel and developing fields (e.g., spatial capture-recapture, occupancy modeling, integrated population models). The study of mammals also poses challenges in terms of fully characterizing dynamics in natural conditions. Ongoing climate change threatens to affect global ecosystems, and mammals provide visible and charismatic subjects for research on local and regional effects of such change as well as predictive modeling of the long-term effects on ecosystem function and stability. Although much remains to be done, the population ecology of mammals continues to be a vibrant and rapidly developing field. We anticipate that the next quarter century will prove as exciting and productive for the study of mammals as has the recent one.

Ecological change points: The strength of density dependence and the loss of history

Change points in the dynamics of animal abundances have extensively been recorded in historical time series records. Little attention has been paid to the theoretical dynamic consequences of such change-points. Here we propose a change-point model of stochastic population dynamics. This investigation embodies a shift of attention from the problem of detecting when a change will occur, to another non-trivial puzzle: using ecological theory to understand and predict the post-breakpoint behavior of the population dynamics. The proposed model and the explicit expressions derived here predict and quantify how density dependence modulates the influence of the pre-breakpoint parameters into the post-breakpoint dynamics. Time series transitioning from one stationary distribution to another contain information about where the process was before the change-point, where is it heading and how long it will take to transition, and here this information is explicitly stated. Importantly, our results provide a direct connection of the strength of density dependence with theoretical properties of dynamic systems, such as the concept of resilience. Finally, we illustrate how to harness such information through maximum likelihood estimation for state-space models, and test the model robustness to widely different forms of compensatory dynamics. The model can be used to estimate important quantities in the theory and practice of population recovery.

Accounting for sampling error when inferring population synchrony from time-series data: a Bayesian state-space modelling approach with applications

Background

Data collected to inform time variations in natural population size are tainted by sampling error. Ignoring sampling error in population dynamics models induces bias in parameter estimators, e.g., density-dependence. In particular, when sampling errors are independent among populations, the classical estimator of the synchrony strength (zero-lag correlation) is biased downward. However, this bias is rarely taken into account in synchrony studies although it may lead to overemphasizing the role of intrinsic factors (e.g., dispersal) with respect to extrinsic factors (the Moran effect) in generating population synchrony as well as to underestimating the extinction risk of a metapopulation.

Methodology/Principal findings

The aim of this paper was first to illustrate the extent of the bias that can be encountered in empirical studies when sampling error is neglected. Second, we presented a space-state modelling approach that explicitly accounts for sampling error when quantifying population synchrony. Third, we exemplify our approach with datasets for which sampling variance (i) has been previously estimated, and (ii) has to be jointly estimated with population synchrony. Finally, we compared our results to those of a standard approach neglecting sampling variance. We showed that ignoring sampling variance can mask a synchrony pattern whatever its true value and that the common practice of averaging few replicates of population size estimates poorly performed at decreasing the bias of the classical estimator of the synchrony strength.

Conclusion/Significance

The state-space model used in this study provides a flexible way of accurately quantifying the strength of synchrony patterns from most population size data encountered in field studies, including over-dispersed count data. We provided a user-friendly R-program and a tutorial example to encourage further studies aiming at quantifying the strength of population synchrony to account for uncertainty in population size estimates.

Contrasting effects of climate on grey heron, malleefowl and barn owl populations

Context The population dynamics of many wildlife species are associated with fluctuations in climate. Food and abundance may also influence wildlife dynamics. Aims The present paper aims to evaluate the relative effects of climate on the annual instantaneous population growth rate (r) of the following three bird species: grey heron and barn owl in parts of Britain and malleefowl in a part of Australia. Methods A priori hypotheses of mechanistic effects of climate are derived and evaluated using information theoretic and regression analyses and published data for the three bird species. Climate was measured as the winter North Atlantic Oscillation (NAO) for herons and owls, and rainfall and also the Southern Oscillation Index (SOI) for malleefowl. Key results Population dynamics of grey heron were positively related to the winter NAO, and of malleefowl were positively related to annual rainfall and related in a non-linear manner to SOI. By contrast, population dynamics of barn owl were very weakly related to climate. The best models for the grey heron differed between time periods but always included an effect of the NAO. Conclusions The annual population growth rate of grey heron, malleefowl and barn owl show contrasting relationships with climate, from stronger (heron and malleefowl) to weaker (barn owl). The results were broadly consistent with reported patterns but differed in some details. Interpretation of the effects of climate on the basis of analyses rather than visual assessment is encouraged. Implications Effects of climate differ among species, so effects of future climate change may also differ.

Detecting density dependence in recovering seal populations

Time series of abundance estimates are commonly used for analyses of population trends and possible shifts in growth rate. We investigate if trends in age composition can be used as an alternative to abundance estimates for detection of decelerated population growth. Both methods were tested under two forms of density dependence and different levels of environmental variation in simulated time series of growth in Baltic gray seals. Under logistic growth, decelerating growth could be statistically confirmed after 16 years based on population counts and 14 years based on age composition. When density dependence sets in first at larger population sizes, the age composition method performed dramatically better than population counts, and a decline could be detected after 4 years (versus 10 years). Consequently, age composition analysis provides a complementary method to detect density dependence, particularly in populations where density dependence sets in late.

Effects of Snail Density on Growth, Reproduction and Survival of Biomphalaria alexandrina Exposed to Schistosoma mansoni

The effects of snail density on Biomphalaria alexandrina parasitized with Schistosoma mansoni were investigated. Laboratory experiments were used to quantify the impact of high density on snail growth, fecundity, and survival. Density-dependent birth rates of snails were determined to inform mathematical models, which, until now, have assumed a linear relationship between density and fecundity. The experiments show that the rate of egg-laying followed a negative exponential distribution with increasing density and this was significantly affected by exposure to parasitic infection. High density also affected the weight of snails and survival to a greater degree than exposure to parasitic infection. Although snail growth rates were initially constrained by high density, they retained the potential for growth suggesting a reversible density-dependent mechanism. These experimental data can be used to parameterise models and confirm that snail populations are regulated by nonlinear density-dependent mechanisms.

Shifting trends: detecting environmentally mediated regulation in long-lived marine vertebrates using time-series data

Assessing the status and trends in animal populations is essential for effective species conservation and management practices. However, unless time-series abundance data demonstrate rapid and reliable fluctuations, objective appraisal of directionality of trends is problematic. We adopted a multiple-working hypotheses approach based on information-theoretic and Bayesian multi-model inference to examine the population trends and form of intrinsic regulation demonstrated by a long-lived species, the southern elephant seal. We also determined the evidence for density dependence in 11 other well-studied marine mammal species. (1) We tested the type of population regulation for elephant seals from Marion Island (1986-2004) and from 11 other marine mammal species, and (2) we described the trends and behavior of the 19-year population time series at Marion Island to identify changes in population trends. We contrasted five plausible trend models using information-theoretic and Bayesian-inference estimates of model parsimony. Our analyses identified two distinct phases of population growth for this population with the inflexion occurring in 1998. Thus, the population decreased between 1986 and 1997 (-3.7% per annum) and increased between 1997 and 2004 (1.9% per annum). An index of environmental stochasticity, the Southern Oscillation Index, explained some of the variance in r and N. We determined analytically that there was good evidence for density dependence in the Marion Island population and that density dependence was widespread among marine mammal species (67% of species showed evidence for population regulation). This approach demonstrates the potential functionality of a relatively simple technique that can be applied to short time series to identify the type of regulation, and the uncertainty associated with the phenomenon, operating in populations of large mammals.

Using reproductive value to estimate key parameters in density-independent age-structured populations

The dynamics of reproductive value are used to provide a simple derivation of Tuljapurkar's approximation for the long-run growth rate and environmental variance of lnN, in a density-independent age-structured population in a random environment. With no environmental autocorrelation, the dynamics of total population size, N, generally shows time lags and autocorrelation caused by life history, which may strongly bias estimates of environmental variance obtained by ignoring age structure. In contrast, the total reproductive value, V, is Markovian and obeys a first-order autoregressive process. This suggests a simple method for estimating the environmental variance, and avoiding potentially large bias due to age-structure fluctuations, by converting a multivariate time series of age structure to a univariate time series of lnV. We illustrate the method by estimating the long-run growth rate and the environmental variance in an exponentially growing population of Bighorn Sheep.

Estimating Density Dependence from Time Series of Population Age Structure

Population fluctuations are caused by demographic and environmental stochasticity, time lags due to life history, and density dependence. We model a general life history allowing density dependence within and among age or stage classes in a population undergoing small or moderate fluctuations around a stable equilibrium. We develop a method for estimating the overall strength of density dependence measured by the rate of return toward equilibrium, and we also consider a simplified population description and forecasting using the density-dependent reproductive value. This generality comes at the cost of requiring a time series of the population age or stage structure instead of a univariate time series of adult or total population size. The method is illustrated by analyzing the dynamics of a fully censused population of red deer (Cervus elaphus) based on annual fluctuations of age structure through 21 years.

Population growth rate and carrying capacity for springtails Folsomia candida exposed to ivermectin

Forecasting the effects of stressors on the dynamics of natural populations requires assessment of the joint effects of a stressor and population density on the population response. The effects can be depicted as a contour map in which the population response, here assessed by population growth rate, varies with stress and density in the same way that the height of land above sea level varies with latitude and longitude. We present the first complete map of this type using as our model Folsomia candida exposed to five different concentrations of the widespread anthelmintic veterinary medicine ivermectin in replicated microcosm experiments lasting 49 days. The concentrations of ivermectin in yeast were 0.0, 6.8, 28.8, 66.4, and 210.0 mg/L wet weight. Increasing density and chemical concentration both significantly reduced the population growth rate of Folsomia candida, in part through effects on food consumption and fecundity. The interaction between density and ivermectin concentration was "less-than-additive," implying that at high density populations were able to compensate for the effects of the chemical. This result demonstrates that regulatory protocols carried out at low density (as in most past experiments) may seriously overestimate effects in the field, where densities are locally high and populations are resource limited (e.g., in feces of livestock treated with ivermectin).

Population growth rates: issues and an application

Current issues in population dynamics are discussed in the context of The Royal Society Discussion Meeting 'Population growth rate: determining factors and role in population regulation'. In particular, different views on the centrality of population growth rates to the study of population dynamics and the role of experiments and theory are explored. Major themes emerging include the role of modern statistical techniques in bringing together experimental and theoretical studies, the importance of long-term experimentation and the need for ecology to have model systems, and the value of population growth rate as a means of understanding and predicting population change. The last point is illustrated by the application of a recently introduced technique, integral projection modelling, to study the population growth rate of a monocarpic perennial plant, its elasticities to different life-history components and the evolution of an evolutionarily stable strategy size at flowering.