Jonas Knape

Are patterns of density dependence in the Global Population Dynamics Database driven by uncertainty about population abundance

Density dependence in population growth rates is of immense importance to ecological theory and application, but is difficult to estimate. The Global Population Dynamics Database (GPDD), one of the largest collections of population time series available, has been extensively used to study cross-taxa patterns in density dependence. A major difficulty with assessing density dependence from time series is that uncertainty in population abundance estimates can cause strong bias in both tests and estimates of strength. We analyse 627 data sets in the GPDD using Gompertz population models and account for uncertainty via the Kalman filter. Results suggest that at least 45% of the time series display density dependence, but that it is weak and difficult to detect for a large fraction. When uncertainty is ignored, magnitude of and evidence for density dependence is strong, illustrating that uncertainty in abundance estimates qualitatively changes conclusions about density dependence drawn from the GPDD.

Effects of weather and climate on the dynamics of animal population time series.

Weather is one of the most basic factors impacting animal populations, but the typical strength of such impacts on population dynamics is unknown. We incorporate weather and climate index data into analysis of 492 time series of mammals, birds and insects from the global population dynamics database. A conundrum is that a multitude of weather data may a priori be considered potentially important and hence present a risk of statistical over-fitting. We find that model selection or averaging alone could spuriously indicate that weather provides strong improvements to short-term population prediction accuracy. However, a block randomization test reveals that most improvements result from over-fitting. Weather and climate variables do, in general, improve predictions, but improvements were barely detectable despite the large number of datasets considered. Climate indices such as North Atlantic Oscillation are not better predictors of population change than local weather variables. Insect time series are typically less predictable than bird or mammal time series, although all taxonomic classes display low predictability. Our results are in line with the view that population dynamics is often too complex to allow resolving mechanisms from time series, but we argue that time series analysis can still be useful for estimating net environmental effects.

ESTIMABILITY OF DENSITY DEPENDENCE IN MODELS OF TIME SERIES DATA

Estimation of density dependence from time series data on population abundance is hampered in the presence of observation or measurement errors. Fitting state-space models has been proposed as a solution that reduces the bias in estimates of density dependence caused by ignoring observation errors. While this is often true, I show that, for specific parameter values, there are identifiability issues in the linear state-space model when the strength of density dependence and the observation and process error variances are all unknown. Using simulation to explore properties of the estimators, I illustrate that, unless assumptions are imposed on the process or observation error variances, the variance of the estimator of density dependence varies critically with the strength of the density dependence. Under compensatory dynamics, the stronger the density dependence the more difficult it is to estimate in the presence of observation errors. The identifiability issues disappear when density dependence is estimated from the state-space model with the observation error variance known to the correct value. Direct estimates of observation variance in abundance censuses could therefore prove helpful in estimating density dependence but care needs to be taken to assess the uncertainty in variance estimates.

Stochastic demography and population dynamics in the red kangaroo Macropus rufus

1. Many organisms inhabit strongly fluctuating environments but their demography and population dynamics are often analysed using deterministic models and elasticity analysis, where elasticity is defined as the proportional change in population growth rate caused by a proportional change in a vital rate. Deterministic analyses may not necessarily be informative because large variation in a vital rate with a small deterministic elasticity may affect the population growth rate more than a small change in a less variable vital rate having high deterministic elasticity. 2. We analyse a stochastic environment model of the red kangaroo (Macropus rufus), a species inhabiting an environment characterized by unpredictable and highly variable rainfall, and calculate the elasticity of the stochastic growth rate with respect to the mean and variability in vital rates. 3. Juvenile survival is the most variable vital rate but a proportional change in the mean adult survival rate has a much stronger effect on the stochastic growth rate. 4. Even if changes in average rainfall have a larger impact on population growth rate, increased variability in rainfall may still be important also in long-lived species. The elasticity with respect to the standard deviation of rainfall is comparable to the mean elasticities of all vital rates but the survival in age class 3 because increased variation in rainfall affects both the mean and variability of vital rates. 5. Red kangaroos are harvested and, under the current rainfall pattern, an annual harvest fraction of c. 20% would yield a stochastic growth rate about unity. However, if average rainfall drops by more than c. 10%, any level of harvesting may be unsustainable, emphasizing the need for integrating climate change predictions in population management and increase our understanding of how environmental stochasticity translates into population growth rate.

On observation distributions for state space models of population survey data

1. State space models are starting to replace more simple time series models in analyses of temporal dynamics of populations that are not perfectly censused. By simultaneously modelling both the dynamics and the observations, consistent estimates of population dynamical parameters may be obtained. For many data sets, the distribution of observation errors is unknown and error models typically chosen in an ad-hoc manner. 2. To investigate the influence of the choice of observation error on inferences, we analyse the dynamics of a replicated time series of red kangaroo surveys using a state space model with linear state dynamics. Surveys were performed through aerial counts and Poisson, overdispersed Poisson, normal and log-normal distributions may all be adequate for modelling observation errors for the data. We fit each of these to the data and compare them using AIC. 3. The state space models were fitted with maximum likelihood methods using a recent importance sampling technique that relies on the Kalman filter. The method relaxes the assumption of Gaussian observation errors required by the basic Kalman filter. Matlab code for fitting linear state space models with Poisson observations is provided. 4. The ability of AIC to identify the correct observation model was investigated in a small simulation study. For the parameter values used in the study, without replicated observations, the correct observation distribution could sometimes be identified but model selection was prone to misclassification. On the other hand, when observations were replicated, the correct distribution could typically be identified. 5. Our results illustrate that inferences may differ markedly depending on the observation distributions used, suggesting that choosing an adequate observation model can be critical. Model selection and simulations show that for the models and parameter values in this study, a suitable observation model can typically be identified if observations are replicated. Model selection and replication of observations, therefore, provide a potential solution when the observation distribution is unknown.

Individual heterogeneity and senescence in silvereyes on Heron Island.

Individual heterogeneity and correlations between life history traits play a fundamental role in life history evolution and population dynamics. Unobserved individual heterogeneity in survival can be a nuisance for estimation of age effects at the individual level by causing bias due to mortality selection. We jointly analyze survival and breeding output from successful breeding attempts in an island population of Silvereyes (Zosterops lateralis chlorocephalus) by fitting models that incorporate age effects and individual heterogeneity via random effects. The number of offspring produced increased with age of parents in their first years of life but then eventually declined with age. A similar pattern was found for the probability of successful breeding. Annual survival declined with age even when individual heterogeneity was not accounted for. The rate of senescence in survival, however, depends on the variance of individual heterogeneity and vice versa; hence, both cannot be simultaneously estimated with precision. Model selection supported individual heterogeneity in breeding performance, but we found no correlation between individual heterogeneity in survival and breeding performance. We argue that individual random effects, unless unambiguously identified, should be treated as statistical nuisance or taken as a starting point in a search for mechanisms rather than given direct biological interpretation.

The importance of individual developmental variation in stage‐structured population models

Population stage structure is fundamental to ecology, and models of this structure have proven useful in many different systems. Many ecological variables other than stage, such as habitat type, site occupancy and metapopulation status are also modelled using transitions among discrete states. Transitions among life stages can be characterised by the distribution of time spent in each stage, including the mean and variance of each stage duration and within-individual correlations among multiple stage durations. Three modelling traditions represent stage durations differently. Matrix models can be derived as a long-run approximation from any distribution of stage durations, but they are often interpreted directly as a Markov model for stage transitions. Statistical stage-duration distribution models accommodate the variation typical of cohort development data, but such realism has rarely been incorporated in population theory or statistical population models. Delay-differential equation models include lags but no variation, except in limited cases. We synthesise these models in one framework and illustrate how individual variation and correlations in development can impact population growth. Furthermore, different development models can yield the same long-term matrix transition rates but different sensitivities and elasticities. Finally, we discuss future directions for estimating realistic stage duration models from data.

Using uncertainty estimates in analyses of population time series

Recording and monitoring wildlife is crucial for the conservation of wild species and the protection of their environment. The most common type of information reported from a monitoring scheme is a time series of population abundance estimates, but the potential of such data for analyzing population dynamics is limited due to lack of information on sampling error. Recent work has shown that replicating the sampling process and analyzing replicates jointly in a dynamical model can considerably increase estimation efficiency compared to analyzing population estimates alone. This method requires that independent replicates are available, and model fitting can be complex in general. Often, however, population estimates are accompanied by standard errors, or standard errors may be estimated from raw data using a sampling model. We evaluate a method where standard errors are used in combination with population estimates to account for sampling variability in state-space models of population dynamics. The method is simple and lends itself readily to data derived from many sampling procedures but ignores uncertainty in the standard errors themselves. We simulate data from a Gaussian state-space model where several observations, which may come from different sites, are available for the population at each time. Fitting the simulated data, we show that the method yields similar or even better results than a method utilizing all observations, even when there are few observations at each time. This holds under a range of simulation settings involving heteroscedastic observation error, site effects, and correlation among observations. We illustrate the approach on real data from the North American Breeding Bird Survey and show that it performs well in comparison to a more difficult maximum-likelihood analysis of the full data under non-Gaussian sampling error.

Fitting complex population models by combining particle filters with Markov chain Monte Carlo

We show how a recent framework combining Markov chain Monte Carlo (MCMC) with particle filters (PFMCMC) may be used to estimate population state-space models. With the purpose of utilizing the strengths of each method, PFMCMC explores hidden states by particle filters, while process and observation parameters are estimated using an MCMC algorithm. PFMCMC is exemplified by analyzing time series data on a red kangaroo (Macropus rufus) population in New South Wales, Australia, using MCMC over model parameters based on an adaptive Metropolis-Hastings algorithm. We fit three population models to these data; a density-dependent logistic diffusion model with environmental variance, an unregulated stochastic exponential growth model, and a random-walk model. Bayes factors and posterior model probabilities show that there is little support for density dependence and that the random-walk model is the most parsimonious model. The particle filter Metropolis-Hastings algorithm is a brute-force method that may be used to fit a range of complex population models. Implementation is straightforward and less involved than standard MCMC for many models, and marginal densities for model selection can be obtained with little additional effort. The cost is mainly computational, resulting in long running times that may be improved by parallelizing the algorithm.

Estimates from non‐replicated population surveys rely on critical assumptions

Summary N-mixture and occupancy models are often used to account for non-detections in population surveys. The consensus has been that the methods require data that are replicated in space, as well as within a short period of time while the population at each site remains closed, in order for parameters such as detection probabilities and expected abundances to be identifiable. The requirement of replication prohibits the use of N-mixture and occupancy models for many surveys in practice. Recently, some studies have argued that N-mixture and occupancy models for surveys with only one visit at each site are identifiable when covariates for both detection probabilities and expected abundances, with at least one distinct covariate for each, are available (Journal of Plant Ecology, 5, 2012, 22; Environmetrics, 23, 2012, 197). We investigate the reasons for why detection probabilities have traditionally been considered unestimable from non-replicated counts and how the new methods sidestep these issues. We further use simulations to investigate properties of the new estimators. We show that detection probabilities of the single-visit models with covariates are non-identifiable and that absolute abundances cannot be estimated when particular link functions are employed (log links for both expected abundance and detection probability). Further, assumptions about the range within which detection probabilities vary are necessary to render estimability. The possibility of estimating abundance from single-visit surveys therefore implicitly hinges on knowledge about the link functions. Simulations show that estimates of abundance can be highly variable and sensitive to the choice of link function. We further show how a reduced parameterization of an N-mixture model for surveys repeated over time, without replication under closure but where detection probabilities are constant over time, corresponds to a Poisson model. Non-robust estimation can result in misleading conclusions about population abundance. When estimating abundance from count data that are not replicated, it is therefore important to be aware of how imprecise estimators may be and how sensitive they are to model assumptions.