Data set

We compiled a set of high-quality, long-term population dynamics time-series data for 1,198 species spanning a wide range of taxa, biomes, and life histories. These time-series data were obtained from various online, text and primary sources, but primarily from the Global Population Dynamics Database (GPDD) which provides data for > 5000 populations and > 1400 species. Other sources were used where the data were either superior to, or were unavailable from the GPDD. Inherent differences between the many data sources and ambiguities and inconsistencies within the GPDD lead us to develop a strict set of filtering criteria for the objective removal of poor-quality time-series data ( Brook et al. 2005 ). This objective standardization protocol resulted in one time series for each species (avoiding overrepresentation of a small number of well-studied taxa) (c.f., Woiwod and Hanski 1992, Sibly et al. 2005 ) ensuring a minimum quality and 8 time-step transitions (at least 10 censuses are recommended to parameterize density-independent models) ( Morris and Doak 2002 ). The final data set comprised 639 invertebrate, 529 vertebrate and 30 plant species, and had a mean duration of 22 year-to-year transitions.

Population Dynamics Models and Fitting

There are many potential mathematical simplifications of complex population dynamics; however, for simplicity and generality we used an a priori model-building strategy to arrive at a set of five population dynamics models commonly used to describe phenomenological time-series data ( Saether et al. 2002, Turchin 2003, Sibly et al. 2005 ). The model set was based on variants of the generalized -logistic population growth model:

where N_t = population size at time t, r = realized population growth rate, r_m = maximal intrinsic population growth rate, K = carrying capacity, and θ permits a nonlinear relationship between rate of increase and abundance. The term _t has a mean of zero and a variance (σ²) that reflects environmental variability in r.

            All models below were fitted assuming process error, and hence initial population size was not estimated as a separate parameter. Density-independent model variants used were (1) non-directional population fluctuations with a normally distributed error term (random walk) ( Foley 1994, Shenk et al. 1998 ) where r_m = 0 with a single parameter estimated: σ; and (2) the standard geometric Malthusian growth model ( May 1975 ) with a normally distributed error term (exponential drift; θ = -, r_m and σ estimated). Density-dependent model variants used were (3) a stochastic form of the Ricker-logistic model ( Dennis and Taper 1994 ), which is a special case of the autoregressive model of Ellner and Turchin ( 1995 ) with r_m, K, θ = 1, and σ; (4) the stochastic Gompertz-logistic model where density dependence is proportional to the log of abundance ( Reddingius 1971, Pollard et al. 1987 ), with r_m, log_e[K], log_e[N_t], θ = 1, and σ; and (5) the generalized θ-logistic growth model ( Gilpin and Ayala 1973 ) with r_m, K, θ, and σ.

            For each species, we used maximum-likelihood estimation to fit model parameters (via linear regression for random walk, exponential, Ricker-logistic and Gompertz-logistic, and non-linear regression using Newton optimization for the -logistic model). An example of the fitting process using time series from a mammal (wild boar – Sus scrofus) ( Jedrzejewska et al. 1997 ) is shown in Fig. A1.

Of the population dynamical models that can be fitted to count data, we chose a suite of models that covered an appropriate range of complexity and assumptions. However, many other possible models exist. For instances, the Beverton-Holt model ( Beverton and Holt 1993 ) is commonly used for fish populations, but this three-parameter model is approximated by the -logistic model and so was not considered here. However, our chosen set clearly ignores delayed density-dependent processes (i.e., lags), which may be a common feature of phenomenological data ( Turchin 1990, Turchin and Taylor 1992, Solow 2001, Turchin 2003 ). Therefore, in a separate evaluation we used a model set which included random walk, direct Gompertz-logistic density dependence, delayed Gompertz-logistic density dependence and the combination of the latter two models.

Statistical Analyses

Information theory using Kullback-Leibler information was used to assign relative strengths of evidence (AIC_c weights) to each model described above ( Burnham and Anderson 2002 ), permitting full multi-model inference. For Neyman-Pearson hypothesis testing, we chose three classic tests: Bulmer’s ( 1974 ) R, Pollard et al.’s ( 1987 ) randomization (RAN), and Dennis and Taper’s ( 1994 ) parametric bootstrap likelihood-ratio (PBLR). For model selection, we used: (1) the Bayesian Information Criterion (BIC), a dimension-consistent index of parsimony which is determined by providing an asymptotic correction on the log-likelihood based on sample size and the number of fitted parameters ( Schwarz 1978, Zeng et al. 1998 ), and (2) Turchin’s ( 2003 ) cross-validation (C-V), a numerically intensive, jack-knife analog of model selection which generates a response surface and applies sequential-blocks cross-validation to define model complexity ( Turchin 1996 ), incorporating direct and delayed density-dependent models by implicitly assuming the exponential model as a starting point and removing data sequentially (with replacement) to determine the ability of the fitted model to fit the missing datum.

Evaluating Potential Caveats

A possible prejudice when using the model-selection approach in this instance is the potential biased loading of models in an a priori set toward a particular prediction. We chose to model only two density-independent models (random walk and exponential) and three density-dependent models (Ricker-logistic, Gompertz-logistic and -logistic). Theory argues that the strength of evidence for a specific hypothesis encapsulated by multiple models should not change relative to the number of models evaluated because weights are adjusted relative to all candidate models in the set ( Burnham and Anderson 2002 ). Nonetheless, to provide a direct empirical test of this assumption, we evaluated two separate, dichotomous comparisons where only one density-independent (H₀) and one density-dependent (H_A) models were represented. These included the information-theoretic equivalent of Pollard’s randomization test ( Pollard et al. 1987 ) which compares support for the random walk (density-independent) model to the Gompertz-logistic (density-dependent) model, and the two models evaluated in Dennis and Taper’s ( 1994 ) parametric bootstrap likelihood-ratio test, which compares the exponential (density-independent) model with the Ricker-logistic (density-dependent) model (although the latter can also use a stochastic Gompertz-logistic model).

Simulation approaches have been used extensively to evaluate how well methods such as AIC and BIC model selection perform in detecting density dependence in time-series data. However, simulations will inherently favor the BIC approach because the true model is known and almost universally included in the model set ( Anderson and Burnham 1999 ). Thus, testing the ability of model-selection criteria to select the model that generated a set of simulated data often ends up asking the wrong question ( Fox and Ridsdillsmith 1995 ) – we should instead ask what are the best approximating models for real ecological data sets ( Burnham and Anderson 2002 ). Simulations may not be appropriate for determining the presence of a phenomenon such as density dependence when it is embedded in a complex ecological reality ( Fox and Ridsdill-Smith 1996 ).

The implicit assumption for tests of density dependence using time series is that populations under evaluation are not age structured and have non-overlapping generations ( den Boer and Reddingius 1989 ). However, indices of population abundance can behave as if governed by first-order difference equations ( Dennis and Taper 1994 ), so the use of time-series data for even long-lived taxa is reasonable. The detection of density dependence in time-series data is also influenced by sample size (number of time steps) ( Solow and Steele 1990 ). This has resulted in claims that particular data sets are more or less amenable to the detection of density dependence based on the number of time steps monitored ( Dennis and Taper 1994 ). Further, populations undergoing monotonic decline or increase due to deterministic perturbations may mask density-dependent processes because equilibria are never approached. We examined these issues by comparing the cumulative AIC_c weights for the density-dependent models with the number of time steps monitored (q) for each species, and by examining the per-generational rate of population change relative to the number of generations monitored. Most species had inadequate data describing generation length, so age at first breeding was used as an index of this parameter ( Brook et al. 2005 ).

The variance in population rate of change should be directly proportional to the probability of detecting density dependence because populations that fluctuate greatly are more likely to have gone through the full range of population densities between quasi-extinction and carrying capacity ( Dennis and Taper 1994, Zeng et al. 1998 ). We used least-squares regression to compare the complementary log-log transformation of the cumulative AIC_c weights for the density-dependent models to the log variance of r (logged because of the large right skew).

Error in abundance estimates can inflate the probability of detecting density dependence in time-series data because the random variation due to sampling effort may spuriously emulate regulatory patterns ( Shenk et al. 1998 ). To examine the influence of sampling error on our results, we queried our data set for the highest-quality data and found that of the 1,198 species’ time series, 83 (all vertebrates) represented relatively high-precision, direct-count data (e.g., mark-recapture estimates of abundance, entire colony counts) in contrast to indirect estimates of abundance (e.g., catch per unit effort, harvest indices). Assuming that the direct-count data were indeed more precise (i.e., indirect estimates of abundance also incorporate the error associated with the technique’s capacity to provide an index of population abundance), we also compared the results of different methods used to evaluate density dependence for these 83 species relative to the remaining 446 vertebrates to measure the potential impact of sampling error.

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