An updated perspective on the role of environmental autocorrelation in animal populations

Abstract

Ecological theory predicts that the presence of temporal autocorrelation in environments can considerably affect population extinction risk. However, empirical estimates of autocorrelation values in animal populations have not decoupled intrinsic growth and density feedback processes from environmental autocorrelation. In this study, we first discuss how the autocorrelation present in environmental covariates can be reduced through nonlinear interactions or by interactions with multiple limiting resources. We then estimated the degree of environmental autocorrelation present in the Global Population Dynamics Database using a robust, model-based approach. Our empirical results indicate that time series of animal populations are affected by low levels of environmental autocorrelation, a result consistent with predictions from our theoretical models. Claims supporting the importance of autocorrelated environments have been largely based on indirect empirical measures and theoretical models seldom anchored in realistic assumptions. It is likely that a more nuanced understanding of the effects of autocorrelated environments is necessary to reconcile our conclusions with previous theory. We anticipate that our findings and other recent results will lead to improvements in understanding how to incorporate fluctuating environments into population risk assessments.

Appendices

Appendix A: Curating the GPDD

We chose a high quality subset of the over 5000 GPDD datasets using the following criteria: there must have been at least 15 observations in the time series, the qualitative GPDD reliability rating must have been 4 or 5 (out of a maximum rating of 5), and the data must not have been constant over the first 3 years. In addition, we only allowed datasets where sampling units indicated nonharvest indices, as harvests may not reflect true population abundances. All data were transformed by adding 1 to all observations, in order to remove any 0’s. We tested the effects of this data transformation by analyzing a subset of the 445 datasets where this transformation was not applied, but fewer datasets (166) were available for this analysis. The MainID’s of the GPDD datasets used in our analysis were 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 56 57 58 59 60 61 62 63 64 65 66 67 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1211 1213 1248 1250 1254 1261 1268 1272 1275 1276 1347 2770 2771 2772 2773 2774 2775 2776 2777 2778 2779 2780 2781 2787 2789 2791 2798 2800 2802 2805 2810 2815 2829 2833 2840 2842 2844 2852 2857 2867 2869 2879 2887 2891 2901 2903 2905 2915 2917 2922 2926 2935 2937 2939 2958 2960 2962 2974 2976 2986 2991 3001 3003 3017 3021 3028 3030 3037 3043 3045 3051 3056 3059 3068 3070 3072 3084 3092 3108 3110 3112 3118 3123 3127 3133 3135 3139 3144 3157 3159 3161 3170 3177 3182 3190 3214 3216 3218 3223 3225 3233 3237 3249 3251 3253 3260 3263 3265 3268 3275 3283 3295 3297 3306 3312 3316 3323 3330 3356 3358 3360 3372 3378 3382 3393 3395 3406 3426 3428 3430 3437 3442 3445 3455 3466 3468 3470 3477 3480 3482 3484 3489 3496 3508 3521 3533 3535 3539 3546 3557 3559 3567 3575 3577 3583 3585 3592 3623 3625 3627 3639 3641 3650 3664 3666 3673 3676 3678 3680 3683 3688 3693 3706 3708 3716 3718 3723 3726 3739 3741 3757 3774 3776 3784 3787 3795 3797 3799 3811 3814 3825 3827 3829 3838 3840 3849 3853 3864 3866 3876 3882 5019 5020 6057 6104 6105 6106 6107 6108 6109 6110 6111 6112 6113 6114 6115 6116 6117 6118 6119 6120 6121 6122 6123 6124 6125 6126 6127 6128 6129 6130 6131 6132 6133 6134 6135 6136 6137 6138 6139 6140 6141 6142 6143 6592 6728 6729 6730 6731 6732 6733 6734 6735 6736 6737 6738 6739 6740 6741 6742 6743 6744 6745 6788 6789 6790 6791 6792 6793 6794 6795 6796 6797 6798 6799 6800 6801 6802 6803 6804 6805 6806 6807 6808 6809 6810 6811 6812 6813 6814 6815 6816 6817 6818 6819 6820 6821 6822 6823 6824 6825 6826 6827 6828 6829 6830 6831 6832 6833 6834 6835 6836 6837 6838 6839 6840 6841 6842 6843 6844 6845 6846 6847 6848 6849 6850 6851 6852 6853 6854 6855 6856 6857 6858 6859 6860 6861 6862 7065 9191 9192 9193 9245 9246 9247 9248 9249 9279 9280 9302 9338 9339 9340 9341 9342 9343 9344 9345 9346 9347 9348 9365 9391 9393 9425 9517 9518 9519 9536 9542 9684 9833 9834 9911

Appendix B: Parameter estimation and model selection

Parameter estimation

Parameter estimation was performed using both maximum likelihood (ML) and restricted maximum likelihood (ReML) methods. We used one-step predictions to fit population dynamics models with PEA using abundance data from the GPDD. AICc was used to perform model selection from among the candidate models using the model’s ML estimates. However, our reported values of $\hat {\phi }$ used ReML parameter estimates from the best AICc models. ReML estimation has been shown to perform better than traditional ML when estimating variance components, but is not valid for model selection using information criterion (Staples et al. 2004). ML and ReML estimations were performed using the generalized least squares (gls) function from the NLME package in the R software environment (Pinheiro et al. 2011; R Development Core Team 2012). We assumed a multivariate normal distribution for the observed per capita growth rate, $\ln \left (\frac {N(t)}{N(t-1)}\right )$, where the mean vector was given by the predicted per capita growth rate for the corresponding form of density dependence. The covariance matrix was given by σ ² R, where the correlation matrix, R, was given by the correlation structure for the appropriate ARMA(1,1) model used. For k observations of the per capita growth rate, the ARMA(1,1) correlation structure with AR(1) parameter ϕ and MA(1) parameter θ is given by

$$ \mathbf{R} = \frac{(1 + \theta\phi)(\theta + \phi)}{1 + 2\theta\phi + \theta^{2}}\left[ \begin{array}{llll} 1 & \phi^{0} & {\ldots} & \phi^{k-2}\\ \phi^{0} & 1 & {\ldots} & \phi^{k-3}\\ {\vdots} & {\vdots} & {\ddots} & \vdots\\ \phi^{k-2} & \phi^{k-3} & {\ldots} & 1 \end{array} \right], $$

(B1)

where the AR(1) model corresponds to a special case where θ = 0. For example, the log-likelihood for the Gompertz lag 1 density dependence model was then given by,

$$\begin{array}{@{}rcl@{}} \ln(L) &=& -\frac{k \ln(2\pi)}{2} - \frac{k \ln\left| \sigma^{2}\mathbf{R} \right|}{2} - \\ &&\frac{1}{2\sigma^{2}}\left( \ln \left( \frac{N(t)}{N(t-1)}\right) - a + b \ln(N(t-1)) \right)^{T}\\ &&\times \mathbf{R}^{-1} \left( \ln \left( \frac{N(t)}{N(t-1)}\right) - a + b \ln\left( N(t-1)\right) \right).\\ \end{array} $$

(B2)

Appendix C: Covariate analysis

We tested the explanatory power of a suite of covariates, available in the GPDD, on $\hat {\phi }$. We explored whether environmental factors, species information, and data quality metrics could explain variation in $\hat {\phi }$. Specifically, we tested whether time series length, latitude, and longitude of the data collection location, GPDD reliability index, and sampling frequency were significant predictors of $\hat {\phi }$ at the α = 0.10 level. We ran an additional regression looking at species taxa as predictors of $\hat {\phi }$ at the α = 0.10 level. We also tested whether species status as specialists or generalists could explain variation in $\hat {\phi }$. We used definitions from Murdoch et al. (2002) to determine specialist/generalist status in 48 time of the GPDD series. We predicted that generalists, with potentially more trophic interactions would have lower levels of PEA than specialists who may have fewer, but stronger, trophic interactions consistent with our results from the SLLM model in the main text.

Results

Our exploratory regression analysis on $\hat {\phi }$ was used to determine whether covariates could explain results from our analysis. We used the arctanh transformation on $\hat {\phi }$, defined by $Z=\text {arctan} (\hat {\phi })$, in order to satisfy the assumption of normally distributed residuals. We performed a test on all sampling units reported in the GPDD. We found that only the Breeding females category was a statistically significantly predictor (t test, p value = 0.0293), though when using the Bonferonni correction for multiple tests this is no longer a significant predictor. The two species in this category were Tringa nebularia and Melospiza melodia. We also examined whether sample size and GPDD reliability index influenced estimates. We found that $\hat {\phi }$ tended to decrease with increases in sample size and the reliability index, though these effects were not statistically significant.

Status as specialist ($\hat {\phi }=0.11$) or generalist ($\hat {\phi }=0.03$) was not found to be a significant explanatory value of $\hat {\phi }$ at the α = 0.1 level (p value = 0.44). However, the results were consistent with our predictions with the specialist having slightly higher mean levels of autocorrelation, consistent with predictions from the SLLM.

Appendix D: Alternative error models

Past work has formulated autocorrelation models several different ways. Here, we discuss the differences in two alternative model formulations of temporally autocorrelated processes and contrast their properties to methods used by past studies examining the TPA.

A potentially important distinction in autocorrelated time series models is between long- and short-memory processes. Short-memory processes have autocorrelation functions that decay exponentially to 0 as k→∞ (e.g., exp(−ϕ k)), while long-memory processes have autocorrelation functions that converge to 0 according to slow-decaying power-law functions as k→∞ (e.g., k ^−β) (Shumway and Stoffer 2006). Thus, though both processes may have the same degree of autocorrelation and go to 0, short-memory processes go to 0 faster than long-memory models. Short-memory models depend only on recent realizations of the process while long-memory models exhibit autocorrelation over many past realizations of the process. The relevance of this distinction is that incorporating PEA in long- and short-term memory models with the same degree of autocorrelation can lead to different predictions about species persistence in otherwise identical models (Cuddington and Yodzis 1999; Fowler and Ruokolainen 2013). These findings suggest that distinguishing between long- and short-memory PEA may have practical implications for population modeling.

While it is often convenient to think of population abundances as functions of time, previous discussions of long-memory processes in the ecological literature have often used the frequency domain representation of time series (Halley 1996; Cuddington and Yodzis 1999; Vasseur and Yodzis 2004). This approach decomposes a time dependent signal into an infinite sum of sine waves with different frequencies through the Fourier transform. The transformation calculates the amplitude for each sine wave, giving the relative contribution of that frequency to the original series. This frequency representation can then be used as a convenient diagnostic tool when determining the appropriateness of a particular time dependent model (Box et al. 2011).

The power spectral density function, denoted as S(f), gives the distribution of the frequencies f present in a time series. The most commonly used long-memory process is the inverse power law model (1/f model) (Johnson 1925). For this model, the spectral density function is given by

$$ S(f)=\text{constant} /f^{\beta}, $$

(D1)

where β controls the degree of autocorrelation. The 1/f model has also been proposed several times as a general model of environmental variation for population dynamics due to its apparent ubiquity in natural phenomena (Montroll and Shlesinger 1982; Bak et al. 1987; Halley 1996). The mathematical representation of the autocorrelation in a time series model and the function S(f) are transforms of each other and thus are mathematically equivalent.

Long-memory models are typically investigated within the framework of frequency domain methods while short memory processes are usually studied using traditional time domain tools, which tend to be more convenient for model estimation. We used the fractionally differenced model (FDM) to model time series with long-memory. In the FDM, the current state of the autocorrelated process is given as the sum of all the past contributions to the process and an independent random shock (Hosking 1981). Autocorrelation is controlled by a single coefficient, d, which determines the relative contributions of all past states. The mathematical representation of the model is

$$ E(t) = \sum\limits_{k=1}^{\infty} \frac{-{\Gamma}(k-d)}{\Gamma(k+1){\Gamma}(-d)}E(t-k)+ W(t), $$

(D2)

where E(t) is the state of a process at time t, Γ(x) is a gamma function, d is the fractional differencing parameter where −0.5<d<0.5, and W(t) is a normal distribution with mean equal to zero and variance of σ ². In this model, the state at the current time step depends on all previous time lags through an infinite series representation, leading to the long-memory property. The differencing degree d controls the degree of autocorrelation and corresponds approximately to β/2 in the 1/f model (Hosking 1981). Thus, the FDM and the 1/f model are approximately equivalent for −1<β<1.

Because previous methods have often used frequency domain approaches, we determined whether estimates of the total population autocorrelation (TPA) made to the GPDD dataset using an AR(1) model would be different from previous studies that were based on the estimated spectral exponent, a long-memory model used in several past studies of TPA (Pimm and Redfearn 1988; Inchausti and Halley 2001). We were also interested in whether the FDM model would be more consistent with the long-memory process that is spectral exponent than the AR(1) model. We first estimated the spectral exponent by fitting a linear regression to the Fourier transform of the log-transformed abundances in our GPDD dataset consistent with previous methods. The transform was calculated with the fft function in the R software environment (R Development Core Team 2012). The frequency amplitude of the transformed time series, S(f), was then fit to a model of the form, ln(S(f))=a−βln(f, where f is the observed sampling frequency of the time series signal, β is the spectral exponent, and a is an intercept term. We also estimated the FDM and AR(1) coefficients in Eqs. D2 and 3 using the arfima package in R (Veenstra 2012) from the log-transformed abundance time series in order to compare them to $\hat {\beta }$.

We evaluated the consistency of the AR(1) model and the FDM with $\hat {\beta }$ by regressing each models autocorrelation estimates (ϕ in Eq. 3 and d in Eq. D2) against $\hat {\beta }$ using standard major axis linear regression due to the higher variance in estimates of β. For the AR(1) model, we fit, $\hat {\beta }/2 = a_{\phi } + b_{\phi } \hat {\phi } + W$, and for the FDM model we fit $\hat {\beta }/2 = a_{d} + b_{d} \hat {d} + W$, where W was a random variable assumed to be iid normally distributed. Consistency with the 1/f ^β model was determined by whether the confidence intervals of the intercept parameters (a) contained 0 and the intervals of the slope parameters (b) contained 1. This comparative method was used rather than directly performing model selection on AR(1) and FDM models of PEA due to the inability for information theoretic methods to reliably distinguish between long- and short-memory models (Wagenmakers et al. 2004). We also determined if the properties of the data sets used here were consistent with previous studies on the GPDD by comparing our estimates of the TPA in our GPDD datasets using the 1/f model to previous estimates of TPA in the GPDD (e.g., Inchausti and Halley 2001).

Results

We found levels of the spectral exponent with our dataset, $\hat {\beta }=0.94$ that were similar to previous estimates of $\hat {\beta }=1.02$ (Inchausti and Halley 2002), indicating that the data set used here is comparable to previous work on the GPDD. We also compared autocorrelation estimates from both the AR(1) and FDM error processes to $\hat {\beta }$ in order to determine model consistency with previous estimates from the GPDD. In the comparison of the AR(1) and to 1/f noise, we did not include estimates that indicated nonstationary time series, which led to 387 data sets in the analysis. In the comparison of the FDM model and 1/f model, this same criterion led to 244 data sets being included in the analysis. For the AR(1) model, we found an intercept value of $\hat {a}_{\phi }=-0.0.072$, 95 % CI =(−0.101,−0.045), and slope of $\hat {b}_{\phi }=1.18$, 95 % CI = (1.109, 1.26). For the FDM model, we found an intercept value of $\hat {a}_{d}=0.075$, 95 % CI = (0.069, 0.081), and slope of $\hat {b}_{d}=0.086$, 95 % CI =(0.78,0.95). These results indicate that there may be bias in both models with regards to comparisons with the spectral exponent. However, the AR(1) model does make estimates that are broadly consistent with the spectral exponent, an interesting result. Despite the important differences in the properties of the spectral density of the AR(1) and the spectral exponent, our results suggest that parameter estimation using the AR(1) model is comparable to estimates that would be obtained with a long-memory process population environmental autocorrelation (PEA) model.

Appendix E: PEA in the carrying capacity

We performed a simulation analysis in order to determine if our analysis would provide biased estimates when the PEA occurs in the carrying capacity rather the maximum per capita rate of increase. The carrying capacity in the lag 1 Ricker model, given by K = a/b in Eq. 2, was assumed to be an AR(1) random variable. We set a = ln(1.5) and E[K]=100 for all simulations. We tested 15 different autocorrelation levels evenly spaced in the interval [−0.95, 0.95] over three different levels of variability in K, where variation was measured used the coefficient of variation (standard deviation of K divided by the mean). The CV values we tested were 0.5, 1, and 2. For values of the CV higher than 2, we found that some values of our simulated K became negative and were therefore unrealistic. For each combination of autocorrelation and CV value, we estimated the autocorrelation from a time series with length of 10⁵ following the methods of the main text that assume that the variation occurs in the intrinsic growth rate.

We found that estimation error was negligible at the low CV levels of 0.5 and 1 (Fig. 6). At CV =2, there was estimation bias at very low negative levels of the autocorrelation but the bias was low for positive levels of autocorrelation. Overall, we found that the influence of this kind of model specification to be low, suggesting that our results are not biased by the way that the environmental variation enters into the population dynamics. These results are not too surprising as variation in the carrying capacity is often difficult to distinguish from variation in the maximum per capita rate of increase with differences emerging primarily at very low or very high abundances (Ferguson, unpublished results).

Fig. 6

The impact on estimated PEA when the PEA arises in the population carrying capacity but the error model is in the intrinsic rate of reproduction. The degree of estimation error increases as a function of the coefficient of variation (CV) in the carrying capacity and is larger for negative values of the PEA

Appendix F: Spectral mimicry

In the main text, we used the known distributional form of stationary AR time series to ensure that the simulated time series had identical properties except for the degree of autocorrelation. Although this allowed us to control all the statistical moments, alternative methods, such as spectral mimicry, exist that allow the generation of time series using the same realization of the generated data with only the degree of autocorrelation changing. We used this method to check if our approach properly accounted for changes in statistical properties with autocorrelation.

Spectral mimicry (Cohen et al. 1999) can vary the autocorrelation in the same realization of a stochastic process, thereby controlling for the effects of not scaling of process variance with autocorrelation or other unintended mismatches that may arise between simulated series. The method works by first generating a series from a stochastic model such as a standard normal distribution. In order to turn this iid stochastic process into an autocorrelated one, a new reference series with the desired degree of autocorrelation is generated. The original process is then reordered such that it’s sequence of order statistics matches the reference series. This reordered version has the same realized values as the original process, however, the autocorrelation is equal to the reference series. Because the new series has the same values as the original series, just in a different order, the statistical properties of the ensemble must be equivalent.

We reproduced Figs. 2 and 3 using datasets generated with spectral mimicry in order to confirm that our method of generating autocorrelated data did not lead to any differences between time series other than the degree of autocorrelation. The reproduced figures (Figs. 7 and 8) show no differences with the figures in the main text (Figs. 2 and 3).

Fig. 7

The effect of two different nonlinear interactions on an autocorrelated time series, C(t), when data is generated using the spectral mimicry method. The dashed black line is the autocorrelation in the untransformed environmental covariate, while the colored lines are the population environmental autocorrelation (PEA) values in E(t) for different transformations. Panel a corresponds to power transforms of the form E(t)=−C(t)^η. Panel b corresponds to the logistic function, $E(t) = \frac {e^{\zeta C(t)}}{1 + e^{\zeta C(t)}}$

Fig. 8

Two applications of the SLLM to a number, n, of autocorrelated time-series. In both panels the x-axis denotes the autocorrelation in the environmental covariate while y-axis is the population environmental autocorrelation (PEA) that affects the per capita growth rate once the SLLM has been applied. In panel a, all n covariates have the same autocorrelation. In panel b, n−1 of the covariates have an autocorrelation of 0.5, while the remaining covariate has the autocorrelation value given on the x-axis. In both panels, the dashed line denotes the covariate’s autocorrelation, for comparison. The overall effect of the SLLM depends on the number of environmental covariates, n, that limit the population. In general, the SLLM tends to reduce the magnitude of the PEA relative to what is expected from a single limiting factor. However, as shown in the right panel, the effect can increase the PEA if a single limiting environmental factor has a high enough autocorrelation