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Keywords:

  • Common Birds Census;
  • density dependence;
  • environmental stochasticity;
  • Markov chain Monte Carlo methods;
  • observation error;
  • population prediction interval;
  • stochastic population dynamics

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Models
  5. Methods
  6. Results
  7. Discussion
  8. Concluding remarks
  9. Acknowledgements
  10. References
  11. Supporting Information

1.  One of the greatest challenges in ecology is to develop tools that can give reliable projections of future population fluctuations as well as to quantity uncertainties. The population prediction interval (PPI), i.e. the stochastic interval that includes a given population size with a certain probability, is affected by changes in expected population size, e.g. due to density regulation, fluctuations in population size because of demographic and environmental stochasticity, uncertainties and biases in parameter estimates, and observation error in estimates of population size.

2.  The aim of this study was to examine how PPI can be used to obtain reliable projections of future population fluctuations. Our approach is to split long time series into two parts: the first part is used for parameter estimation and the second part is used for comparing population predictions with actual population sizes after a certain period of time.

3.  Here we use the Common Birds Census – data from the UK for several species of passerines. Unbiased predictions will give a uniform distribution of the recorded population sizes across the PPI when transformed to a scale defined by the quantiles of the PPI. However, deviations from a uniform distribution reveal biases in the predictions. For instance, if there is a predominance of recorded population sizes in the upper quantiles of the PPI, this shows that our predictions underestimate future population sizes.

4.  Unbiased predictions required models that included both partitioning of stochastic influences into demographic and environmental stochasticity as well as observation error.

5.  Precision in the population predictions was improved when including observation error as well as density dependence.

6.  We recommend that predictions of future fluctuations of small passerine populations are based on models that include density dependence as well as observation error and estimates of the demographic variance that is obtained from individual-based demographic data or based on species-specific life-history characteristics.

7.  These results show that constructing PPI by stochastic simulations may be a useful tool for obtaining reliable population projections.


Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Models
  5. Methods
  6. Results
  7. Discussion
  8. Concluding remarks
  9. Acknowledgements
  10. References
  11. Supporting Information

Developing ecology as a quantitative science requires tools that can be used to describe and forecast population fluctuations in a reliable way. This problem has received particularly large attention in population viability analyses, in which the central focus is to predict extinction or quasi-extinction of rare or endangered species (Beissinger & McCullough 2002; Morris & Doak 2002). Ideally, population predictions should be based on models that at each age- or stage-class describe the effects on the population growth of density dependence as well as stochastic effects arising from demographic and environmental stochasticity (Lande, Engen & Sæther 2003). In most cases, such an approach will not be feasible because this will require access to long-term individual-based demographic data, which are available only for a few natural populations. Even in presence of such high-quality data, predictions may be uncertain due to high uncertainty in the large number of parameters necessary to parameterize such complex models (Lande et al. 2006).

Reliable population projections require that several conditions are fulfilled (Holmes 2004). First, the expected changes in dynamics must be correctly modelled. In most cases, this will require unbiased estimates of the specific growth rate at small population size (Taylor 1995; Fieberg & Ellner 2000, 2001; Morris & Doak 2002), and that the carrying capacity K and the form of density regulation are correctly estimated in populations subject to density dependence. Especially in age-structured populations this represents a considerable challenge (Fieberg & Ellner 2001; Wilcox & Possingham 2002; Lande et al. 2006). Second, the stochastic influences on the population dynamics must be correctly estimated and modelled. Environmental stochasticity will in general influence the dynamics of even large populations and hence strongly affect the accuracy of any population projection (Ludwig 1999). In addition, demographic stochasticity, i.e. random variation among individuals in annual fitness (Lande et al. 2003), will provide huge contributions to the dynamics at smaller population sizes (Lande 1998). Third, large uncertainty in parameter estimates will reduce our ability to precisely predict future population fluctuations (Ellner & Fieberg 2003; Holmes et al. 2007; Ellner & Holmes 2008). Fourth, errors in estimating population size will result in biased estimates of critical parameters such as the strength of density dependence and environmental variances (Bulmer 1975; Freckleton et al. 2006). Such effects of sampling errors in the population estimates must be accounted for (e.g. De Valpine & Hastings 2002; Buckland et al. 2004; Clark & Bjørnstad 2004; Clark 2005; Dennis et al. 2006; Lele 2006; Lillegård et al. 2008) to obtain unbiased population projections.

We have previously proposed (Engen & Sæther 2000; Sæther et al. 2000; Engen, Sæther & Møller 2001; Sæther & Engen 2002a) that the concept of population prediction interval (PPI) can be useful for making predictions of future population sizes. A PPI is defined as the stochastic interval that includes a given population size with probability (1 − α), so that α is the probability that the variable we want to predict is not contained in the stochastic interval. The width of the PPI (Fig. 1) is affected by changes in expected population size, e.g. due to density regulation, fluctuations in population size because of demographic and environmental stochasticity and uncertainties in parameter estimates and biases in estimates of population size (Sæther et al. 2007b) . Thus, the PPI can be used to identify parameters that influence the precision of future population projections (Sæther et al. 2000, 2002a, b, 2007b).

image

Figure 1.  Population prediction interval (PPI) for the population of Blackbird showing relatively stationary fluctuations in plot 790 (a) and for a declining population of Blackbird in plot 894 (b). We used the time series for the first 10 years (black circles and solid line) to estimate the parameters in (a) a logistic model of density regulation (eqn 2) and (b) a density-independent model (eqn 1) in plot 790 and 894, respectively. Strippled lines show quantiles of the PPIs based on estimated parameters and the last population sizes in the part of time series used for estimating parameters (see Methods). The actual population sizes in the years covered by the PPI are shown with open circles and solid lines. The dotted line shows the quantile in which the recorded population size was located after 5 years.

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How can we validate the accuracy of the PPI? Let us assume that our model in the probabilistic sense makes correct predictions of future population fluctuations. At a given time step, the population trajectories will then be uniformly distributed across the PPI (Fig. 2), i.e. the proportion of population sizes falling below the α-quantile of the PPI equals α. Assuming no age-structure effects, deviations from a uniform distribution as depicted in Fig. 2 will then reveal biases in the predictions. If there is a predominance of population estimates in the upper quantiles of the PPI, this shows that our predictions underestimate future population sizes. This occurs if we have estimated too small population growth rates in a density-independent model or if we underestimate the carrying capacity in a model with density dependence. In contrast, skewed distributions of the recorded population sizes towards smaller quantiles indicate an overestimation of future population sizes, which may be due to overestimates of specific population growth rates or, in populations subject to density regulation, estimates of too large carrying capacities. Similarly, if the distribution resembles a normal distribution, this shows that we have overestimated stochastic influences on the population dynamics, caused either by demographic or environmental stochasticity. A similar effect will also appear if the observation error is underestimated because this will result in an overestimate of the stochastic components (Lillegård et al. 2008). Finally, in populations subject to density regulation, we may also encounter a dome-shaped downward distribution, which indicates that we have overestimated the strength of density regulation or underestimated the stochastic components resulting in a too narrow PPI.

image

Figure 2.  Different hypothetical forms of distributions of quantiles (Fig. 1) of the population prediction interval (PPI) at which recorded population sizes are located after a period of t years. If our modelling gives a reliable prediction of future population sizes, we would expect a uniform distribution of the quantiles (I). An overrepresentation of the distributions of recorded population sizes towards lower quantiles (II) indicates over-estimation of future population sizes (e.g. due to overestimates of the population growth rate). In contrast, a skewed distribution towards higher quantiles (III) indicates underestimation of future population sizes. An upward parabolic distribution indicates overestimation of uncertainty in future population sizes (IV), whereas a downward parabolic distribution with few intermediate quantiles (V) indicates underestimation of uncertainty in future population sizes, e.g. because of overestimation of the strength of density dependence. For further explanation, see text.

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Reliable predictions of future population fluctuations are not only influenced by bias in the PPI, as depicted in Fig. 2, but are also affected by the precision of the projections. Following Sæther et al. (2007b), the width of the PPI will indicate how precisely we can predict future population sizes at a given time step (Fig. 1). A narrow PPI shows that we are likely to predict future population size quite precisely within a small range of population sizes. We are then facing a classical problem in statistics, which is to identify the minimum number of critical parameters necessary for obtaining reliable predictions (Burnham & Anderson 2002). In general, bias decreases and variance increases with the dimension of the model, and lead to the formulation by Box & Jenkins (1970, p. 17) of the famous principle of parsimony that should result in a model with the smallest number of parameters for adequately representing the data. A classical trade-off in model selection can then appear: complex models involving many parameters may be needed to remove the bias, but such complex models will have a small predictive power. Thus, adding more complexity to models due to adding more parameters could lead to less bias but lower precision in the PPI.

In this study, we will use the Common Birds Census (CBC) data from the UK to estimate parameters in different population models from the first 10 years of data. We then use these parameters to construct PPI 5 years ahead in time. For each time series, we then find the quantile in which the actual population size is located. Because this data set contains large numbers of time series, we can then use these distributions of quantiles to examine whether any of the population models gives unbiased predictions (pattern I in Fig. 2) of future population sizes. Furthermore, we can also use this approach to examine the accuracy of the predictions by fitting the beta distribution to the set of observed relative ranks to obtain a uniform distribution at the transformed scale, which enables us to compare the precision in the predictions of different models.

Finally, it is well known that the observer error may bias estimates of most population parameters (Bulmer 1975; Freckleton et al. 2006). Several methods have been employed to account for uncertainties in population estimates when estimating parameters from time series of population fluctuations (De Valpine & Hastings 2002; Dennis et al. 2006). Here we will compare precision and bias in the PPI with and without using a Bayesian Markov chain Monte Carlo (MCMC) approach to account for an observation error.

Models

  1. Top of page
  2. Summary
  3. Introduction
  4. Models
  5. Methods
  6. Results
  7. Discussion
  8. Concluding remarks
  9. Acknowledgements
  10. References
  11. Supporting Information

First, we considered a population model without density regulation, following Engen, Bakke & Islam (1998) and Lande et al. (2003). Writing Nt for the population size at time t the change in population size from year t to t + 1 ΔNt is

  • image

and

  • image

Here r is the specific population growth rate, inline image is the demographic variance and inline image is the environmental variance. The first-order approximation of the mean and variance in ΔXt = Xt+1 − Xt = ln Nt+1 − ln Nt is then:

  • image( eqn\,1a)

and

  • image( eqn\,1b)

Previous analyses of avian population dynamics (Sæther & Engen 2002b) have shown that the logistic model of density regulation (May 1981) may be an appropriate model to describe the density regulation of passerines. Thus, the first-order approximation of the mean and variance for a logistic type of model is then:

  • image( eqn\,2a)

and

  • image( eqn\, 2b)

where β = r/K describes the density regulation. Hence, the stochastic population growth rate at very low population sizes or in cases of no density dependence is inline image, where inline image is the stochastic growth rate at large population sizes. For further details, see Lande et al. (2003, pp. 18–20, 31–33).

The relationship between the observed and the actual number of individuals was modelled (Clark 2007) assuming that the observed number of individuals on a logarithmic scale at time t, Yt, is normally distributed with mean (Xt = ln Nt) and variance inline image (see also Clark & Bjørnstad 2004 and Dennis et al. 2006).

Methods

  1. Top of page
  2. Summary
  3. Introduction
  4. Models
  5. Methods
  6. Results
  7. Discussion
  8. Concluding remarks
  9. Acknowledgements
  10. References
  11. Supporting Information

Data

We analyse data from the CBC of British birds operated by the British Thrust for Ornithology between 1962 and 2001. These data are based on annual censuses of breeding bird populations in plots (c. 260 in any single year) scattered all over Britain in both farmland and woodland habitats. The plots were thoroughly censused on several visits each year by the territory mapping method (see Marchant et al. 1990 for details) that provides an estimate of the number of stationary pairs of birds in the plot. This data set is the largest scale, longest term multispecies data set obtained using such a detailed mapping approach, and is thus the best for conducting this modelling exercise.

We selected time series that had at least 15 years of continuous data. In this study, we included the six passerine species that had the highest number of selected time series: Blackbird Turdus merula L., Blue tit Cyanistes caeruleus L., Chaffinch Fringilla coelebs L., Dunnock Prunella modularis L., Robin Erithacus rubecula L. and Wren Troglodytes troglodytes L.. For time series with at least 15 years of data, we used the first continuous time series of 10 years of population fluctuations to estimate the parameters of the different models and to calculate the PPI for the population after 5 years. We then found the quantile of the PPI that included the actual population size at the 15th year of census.

We also repeated the analyses using time series with at least 20 years of data. We then used the first 15 years of data to estimate the parameters. Fortunately, the distributions of the quantiles that included the actual population size were quite similar (Figs S1 and S2) to the estimates from 10 years of data.

Estimation of parameters

First we estimated the unknown parameters s, β and inline image by maximizing the log-likelihood, assuming no census error (Sæther et al. 2002b). Demographic variance could in principle be estimated from long-term time series spanning a large range of abundances. In practice, demographic variance needs to be estimated from individual-based demographic data (Lande et al. 2003). Due to lack of suitable data, we used three values inline image, inline image and inline image, which covers a large part of the range of variation in inline image in birds (Sæther et al. 2004). Assuming that Xt+1 = ln Nt+1 is normally distributed when conditioned on Xt = ln Nt, and writing f(x; μ, σ2) for the normal probability distribution with mean μ and variance σ2, the log-likelihood function takes the general form:

  • image

where inline image (eqns 1b and 2b), and the mean m(Xt) is given by eqns 1a and 2a for the density-independent and density-dependent models respectively. For the density-independent model, the log-likelihood function can be written as:

  • image( eqn\,\, 3)

and, for the density-dependent model with logistic density regulation, the log-likelihood can be written as:

  • image( eqn\, \,4)

Uncertainties were evaluated by parametric bootstrapping (Efron & Tibshirani 1993) involving simulating the time series using the initial value of the data and the estimated parameters.

We then included observation error (inline image) by adopting a Bayesian approach in combination with MCMC techniques (Gilks, Richardson & Spiegelhalter 1996; Millar & Meyer 2000; Clark & Bjørnstad 2004; Clark 2007). The Bayesian approach is based on two steps. First, we defined full probability distributions P1 and P2 for all unobservable inline image and observed inline image quantities at the log-scale, which were defined by the state and observation model (see Model section). Second, prior probability distributions (priors) were assigned to parameters with distributions that were not directly conditioned on other parameters or data, in this case s, β, inline image and inline image. We chose independent and uninformative priors because we have little a priori information about the size of the parameters. The priors were

  • image

denoted π1 (s), π2 (β), inline image and inline image for s, β, inline image and inline image respectively. Here N represents the normal distribution and IG represents the inverse gamma distribution. Finally, we let the state at the first time step, X1, have normal prior distribution, denoted P0, with mean Y1 and variance 106. The estimates were relatively insensitive to changes in the variance in the prior distribution P0 for X1. For instance, a reduction in prior variance by 7 orders of magnitude had essentially no effects on the results.

If conditioned on the data, the joint posterior distribution of the unknown quantities, inline image and inline image for the density-independent and density-dependent models respectively is according to Bayes’ theorem proportional to

  • image

if there is no density regulation, and

  • image

with density dependence. This distribution is analytically intractable, but was estimated by successive simulations from the posterior distributions using the program WinBUGS (Spiegelhalter et al. 2003). Similar to the models that did not include observation error, we fitted the models with three different levels of demographic variance, inline image, inline image and inline image respectively.

Computation of PPI

Let T denote the last year of population census. Computation of PPI for NT+1, NT+2,... was performed using eqns 1 and 2 for the density-independent and density-dependent models respectively. When no observation error was assumed to be present, we used the sets of parameters obtained by parametric bootstrapping. In the model including observation error σY, the joint posterior distribution of the unknown parameters was used. If the demographic variance inline image is assumed to be known and inline image is one realization of the posterior distribution or a bootstrap sample for the density-independent model, the population size at time T + 1 is predicted by:

  • image( eqn 5)

where ΔXT is normally distributed with mean inline image (eqn 1a) and variance inline image (eqn 1b). Similarly, for the density-dependent model where inline image is one realization of the posterior distribution or a bootstrap sample, we specify ΔXT in eqn 5 as normally distributed with mean inline image (eqn 2a) and variance inline image (eqn 2b).

To estimate the PPI, we simulated 20 time series from each of the 5000 bootstrap or posterior distribution sets of the estimated parameters, giving a total of 100 000 simulated time series per observed series.

To evaluate the deviations from a uniform distribution (Fig. 2), we computed, using the first 10 years of data, the distribution of the quantiles of the PPI in which the recorded population size after 15 years was located F(q15), that is, F(q15) is the rank of the observed value among the simulated values divided by the number of simulations. Hence, for a given model, we will for each species obtain one observation F(q15) for each population. If the model is correct, F(q15) would be an observation from a standard uniform distribution, whereas all records of F(q15) from all available data sets would be a sample of independent uniformly distributed variables. This is true, even if the parameters describing the dynamics differ among the populations.

We characterize the distribution of F(q15) by fitting a beta distribution inline image, where a1 and a2 are shape parameters. An advantage using this distribution is that variation of a1 and a2 can describe the basic patterns depicted in Fig. 2. If a1 = a2 = 1, we get a uniform distribution. For a1 = 1 and a2 > 1 and a1 > 1 and a2 = 1, we get patterns II and III in Fig. 2 respectively. The downward dome-shaped distribution appears when a1 < 1 and a2 < 1. Finally, for a1 > 1 and a2 > 1, we get a peaked distribution (pattern IV in Fig. 2) in which the peak is dependent on a1 + a2. When a1 < a2, the distribution is symmetrical around x = 0·5.

To compare PPI among species and models, we fitted, following Engen et al. (2001), the beta distribution to the set of observed relative ranks and replaced the relative rank of each simulated population size by the integral of the fitted beta distribution up to the observed relative rank. These new relative ranks will then follow a uniform distribution and the coverage of the new prediction intervals will be more accurately computed. The absolute precision in the PPI after time T was expressed as WT = q0·975 − q0·025, and the corresponding relative precision wT = (q0·975 − q0·025)/q0·500, where q0·025, q0·50 and q0·975 are the 2·5%, 50·0% and 97·5% quantiles of the PPI at an absolute scale respectively, after the observed relative rank is replaced by the rank obtained after fitting the beta distribution.

Results

  1. Top of page
  2. Summary
  3. Introduction
  4. Models
  5. Methods
  6. Results
  7. Discussion
  8. Concluding remarks
  9. Acknowledgements
  10. References
  11. Supporting Information

Reliability of the PPI

There was generally no significant difference (χ2-test) in the distributions of the quantiles (Fig. 1) of the PPI in which the recorded population size after 15 years was located F(q15) between estimates based on raw data ignoring census error and estimates based on MCMC methods for any of the models of density regulation (Figs 3 and 4). The only two exceptions involved models assuming inline image (density-independent model in Blue tit and density-dependent model in the Dunnock). Furthermore, F(q15) estimated by MCMC methods was closely correlated with the F(q15) based directly from raw data ignoring census error for the corresponding time series.

image

Figure 3.  Interspecific differences in the distribution of quantiles of the population prediction interval (PPI) in which the recorded population size was located after 5 years predicted by a Brownian population model (eqn 1) using the first 10 years of data F(q15) for different values of the demographic variance inline image. Shaded bars show the distribution based only on estimates based on raw data, whereas black bars give the distribution using MCMC methods to account for observation error.

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image

Figure 4.  Interspecific differences in the distribution of quantiles of the population prediction interval in which the recorded population size was located after 5 years predicted by a logistic model of density regulation (eqn 2) using the first 10 years of data F(q15) for different values of the demographic variance inline image. Shaded bars show the distribution based only on estimates based on raw data, whereas black bars give the distribution using MCMC methods to account for observation error.

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When density dependence was neglected, the distribution of recorded population size after 5 years across the quantiles of the PPI (Fig. 3) deviated significantly from a uniform distribution in all but one species (Chaffinch). In general, actual population sizes after 5 years were much smaller than predicted. This was also revealed by the estimates of the shape parameters of the beta distribution a1 and a2, in general both were larger than 1 (Table 1). This was in all species associated with a decrease in F(q15) with increasing size of the last recorded population size of the 10-year time series, which was used to estimate the parameters. Thus, ignoring density dependence resulted in an overestimate of the future growth of larger populations.

Table 1.   Interspecific variation in the parameters of the beta distribution a1 and a2 fitted to the set of observed relative ranks and replacing the relative rank of each simulated population size by the integral of the fitted beta distribution up to the observed relative rank for different values of demographic variance inline image. These new relative ranks will then follow a uniform distribution. The absolute of precision in the population prediction interval (PPI) after time T was expressed as WT = q0·975 − q0·025, and the corresponding relative precision wT =(q0·975 − q0·025)/q0·500, where q0·025, q0·50 and q0·975 are the 2·5%, 50·0% and 97·5% quantiles of the PPI at an absolute scale respectively. For further description, see text
Speciesinline imageMCMCNo observation error
a1a2Median w5Median W5a1a2Median w5Median W5
Density-independent model
 Blackbird0·001·021·521·5221·091·131·701·9724·13
0·251·321·801·6120·401·191·561·9324·11
0·501·652·161·6521·051·421·661·8723·01
 Blue Tit0·001·011·691·5517·111·171·672·2823·05
0·251·351·961·7317·731·251·552·1920·89
0·501·582·181·9819·801·351·502·2122·01
 Chaffinch0·001·061·232·0225·861·331·402·2529·40
0·251·401·481·8723·811·331·262·2729·09
0·501·721·771·9125·891·421·232·2629·35
 Dunnock0·001·051·492·3317·931·101·303·4324·95
0·251·261·552·5718·291·191·243·5422·33
0·501·471·712·8220·881·331·243·3422·11
 Robin0·001·191·621·7728·521·501·882·1933·45
0·251·501·831·7226·751·521·722·1132·87
0·501·762·071·7727·231·661·711·9929·92
 Wren0·000·951·562·6844·861·111·643·4153·89
0·251·051·522·8244·041·031·443·5953·05
0·501·121·562·8548·281·031·353·5248·88
Density-dependent model
 Blackbird0·000·600·951·2717·450·360·651·1313·74
0·250·771·171·2116·090·390·711·0513·80
0·500·991·461·2215·740·450·821·1214·85
 Blue Tit0·000·750·871·3012·110·510·641·1011·19
0·251·071·221·1210·890·610·741·0210·01
0·501·371·521·1511·610·740·891·0110·54
 Chaffinch0·000·930·711·4820·830·610·481·1116·44
0·251·170·891·2517·810·600·491·1215·82
0·501·521·161·1716·560·690·571·1115·80
 Dunnock0·000·510·752·0715·820·320·541·8413·55
0·250·610·872·0015·810·340·561·9113·34
0·500·741·042·0014·970·390·651·9413·22
 Robin0·000·660·721·8126·450·450·501·2818·94
0·250·820·851·4821·500·440·501·2719·20
0·501·031·031·4620·320·510·561·2319·27
 Wren0·000·730·672·4541·720·540·501·6027·55
0·250·830·722·3041·590·510·491·6227·88
0·500·940·812·2135·810·550·511·5926·62

Applying a logistic model of density regulation, but still assuming inline image, caused a downward dome-shaped distribution of the qN15 (Fig. 4) with a1 < 1 and a2 < 1 for the density-dependent model in all but one species (Table 1). There was no significant relationship between F(q15) and the last recorded population size in any species.

These analyses assume that the stochastic effects on the population fluctuations are caused only by environmental stochasticity. Including demographic stochasticity in the models strongly affected the population predictions (Figs 3 and 4). Assuming a fixed demographic variance inline image, the width of the prediction intervals became too wide, resulting in a significant (< 0·01) deviation from a uniform distribution of qN15 (Fig. 3) with a1 >> 1 and a2 >> 1 for the density-independent model in all species (Table 1). In the density-dependent model, there was a significant deviation (< 0·01) from a uniform distribution for inline image in three species (Blackbird, Chaffinch and Dunnock), whereas Blackbird and Dunnock deviated significantly (< 0·05) when the demographic variance was reduced to inline image

Our next step was to examine how the reliability of the prediction was affected by variation in different parameters. For the density-independent model, the actual population size was smaller than predicted for larger values of the last population size N10 (Fig. 5). No such effects of N10 was present for the density-dependent model, indicating that growth rates were overestimated at larger population sizes when applying the density-independent model. Otherwise, the quantile of the PPI in which the recorded population size was located could not be predicted from any of the population parameters. The only exception was the Wren for which larger population sizes than predicted were recorded for large values of s and β.

image

Figure 5.  Interspecific differences in the relationship between the quantile of the population prediction interval in which the recorded population size was located after 5 years predicted by a Brownian population model (eqn 1) using the first 10 years of data and the last population size recorded after the first 10 years of study N10 for demographic variance inline image. The lines indicate significant regression lines.

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Precision in the predictions

The precision of the predictions were assessed by the width of the PPI at the transformed scale. Higher precision of the predictions were obtained when assuming density dependence (Table 1). For the density-independent model, PPI based on estimates obtained from MCMC methods were more precise than PPI computed from estimates of parameters without assuming any observation error (Table 1). Furthermore, the level of inline image had only a small effect on the precision in the population predictions (Table 1). This may be related to the interaction among the different variance components (Table 2). The variance component due to observation error and environmental stochasticity both decreased with increasing demographic stochasticity (Table 2). Similarly, the uncertainty in the estimates of the environmental variance also decreased with increasing inline image

Table 2.   Median of the estimates of the variance in the observation error inline image and the environmental variance inline image for different values of demographic variance inline image
SpeciesModelinline imageinline image
inline imageinline imageinline imageinline imageinline imageinline image
  1. Densdep, density-dependent model; Densindep, density-independent model.

BlackbirdDensdep0·00580·00380·00290·02990·01300·0084
Densindep0·00590·00450·00330·04160·02510·0146
Blue TitDensdep0·00600·00360·00280·03750·01380·0093
Densindep0·00710·00470·00350·05340·02550·0148
ChaffinchDensdep0·00630·00390·00290·03440·01490·0093
Densindep0·00580·00490·00340·05030·02870·0164
DunnockDensdep0·00740·00520·00360·06310·03110·0170
Densindep0·00680·00610·00460·10480·06140·0346
RobinDensdep0·00620·00450·00340·04370·02440·0131
Densindep0·00550·00490·00420·06630·04310·0264
WrenDensdep0·00690·00590·00440·07450·05230·0335
Densindep0·00590·00450·00330·13330·09630·0682

Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Models
  5. Methods
  6. Results
  7. Discussion
  8. Concluding remarks
  9. Acknowledgements
  10. References
  11. Supporting Information

Our analyses show that both density dependence and partitioning of stochastic influences into demographic and environmental stochasticity are important to include in population models in order to predict future population fluctuations of small passerine populations (Figs 3 and 4).

Ignoring density dependence will lead to biased predictions because of overestimation of the future population growth of especially larger populations (Fig. 5, Table 1). Furthermore, precision in the population predictions will be improved when including density dependence. Finally, including observation error in the models improves precision in density-independent predictions (Table 1) and reduces bias for density-dependent models (Fig. 4, Table 1).

Our approach is based on the assumption that the parameters estimated from the first part of the time series can be used to describe the population dynamics during the following period of 5 years. This is not a valid assumption if the environmental conditions changes throughout the study period. In fact, the composition of the bird communities especially in British farmlands have changed during the past decades, most likely due to changes in agricultural practices (Benton et al. 2002; Newton 2004). However, we found no significant differences in the distribution of quantiles between woodland and farmland plots for any of the models. Furthermore, the distributions of the quantiles were also quite independent of the initial time period used for estimating the parameters (Figs S1 and S2). This indicates that parameter drift (Ellner 2003; Halley 2003) had small influence on the bias in predictions.

Statistical analyses of time series of population fluctuations have often treated population estimates as being error-free (Royama 1992), which can lead to serious bias in the estimates of parameters such as the strength of density dependence or environmental variance (Bulmer 1975; Freckleton et al. 2006). In this study, we apply a Bayesian MCMC approach to estimate the population parameters in the face of unknown levels of error. Such techniques are now commonly used in population ecology (Clark & Bjørnstad 2004; Clark 2005, 2007; Thomas et al. 2005; Newman et al. 2006; Sæther et al. 2007b, 2008). Although computationally intensive, this approach avoids assumptions about normal distribution of errors and linear effects (Schnute 1994; De Valpine & Hastings 2002) as is the case for models analysed by maximum likelihood methods. It is, however, important to notice that the estimates of the variance of the observation error inline image are affected by how the stochastic effects on population dynamics are modelled. If population dynamics are modelled without any demographic stochasticity, estimates of inline image will be much larger than when demographic variance is included in the model (Table 2). Similarly, assumptions about the magnitude of the demographic stochasticity will also affect the estimates of the environmental variance (Engen et al. 2001). Because the effects of demographic stochasticity increase with decreasing population size (Lande et al. 2003), estimates of inline image are dramatically reduced by increasing inline image in such small populations as in the present study (Table 2).

Assumptions about the magnitude of demographic stochasticity influenced both the bias in the population predictions, evaluated by the distribution of the quantiles of the PPI at which the recorded population sizes were located (Figs 3 and 4), as well as the precision in the predictions of density-independent models, determined by the width of the PPI at a given time (Table 1). Ignoring demographic stochasticity resulted in predictions of too large future population sizes and imprecise density-independent population predictions. This effect occurred because assumptions about the demographic stochasticity affected the magnitude of the variance components due to observation error and environmental stochasticity (Table 2). In addition, demographic stochasticity should be included to obtain unbiased density-dependent predictions (Table 1, Fig. 4).

Unfortunately, demographic variance cannot be estimated from short time series of population fluctuations, but requires demographic data on individual variation in annual fitness contributions to future generations (Lande et al. 2003). However, interspecific variation in the demographic variance of birds is related to life-history characteristics and values of inline image in the range 0·20–0·40 seems typical for small temperate passerines (Sæther et al. 2004). Accordingly, applying a density-independent model that included a fixed demographic variance of inline image gave for most of the species a uniform distribution of the quantiles (Fig. 2) and more precise predictions (Table 1, Fig. 2). This suggests that reliable predictions of future population fluctuations of small passerines require models that include estimates of the demographic variance. We suggest that inline image may be an appropriate value for such species. However, knowledge about factors affecting the demographic variance is especially important in developing predictions of future developments of small populations (Ringsby et al. 2006), e.g. of endangered and threatened species.

Ignoring observation errors in population estimates results in biased estimates of many population parameters (Freckleton et al. 2006). Many attempts have now been made to estimate the census error by different state-space models (De Valpine & Hastings 2002; Buckland et al. 2004; Clark & Bjørnstad 2004; Clark 2005, 2007; Dennis et al. 2006; Newman et al. 2006; Sæther et al. 2007b; Lilligård et al. 2008). These studies reveal large variability in the contribution of the observation error to the variation of the recorded population sizes, ranging from very small effects in species such as the moose Alces alces (Clark & Bjørnstad 2004; Clark 2007) and ibex Capra ibex (Sæther et al. 2007b) to time series of population estimates in which most of the recorded variation is likely due to observation error (Dennis et al. 2006). In the present study, the effects of the observation error were large for density-independent predictions (Table 1). One reason for this is that the populations analysed were relatively small, which cause a large effect of the demographic stochasticity on the population fluctuations (Lande et al. 2003). Accordingly, ignoring inline image caused the estimates inline image to increase (Table 2). Although a large number analyses of population dynamics and population variability have been published on this data set (e.g. Pimm 1991; Blackburn et al. 1998; Freckleton et al. 2005), this is the first study to account the effects of observation error on the parameter estimates.

The CBC is based on the mapping method originally developed by Enemar (1959) and conducted according to strict procedures based on international standards (Anonymous 1969, see Appendices 1 and 2 in O’Connor & Marchant (1981) for further details). Two major sources of bias are identified in the use of this method. (i) The accuracy of the censuses may differ considerably among observers due to individual variation in the ability to discover and to correctly identify territorial individuals who are present in the study plot. (ii) A critical feature of the method is to correctly delimit the territory of a stationary individual. Such interpretational problems are known to strongly affect the estimates of the sizes of the populations at the study plots (Williamson 1964; O’Connor & Marchant 1981). Our estimates of inline image (Table 2) indicate considerable interspecific variation in these sources of error. However, the variance component due to observer error was in our modelling approach far less than the component due to environmental stochasticity. This suggests that although labour intensive, the mapping method can be used to infer how environmental changes may affect population trends (Mazzetta, Brooks & Freernan 2007).

Another important feature of the observation error was that it reduced the bias in density-dependent predictions (Fig. 4, Table 1). Ignoring observation error gave in general underestimation of the width of PPI (pattern V in Fig. 2) because the return time to equilibrium [i.e. the inverse of the strength density regulation (May 1974)] was underestimated (e.g. Freckleton et al. 2006). This means that without accounting for observation error when such errors are present, we underestimate the range of variation in future population sizes, which can have serious implications. e.g. when predicting the risk of extinction during a given time period.

Previously, we have estimated the environmental variance from time series of fluctuations in the size of Blue tit populations that nest in nest boxes, which renders observation error in the population estimates small. Our median estimate of inline image for the Blue tit populations included in the present study (inline image), assuming inline image, was within the range of estimates for 18 populations of this species across Central Europe (Sæther et al. 2007a) and was even less than the median value of inline image in the previous study (inline image). Because underestimation of the observation error will result in too large a value of inline image, this gives us further faith in the robustness of parameter estimates obtained by the present approach.

Dennis et al. (2006) provided estimates of large observation error for a time series of fluctuations in the size of an American Redstart population, which was part of the Breeding Bird Survey monitoring programme in the USA. Their analyses were based on a state-space model using maximum likelihood estimation, assuming log-normal distribution of the noise and a loglinear form of density dependence. When we reanalysed their data using our Bayesian approach with a loglinear form of density regulation (May 1981), we got a much larger observation error (inline image) than for the species included in the present study, However, this estimate was less than Dennis et al.’s, resulting in larger contribution of environmental stochasticity to the population fluctuations by our approach. Furthermore, we also estimated stronger density dependence in this time series than Dennis et al. (2006). As a consequence, we found larger annual fluctuations in the underlying population process than the more smoothed fluctuations depicted in Fig. 1 of Dennis et al. (2006). This shows that estimates of observation error are closely related to how the stochastic effects and expected dynamics are modelled as well as the underlying statistical model.

An important topic in population ecology is the importance of density regulation on population dynamics. In this study, models that did not include density dependence gave biased population predictions of future population sizes (Table 1, Fig. 4), mainly due to overestimate of future population sizes of larger populations (Fig. 5). Although our data sets contain a large number of time series of fluctuations of small populations, failure to include density dependence, which will cause the population to return to some equilibrium population size (May 1974), resulted in biased population predictions especially of larger populations.

In the Wren, we failed to obtain reliable population projections because a large proportion of actual population sizes were located towards both ends of the PPIs (Figs 3 and 4), indicating that we have overestimated the strength of density dependence (pattern V in Fig. 2). In particular, we predicted too small sizes of the larger populations of this species. This may be due to bias in the estimates of density regulation β. In a simulation study, Lillegård et al. (2008) showed that hierarchical Bayesian MCMC methods tended to overestimate the strength of density regulation. This may explain that larger populations, for which density regulation is expected to be strongest, were difficult to predict in this species. Accordingly, this small species has been shown too vulnerable to bad winter weather, resulting in large population fluctuations (Robinson, Baillie & Crick 2007).

Concluding remarks

  1. Top of page
  2. Summary
  3. Introduction
  4. Models
  5. Methods
  6. Results
  7. Discussion
  8. Concluding remarks
  9. Acknowledgements
  10. References
  11. Supporting Information

Our analyses show that density-independent model with specific assumptions about the magnitude of the demographic stochasticity tend to give unbiased predictions of future fluctuations in the size of small populations of passerines. The predictions from such models are very uncertain, especially of larger populations. Some improvement can be achieved by including observation error, e.g. by using a Bayesian MCMC approach. To improve precision, we recommend including specific models of the density dependence, but this may result in biased predictions because of overestimation of the strength of density regulation, resulting in underestimation of future population fluctuations. This bias can be dramatically reduced by including observation error and estimates of the demographic variance in the models. These results also extend previous results (Fieberg & Ellner 2000; Sæther et al. 2000; Engen et al. 2001; Holmes & Fagan 2002; Sæther & Engen 2002a; Holmes et al. 2007; Ellner & Holmes 2008) that constructing PPI by stochastic simulations may be a very useful tool for population projections.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Models
  5. Methods
  6. Results
  7. Discussion
  8. Concluding remarks
  9. Acknowledgements
  10. References
  11. Supporting Information

We thank two anonymous reviewers for excellent comments and teaching us the meaning of the expression ‘Referee from Hell’. We are grateful for financial support from the Research Council of Norway (Strategic University Program in Conservation Biology) and the Norwegian University of Science and Technology to CCB. R.P. Freckleton was funded by a Royal Society University Research Fellowship.

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  3. Introduction
  4. Models
  5. Methods
  6. Results
  7. Discussion
  8. Concluding remarks
  9. Acknowledgements
  10. References
  11. Supporting Information
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Supporting Information

  1. Top of page
  2. Summary
  3. Introduction
  4. Models
  5. Methods
  6. Results
  7. Discussion
  8. Concluding remarks
  9. Acknowledgements
  10. References
  11. Supporting Information

Fig. S1. Interspecific differences in the distribution of quantiles of the Population Prediction Interval (PPI) in which the recorded population size was located after 5 years predicted by a Brownian population model (eqn. 1) using the first fifteen years of data F(q15) for different values of the demographic variance σ2d. Shaded bars show the distribution based only on estimates based on raw data, whereas black bars give the distribution using MCMC-methods to account for observation error.

Fig. S2. Interspecific differences in the distribution of quantiles of the Population Prediction Interval (PPI) in which the recorded population size was located after 5 years predicted by a logistic model of density regulation (eqn. 2) using the first fifteen years of data F(q15) for different values of the demographic variance σ2d. Shaded bars show the distribution based only on estimates based on raw data, whereas black bars give the distribution using MCMC-methods to account for observation error.

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