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Keywords:

  • spectrum;
  • spectral exponent;
  • red noise;
  • population dynamics;
  • climate change;
  • extinction risk;
  • Ricker;
  • population models;
  • Soay sheep

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

1. The spectral colour of population dynamics and its causes have attracted much interest. The spectral colour of a time series can be determined from its power spectrum, which shows what proportion of the total variance in the time series occurs at each frequency. A time series with a red spectrum (a negative spectral exponent) is dominated by low-frequency oscillations, and a time series with a blue spectrum (a positive spectral exponent) is dominated by high-frequency oscillations.

2. Both climate variables and population time series are characterised by red spectra, suggesting that a population's environment might be partly responsible for its spectral colour. Laboratory experiments and models have been used to investigate this potential link. However, no study using field data has directly tested whether populations in redder environments are redder.

3. This study uses the Global Population Dynamics Database together with climate data to test for this effect. We found that the spectral exponent of mean summer temperatures correlates positively and significantly with population spectral exponent.

4. We also found that over the last century, temperature climate variables on most continents have become bluer.

5. Although population time series are not long or abundant enough to judge directly whether their spectral colours are changing, our two results taken together suggest that population spectral colour may be affected by the changing spectral colour of climate variables. Population spectral colour has been linked to extinction; we discuss the potential implications of our results for extinction probability.


Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

The positive autocorrelation typical in animal population dynamics and its causes have stimulated substantial interest over the past 30 years (Roughgarden 1975; Lawton 1988; Cohen 1995; Akçakaya, Halley & Inchausti 2003; Schwager, Johst & Jeltsch 2006; Ruokolainen et al. 2009). Many climatic variables are also positively autocorrelated, suggesting that a population's environment might be partly responsible for the positive autocorrelation seen in its dynamics. However, no study using field data has directly tested whether more positively autocorrelated populations live in more positively autocorrelated environments. Also, insufficient work in the ecological literature has addressed the related question of how the autocorrelation of environmental variables may be affected by climate change and what the population consequences of these changes may be. These questions have practical implications because the level of autocorrelation in population dynamics affects population extinction probabilities as well as temporal patterns of offtake in the case of exploited populations and temporal patterns of economic or disease burden in the case of pest or vector populations (Reuman et al. 2006, 2008).

Empirical data show that annually censused population dynamics are positively autocorrelated, and consequently described by red power spectra (Pimm & Redfearn 1988; Sugihara 1995; Halley 1996; Inchausti & Halley 2001); we provide definitions to make this statement precise. The power spectrum is a widely used mathematical technique that takes a time series (population or environmental) as input and returns as output a plot that shows the decomposition of the total variance (or power) in the time series into its frequency components (Brillinger 2001). A red time series, by definition, has more variation at low frequencies than at high frequencies. A blue time series has more variation at high frequencies, and a white time series has equal variation at all frequencies in a range. The colour-based terminology used here was coined because red (respectively, blue) light is more dominated by lower (respectively, higher) frequencies than other colours of visible light. Colour can be quantified for a time series by calculating the spectral exponent, defined as the slope of a linear regression line drawn through a log-power-vs.-log-frequency plot of the spectrum; negative slopes correspond to red time series and positive slopes to blue time series, with white noise having a spectral exponent equal to or close to zero. Inchausti & Halley (2002) found that the spectral exponents in annually censused animal populations across several clades and trophic levels were negative: population dynamics, as typically measured by ecologists, are red.

Ascribing the spectral colour of populations to a cause or mechanism has proven more complex than describing the pattern. Early work focussed on simple unstructured deterministic population models to see whether intrinsic dynamics could be the cause of population spectral redness. For example, Cohen (1995) investigated several such population models using a single point in parameter space chosen to be in the models’ chaotic regime, finding that the dynamics predicted by the selected models tended to be blue. Other authors subsequently found, however, that the same models with other parameters produced red spectra (Blarer & Doebeli 1996; White, Begon & Bowers 1996a): simple deterministic models can produce dynamics of a range of colours depending on parameters. Deterministic models alone failed to completely explain the origin of populations’ spectral colour, unless accompanied by an argument that real populations are constrained to certain parameter regimes.

Several modifications of the initial deterministic models were examined, all with the potential to redden spectra. These included the introduction of measurement error (Akçakaya, Halley & Inchausti 2003), a spatial component (White, Bowers & Begon 1996b), delayed stochastic density dependence (Kaitala & Ranta 1996) and age structure (Greenman & Benton 2005). One mechanism that has received much attention is environmental variability (Lawton 1988; Sugihara 1995; Kaitala et al. 1997; Ranta et al. 2000). Climatic variables are also characterised by reddened spectra (Steele & Henderson 1994; Cyr & Cyr 2003; Vasseur & Yodzis 2004). Given populations’ reliance on the surrounding environment, it seems likely that their spectral redness can, at least in part, be traced back to the redness of climate.

If environmental colour were to have any influence on population spectral colour, population dynamics should be redder in redder environments (Roughgarden 1975; Kaitala et al. 1997). To investigate this link, both laboratory experiments (Petchey 2000; Laakso, Löytynoja & Kaitala 2003b) and theoretical studies (Roughgarden 1975; May 1981; Kaitala et al. 1997; Laakso, Kaitala & Ranta 2001, 2003a; Greenman & Benton 2005; Ruokolainen, Fowler & Ranta 2007) have been undertaken, tentatively concluding that some of the environmental spectral colour is likely to propagate through to the population spectra, ‘tinging’ the dynamics with a similar colour. Figure 1 provides a summary presentation of some prior modelling results demonstrating this effect using the well-known Ricker model (Methods). A similar pattern generally arises in other simple univariate models, such as the Hassell and Maynard Smith models (Appendix S1). It is important, however, to augment prior modelling (Roughgarden 1975; May 1981; Kaitala et al. 1997; Greenman & Benton 2005) and experimental (Laakso et al. 2003b) results summarised here with tests based on field data. Although the use of observational field data makes it difficult or impossible to establish a causal relationship between climate and population spectral colour, field data can be used to test for correlations that such a causal relationship would produce. Modelling and experimental studies have explored causation in a context where it is possible to do so whereas observational field studies are now necessary to see to what degree predicted consequences of the causal hypothesis actually pertain in a broad way to real systems.

image

Figure 1.  The impact of environmental spectral colour on population spectral colour in a stochastic formulation of the Ricker model (Methods). Panel a is the bifurcation plot for the deterministic skeleton of the model, indicating the growth rate (r) values and respective line types used for the following panels. Panel b is with weak environmental noise (σ = 0.01; see Methods), and panel c is with strong environmental noise (σ = 0.1). Results show that environmental spectral colour tinges population spectral colour, to an extent that depends on growth rate and the strength of environmental noise.

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Environmental noise colour has an influence on population extinction risk, but results so far indicate that this influence can be complex and contingent on the details of population dynamics. Prompted by the positive autocorrelation reported for both climatic variables and populations, Lawton (1988; later supported by Halley 1996; Pike et al. 2004 and Inchausti & Halley 2003, the latter using empirical data and the concept of ‘quasi extinction’, a 90% reduction in population size) argued that red noise should increase the risk of extinction, based on the intuition that populations would then suffer long runs of adverse conditions. In apparent contradiction to this intuition, Ripa & Lundberg (1996) claimed that red noise decreases extinction risk. Subsequent studies (Petchey, Gonzalez & Wilson 1997; Heino 1998) expressed a more nuanced view. Theoretical studies have not reached a consensus predominantly because of differences in population model and parameter choice (Ripa & Lundberg 1996, 2000; Heino 1998; Ripa & Heino 1999), environmental noise model (Heino 1998; Halley & Kunin 1999; Cuddington & Yodzis 1999), variance used (Heino, Ripa & Kaitala 2000; Schwager, Johst & Jeltsch 2006) and the time-scales on which extinctions are scored (Halley & Kunin 1999; Heino, Ripa & Kaitala 2000). It is difficult to systematically explore the relationship between colour and extinction risk with models given the variety of modelling choices that must be made. We return to the relationship between spectral colour and extinction risk in the Discussion.

The relationships between the spectral colours of climate and populations and the associated population extinction risk need to be viewed in a context of climate change. Climate patterns throughout the world are changing rapidly, as evidenced by increases in average global temperature and in the variability of climatic conditions (IPCC 2007). These changes are conceivably shifting the spectral colour of climatic variables and consequently may be affecting populations’ spectra, if climate and population spectra are causally related. We test the hypothesis that the spectral exponents of climate variables have changed over the last century and combine the results with our observations about how population and climate spectral exponents are related to formulate hypotheses about how population spectral exponents may be influenced by climate change.

Materials and methods

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

Data sources

Two data sets of climate variables were used, respectively, for the purposes of analysing changes in climate spectral exponent over time and for comparison with population time series: a large collection of direct measurements taken from weather stations, and a global coverage, spatially gridded data set derived from measurements by interpolation. These data sets have, respectively, the complementary strengths of greater reliability and coverage that make them suitable to be used for the intended purposes. Weather station data were downloaded from the Global Historical Climatology Network (GHCN; Peterson & Vose 1997). The GHCN provides data from about 7280 stations world-wide, although different stations were active for different periods. Spatially gridded data were downloaded from the Climatic Research Unit (CRU TS 2.1 data set). CRU data have global terrestrial coverage at 0.5° by 0.5° resolution and monthly temporal resolution from 1901 to 2002 (Mitchell & Jones 2005). The interpolation procedure used for the CRU data is described by New, Hulme & Jones (2000).

CRU data span a century and are spatially comprehensive, enabling comparison between population time series and interpolated climate data from the same location. However, the reliability of the interpolated data depends on location and time, being related to the number and proximity of nearby weather stations. A density index of nearby stations for each grid cell at each time is provided with the CRU data set. CRU data were validated against the GHCN data (see Appendix S2) to obtain a threshold value for the station density index above which the CRU data were found to be sufficiently reliable. Only data with reliability above this threshold value were used for comparison with population data, so populations in a time or place with CRU data reliability below the threshold were not used.

The Global Population Dynamics Database (GPDD; NERC Centre for Population Biology & Imperial College 1999) currently holds nearly 5000 animal and plant population time series and is freely accessible. It has been used in several population dynamics studies, some of which investigated population spectral colour (Kendall, Prendergast & Bjørnstad 1998; Inchausti & Halley 2001; Halley & Inchausti 2002; Inchausti & Halley 2003). GPDD data were filtered to remove time series not suitable for our analysis. The filtering process, described in detail in Appendix S3, kept only annual time series with at least 30 continuous data points that were also accompanied by metadata with the geographic coordinates of the location. Other filtering constraints were also applied. One hundred and forty-seven time series remained after filtering (see Appendix S3 for a complete list of time series used).

Preprocessing of weather data

The GPDD data used are annual, whereas the CRU and GHCN data used consist of mean monthly temperatures (i.e. time series with a sampling frequency of 12 per year). In order for the two to have the same temporal resolution, the CRU and GHCN data were preprocessed to derive several variables, all with a sampling frequency of one per year. Mean annual temperatures were obtained by taking the mean of the 12 mean monthly temperature values (January–December). All but one of the populations left over from the filtering process were located in the northern hemisphere, so seasons were defined accordingly, with winter being December–February, spring being March–May, and so on. Mean summer temperature is the mean of the three monthly temperature values corresponding to summer. Similarly, the other seasons are defined as the means of their respective months. Mean seasonal temperature refers to all four time series, collectively.

General methods

The wide prior use of the spectral exponent facilitates direct comparisons of our results with earlier studies. All time series were linearly detrended, and the spectrum was then estimated using an unsmoothed periodogram (spec.pgram function in the R programming language). Before detrending and computing spectra, population numbers, p, were transformed by  log10(p + 1). Climate variables were detrended but not transformed. All computations and graphics were carried out in the R computing environment, version 2.10.0 (2009-10-26; R Development Core Team 2009).

Testing for correlation between climate and population spectral exponents

Using the CRU data, the spectral exponents of mean seasonal and mean annual temperatures were calculated for the same time period and location (rounded to 0.5°) as each of the 147 GPDD population time series. The null hypothesis that the correlation between climate and population spectral exponents was zero was then tested by computing a Pearson correlation coefficient and P value, taking spatial autocorrelation into account as described later.

Testing for change in climate spectral exponent

The GHCN data were filtered to include time series that covered the 1911–1990 period. These years were chosen because they gave a good compromise between length of time period and number of weather stations active throughout that period. Using the most recent years available (until 2002 in the version of the GHCN data set used for this study) would have greatly reduced the number of weather stations available (New, Hulme & Jones 2000). For each half of the time series (1911–1950 and 1951–1990), a maximum proportion of missing values of 0.15 was allowed. Because it can accommodate missing data, for this spectral analysis the Lomb periodogram (Scargle 1982) was used to calculate spectra. Spectral exponents for both halves of the time series (1911–1950 and 1951–1990) were calculated for mean annual and mean seasonal temperatures. The null hypothesis that the spectral exponents of the two halves were the same was tested using a t-test, taking spatial autocorrelation into account as described later. This hypothesis was tested for the whole world and for continental regions separately (see Appendix S4 for region definitions). Although we used both Lomb periodograms and ordinary periodograms in this study, the two methods were used in different analyses and results were kept separate.

Correcting for spatial autocorrelation

The focus of this study is the analysis of temporal variations in climate and populations. Climate and population phenomena have a spatial structure, however, that needs to be accounted for to avoid inflation of Type I error rates (Legendre & Legendre 1998). The software package Sam (Spatial Analysis in Macroecology; Rangel, Diniz-Filho & Bini 2006) was used to calculate effective numbers of degrees of freedom, with which the appropriate reference distributions could then be found for the t-tests mentioned above, and the corrected value of P computed for the correlations between climate and population spectral exponents. We followed the method of Dutilleul (1993). This standard approach does not depend on any a priori assumption on the functional form of spatial autocorrelation, as might be the case when using, for example, generalised least squares methods (e.g. exponential, Gaussian or spherical assumptions; see Dormann et al. 2007).

Set-up of models

We use a stochastic formulation of the Ricker model to help illustrate and explain background information and interpret empirical results. The model is pt+1 = pt exp (r(1 − pt/K) + xt), where K is carrying capacity (K = 1 was used), r is growth rate, and xt is the environmental noise modelled as an autoregressive order 1 (AR1) process. The spectral colour of xt is determined by ρ, its lag-1 autocorrelation (−1 < ρ < 1, ρ > 0 for red noise, ρ < 0 for blue noise). The strength of environmental noise is σ, the standard deviation of the process.

We used a threshold autoregressive model of Grenfell et al. (1998) to help interpret results. The model is defined as xt+1 = a0 + b0xt + ɛ0 for xt ≤ C and xt+1 = a1 + ɛ1 for xt > C, where xt is log population density. The model is diagrammatically depicted in Grenfell et al. (1998). Here, C is a carrying capacity above which winter weather, ɛ1, may cause a substantial crash or a modest increase in very good years. Below C, growth is exponential, with noise that depends on summer weather, ɛ0. Each noise time series ɛi is AR1 with standard deviation σi (σ1 > σ0) and colour ρi, where ρi ranges from −0.9 (very blue noise) to 0.9 (very red noise). Parameter values ai, b0 and σi used were those given in Grenfell et al. (1998), except C was slightly changed from 7.01 to 7.23 to better illustrate the phenomenon of interest, although the original value produced qualitatively similar results.

Results

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

Mean summer and annual temperatures had spectral exponents significantly correlated with population spectral exponents (Table 1), even after accounting for spatial autocorrelation, confirming the hypothesis that redder populations live in redder climates. The correlation coefficients r for separate species groups were generally similar to overall r values and were always positive for mean summer and annual temperatures.

Table 1.  Correlations between the spectral exponents of animal populations and the spectral exponents of mean temperature, for seasonal and annual averages. P is the P value corrected for spatial autocorrelation. Ntotal = 147, N for Aves is 56, for Crustacea 12, for Mammalia 47 and for Osteichthyes 23. The P values for the clade-specific regressions were not significant because of the reduced statistical power that comes from a reduced data set, although r values show that clade-specific patterns were consistent with overall trends
  r P Aves r Crustacea r Mammalia r Osteichthyes r
Winter−0.0550.659−0.0400.508−0.2300.062
Spring0.0600.5900.2260.538−0.0650.123
Summer0.3120.0210.2940.4060.3060.207
Autumn−0.1790.146−0.179−0.105−0.160−0.250
Annual0.1350.0490.1910.2060.2990.202

The change in spectral exponent from 1911–1950 to 1951–1990 was generally statistically significant for most climate variables and geographical regions: most spectral exponents became less red-shifted (see Fig. 2 for mean summer temperatures, Fig. 3 for other examples and Appendix S5 for all climate variables examined). There is a conspicuous exception to the trend: Asia was redder in 1951–1990 than it was in 1911–1950 for all climate variables except for mean autumn temperatures. The spectral exponents for all continents were still typically red, however, in both the first and second halves of the time series examined. Mean summer temperatures are of particular interest because their spectral exponents correlated most strongly with population spectral exponents. For mean summer temperatures, Asia and Australasia became redder, and other regions became conspicuously bluer (Fig. 2).

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Figure 2.  Change in the spectral exponents of mean summer temperature time series from 1911–1950 to 1951–1990. P values (t-test corrected for spatial autocorrelation, N in parentheses) are listed above each box-whisker plot. A positive (respectively negative) difference in spectral exponent denotes a bluer (respectively redder) spectrum during 1951–1990 compared to 1911–1950.

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image

Figure 3.  Other examples of changes in climate spectral exponents (s.e.). The change in the spectral exponents to 1951–1990 (‘after’) from 1911–1950 (‘before’) for annual mean temperature in Europe (a, b) and winter mean temperature in Asia (c, d) as histograms (a, c) and as paired values (b, d).

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Distributions of the spectral exponents of population and climate variables appeared symmetric and unimodal, and quantile–quantile plots indicated that they were not markedly different from normal. These results help justify the use of t-tests and Pearson correlations.

Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

Our results show that the spectral exponents of population time series correlated positively and significantly with the spectral exponents of the mean summer temperatures the populations experienced. The correlation is weak, but this is expected because we analysed a wide range of species, and each could be affected predominantly by different factors only partly related to those considered; a variety of measurement errors will also have weakened the correlation. The fact that a relationship can be detected at all in spite of these heterogeneities is a valuable result that merits analyses in future research using additional data sets.

We also found that mean seasonal and annual temperatures have become bluer over the past century on all continents, except Asia and, for some climate variables, Australasia and North America. This indicates that high frequencies are generally becoming increasingly important relative to low frequencies in the climate variables we examined.

The combination of our two results suggests the possibility that population spectra are in the process of becoming bluer as a consequence of ongoing climate change. Although this conclusion is indirect because population time series are not abundant or long enough to directly examine how their spectral exponents are changing, it is important because it represents a broad possible impact of climate change on population dynamics.

Why summer?

Why does summer mean temperature correlate most significantly of the variables we examined? Many of the populations were at high latitudes, with severe winter weather, suggesting that spectral exponents of winter climatic variables should perhaps correlate more strongly with population spectral exponents than summer climate spectral exponents. We argue here that this expectation is flawed, and we present a possible hypothetical explanation for the importance of summer.

In populations for which bad winter weather causes crashes at high densities, interannual autocorrelation in winter weather is not transmitted to population autocorrelation because a crash caused by the first bad winter makes subsequent bad winters have little effect. In contrast, summer weather maps more directly onto successive years of population growth if it takes multiple years for a population to reach carrying capacity and summer weather affects population growth. This reasoning and the assumptions implicit in it are explained in more detail in Appendix S6.

The hypothesis presented here is supported by a simple model of Grenfell et al. (1998) which quantitatively captures the mechanisms (see Methods for the model definition). Model output (Fig. 4) indicates that the impact of summer noise colour on population spectral colour can indeed be substantially greater than the impact of winter noise colour when growth is slow and affected by summer weather and crashes are rapid and brought about by bad winter weather and high population density. The model thereby supports our explanation of empirical results.

image

Figure 4.  The effect of winter and summer environmental spectral colour on population spectral colour according to the model of Grenfell et al. (1998) (Methods), for which the repercussions of environmental autocorrelation in the two seasons on population spectral exponent can be separately analysed. Model population spectral colour was much more strongly affected by summer spectral colour (ρ0) than by winter spectral colour (ρ1). s.e. = spectral exponent.

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Extinction risk

The impacts that climate and population spectral colours have on extinction risk are complex and have not been settled, as testified by the lack of consensus in the prior theoretical work summarised in the Introduction. Nevertheless, it is important to discuss the link between our results and the large extinction risk literature because extinction risk is one major reason for studying population and climate spectral colour. For this reason, we discuss the link within the context of a family of univariate population models for which the relationship between spectral colour and extinction risk is well understood. For the Ricker model (Fig. 5) and other unstructured population models (Appendix S7), it has been observed that for red-shifted, slow-growing populations, reddening of environmental noise increases extinction risk, whereas for blue-shifted, fast-growing populations, reddening of environmental noise decreases extinction risk (Cuddington & Yodzis 1999; Heino, Ripa & Kaitala 2000; Schwager, Johst & Jeltsch 2006). In particular, for populations that are already red-shifted, becoming less red-shifted is associated with decreased extinction risk. Because most populations typically monitored by ecologists are red-shifted (Inchausti & Halley 2002), and because we have shown that spectra of some environmental variables are getting bluer and this is correlated with bluer population spectra, our results suggest that the observed shifts may broadly contribute to decreased extinction risk. This conclusion is in the context of the univariate population models considered here; the same patterns may not hold for stage-structured or spatially structured models or models with other elaborations. Also, numerous other factors contribute to extinction risk, including aspects of environmental signals such as their mean and variance, and direct human factors such as habitat destruction and population exploitation. Future research quantifying the relative contributions of these and other factors to total extinction risk under different scenarios of population dynamics would be useful.

image

Figure 5.  The relationship between noise and population spectral colour and extinction risk in the stochastic Ricker model of Fig. 1 (Methods). The results suggest that for red-shifted, slow-growing populations, reddening of environmental noise increases extinction risk; in contrast, for blue-shifted, fast growing populations, the opposite is true. Each individual line is labelled by the fixed growth rate (r) value used for all points on the line; line colour corresponds to environmental noise colour (the value of ρ used; see Methods). For 0.7 < r < 1.9, extinction risk was ≤0.0165 for all environmental noise colours; hence lines for these growth rate values are not visible in the plot. The results presented in this Figure are present in the literature in fragmented form (Cuddington & Yodzis 1999; Heino, Ripa & Kaitala 2000; Schwager, Johst & Jeltsch 2006)

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Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

We thank Tim Coulson, C. David L. Orme, F. J. Frank van Veen, Owen R. Jones, Owen L. Petchey, Ana I. Bento, Aurelio F. Malo, Joaquín Hortal, Miguel-Á. Olalla-Tárraga, Georgina M. Mace, Alex Lord, Lisa Signorile, Lawrence Hudson and two anonymous referees for useful suggestions. The work was partly supported by a NERC PhD studentship grant NE/G523447/1 and by the NERC Centre for Population Biology working group on ‘Predicting the effects of climate change on natural populations and communities’.

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  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information
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Supporting Information

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Supporting Information

Appendix S1. Effect of environmental noise on two univariate population models.

Appendix S2. Validation of the CRU data set with GHCN data.

Appendix S3. GPDD filtering process and filtered list of species.

Appendix S4. List of filtered GPDD populations.

Appendix S5. Additional methods.

Appendix S6. Results for all climate variables.

Appendix S7. Why summer?

Appendix S8. Extinction risk in other univariate models.

[Correction added after online publication 6 April 2011: legend for Appendix S8 added.]

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