### Summary

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References
- 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

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References
- 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.

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.

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.

### Results

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References
- 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. *N*_{total} = 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 | −0.055 | 0.659 | −0.040 | 0.508 | −0.230 | 0.062 |

Spring | 0.060 | 0.590 | 0.226 | 0.538 | −0.065 | 0.123 |

Summer | 0.312 | 0.021 | 0.294 | 0.406 | 0.306 | 0.207 |

Autumn | −0.179 | 0.146 | −0.179 | −0.105 | −0.160 | −0.250 |

Annual | 0.135 | 0.049 | 0.191 | 0.206 | 0.299 | 0.202 |

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).

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

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References
- 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.

#### 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.

### Acknowledgements

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References
- 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’.

### Supporting Information

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References
- 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 S8.** Extinction risk in other univariate models.

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

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