## Introduction

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.

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.