## Introduction

A pervasive feature of the long-term ecological time series is that their temporal variability of population abundance typically increases with the length of the census (Pimm 1991; Inchausti & Halley 2001, 2002). This may have important consequences for the estimation of extinction risk (Lawton 1988), although its expected consequences have been subject to debate (Ripa & Lundberg 1996; Cuddington & Yodzis 1999; Halley & Kunin 1999).

The increasing temporal variability of ecological time series may be associated with ‘spectral reddening’ (Inchausti & Halley 2002), which occurs when power is concentrated at low frequencies of the power spectrum, indicating an overriding importance for long-term trends in the variability of time series data. This may be quantified by the spectral exponent, ν, estimated by the magnitude of the *downward* regression slope on a log-log scale. For reddened processes, ν is positive, in a white noise process it is around zero, while for ‘blue noise’ process it is negative.

Ecologically, temporal variability may be thought of in terms of two contrasting processes: white noise and random walk. White noise describes a completely uncorrelated random process fluctuating within a well-defined range of values and has a spectral exponent of zero, indicating that events at all frequencies approximately explain similar amount of the variability of the time series. By contrast, a population executing a random walk progressively accumulates random, uncorrelated increments of population abundance over time, leading to a linear increase of population variability with the length of the data series (Halley & Kunin 1999) and an expected spectral exponent of +2. Long-term (> 30 years) time series of animal populations censused annually typically have reddened dynamics with an overall mean value of +1·02 (SE = 0·04) (Inchausti & Halley 2001, 2002), which is ‘halfway’ between white noise and the random walk, sharing some properties of each. Population dynamics may become ‘reddened’ in several ways, by inherited redness from variation in the environment (Lawton 1988), through species interactions (Miramontes & Rohani 1998; Ripa, Lundberg & Kaitala 1998), stochastic delayed density dependence (Kaitala & Ranta 1996), spatially explicit (White, Begon & Bowers 1996) or age-structured (MacArdle 1989) dynamics. However, in an experimental study of single isolated populations, Petchey (2000) found that all populations had reddened spectra regardless of the colour of the environmental fluctuations, and concluded that it may not be necessary to explain reddened dynamics with extrinsic influences.

In this paper, we examine the effect of various general population-level processes on the spectral colour of the resulting time series. In other words, we explore some within-population mechanisms that generate reddened spectra similar to the ones found for wild animal populations. We assume that the environmental fluctuations are white noise at the annual time scale of population dynamics (e.g. Lovejoy & Schertzer 1986; Swanson 1998). Spectral analysis of historical meteorological data (Pelletier 1997) shows that while environmental variability is strongly reddened on short time scales (ν = 1·37, for days to months), the reddening is much weaker over timescales of interest (ν = 0·37, for seasons to hundreds of years) when examining the dynamics of ecological populations. We considered combinations of environmental variation, demographic stochasticity, intensity of density dependence and age structure to examine the relative importance of these factors in producing the type of reddened variation observed in natural populations. Because natural populations are rarely subject to complete censuses, it is essential to consider the influence of measurement error if we are to compare the degree of spectral reddening generated by the models with the values estimated from long-term time series of wild populations.