## Introduction

All natural populations are influenced by stochastic variation in the environment. In the ecological literature, this stochasticity is commonly assumed to be white noise, meaning that there is no temporal correlation, so that values at two different time steps are completely independent of each other (Chatfield 2003). Recent evidence does indicate that there may be various degrees of correlations in time in the environmental fluctuations (Halley 1996). Steele (1985) suggested that the structure of these autocorrelations provides an important property of environmental variables affecting ecological processes. For instance, the pattern of environmental fluctuations is fundamentally different in marine and terrestrial environments. On land, the variance of most abiotic factors remains uniform over longer time periods and has approximately white noise, whereas in the oceans, many environmental variables are often positively autocorrelated (Halley 1996, 2005) and thus seem more ‘reddened’ than in terrestrial environments (Cyr & Cyr 2003; Vasseur & Yodzis 2004).

The pattern of autocorrelation in the environmental noise can be used not only to characterize differences among habitats, but is also important when making inferences about temporal changes in environmental conditions. For instance, the IPCC Fourth Assessment Report (IPCC 2007) indicates that the Earth's climate is likely to change dramatically over the coming decades, which may alter the autocorrelation structure in several climate variables (Halley 2009). Recent evidence indicates that the pattern of temporal variation in sea surface temperature has changed during the last century (Swanson, Sugihara & Tsonis 2009). Human impact can also influence the pattern of the environmental autocorrelations substantially (Ruokolainen *et al*. 2009).

However, how variation in environmental covariates affects population dynamics is complicated because the variance in population size is determined by the combined effect of environmental stochasticity and strength of density-dependent population regulation, and produce autocorrelations in the population fluctuations (Royama 1992; Ranta, Lundberg & Kaitala 2006). Such temporal (or spatial) autocorrelation is in general the correlation between two variables measured at different times (or locations). This article deals with temporal autocorrelation in the environmental noise, which in a structured population is a noise matrix. Autocorrelation in the noise must not be confused with autocorrelation in population size. If there are no autocorrelations in the noise (white noise), then there may still be large autocorrelations between sizes and of age classes *i* and *j* at times *t* and *t*+*h*.

Such environmental autocorrelation may influence the mean and variance of population size (Roughgarden 1975; Heino, Ripa & Kaitala 2000; Tuljapurkar, Horvitz & Pascarella 2003; Wichmann *et al*. 2005; Tuljapurkar, Gaillard & Coulson 2009; Morris *et al*. 2011). In density-independent populations with no age structure, environmental autocorrelation has no impact on the long-run growth rate of the population. Writing for the population size at time *t*, the long-run growth rate is the limit of as *t* approaches infinity (Dennis, Munholland & Scott 1991). From the relation , it can be seen that the long-run growth rate is the expected value of for a stationary noise process even if there are autocorrelations because expectations in general are additive. Thus, only the environmental variance, here defined on absolute scale as (Lande, Engen & Sæther 2003), influences the long-run growth rate, which can be approximated by , where *r* = ln λ is the deterministic growth rate defined by . However, in age-structured models, environmental autocorrelation also affects the long-run growth rate (Tuljapurkar 1982; Caswell 2001; Doak *et al*. 2005; Tuljapurkar & Haridas 2006). The impact of this on the expected log population size over a time interval *t* is proportional to *t*, whereas the standard deviation is proportional to the square root of *t*. Hence, for extremely long time intervals, the effect on the mean becomes more important than the variance term even if the former is relatively small. However, for intervals, say 20–200 years, the situation may be the opposite, depending on the magnitude of these terms.

Tuljapurkar & Haridas (2006) analysed the effects of environmental autocorrelation on the dynamics of age-structured populations including second-order terms in the long-run growth rate due to environmental autocorrelation. First, there is an effect of approximately of the environmental variance in the age-structured dynamics ( also here defined on the absolute scale not including the factor ) as in the case of no age structure. Typical values of are on the order of 0·01 per year (Sæther & Engen 2002; Lande, Engen & Sæther 2003; Sæther *et al*. 2005). Second, there is also an effect due to temporal autocorrelations and cross-correlations among elements in the projection matrix (Doak *et al*. 2005; Engen *et al*. 2005, 2007; Haridas & Tuljapurkar 2005; Morris *et al*. 2011; Sæther & Engen 2010). This term is usually negative and may be smaller or larger in magnitude than depending on the mean projection matrix and the strength of temporal autocorrelations. Tuljapurkar & Haridas (2006) found that for large environmental autocorrelation, it may even be substantially larger in magnitude than .

The effects of temporal autocorrelations on population dynamics will strongly influence the degree to which the population is buffered against changes in the environment, for example, due to expected changes in climate (IPCC 2007). In general, the effects of environmental stochasticity on population dynamics tend to depend on life history (Sæther *et al*. 2005; Morris *et al*. 2008). For instance, Morris *et al*. (2008) found using elasticity analysis of the stochastic growth rate that responsiveness in population dynamics to a change in the environment was higher among short-lived than among long-lived species, indicating the buffering capacity towards environmental variability increases with life expectancy. However, this effect will be influenced by the correlation structure of the environmental components of age-specific vital rates (Doak *et al*. 2005; Engen *et al*. 2005; Haridas & Tuljapurkar 2005; Sæther & Engen 2010).

In this article, we base our analysis on the result of Cohen (1977, 1979) that any linear combination of age classes asymptotically grows exponentially with the same long-run growth rate as for the total population size. Our approach employs the total reproductive value of the population. The concept of reproductive value was originated by Fisher (1930) and has proved useful in deterministic (Crow & Kimura 1970; Charlesworth 1994) as well as in stochastic (Caswell 1978; Tuljapurkar 1982; Lande, Engen & Sæther 2003; Engen *et al*. 2007, 2009) age-structured population modelling.

Engen *et al*. (2007) derived an estimation method for the long-run growth rate based on yearly total reproductive values of the population, assuming temporally uncorrelated environments. The method was applied by Sæther *et al*. (2007) to analyse the stochastic dynamics of an age-structured Moose (*Alces alces*) population. Here we show that this estimation method for the long-run growth rate is valid also when the environmental noise is temporally correlated, because the mean of differences in log reproductive values includes the term due to temporal autocorrelation. This enables us using the concept of individual reproductive value defined by Engen *et al*. (2009) to drive a simple transparent interpretation of the effect of environmental autocorrelations on the long-run growth rate. We then show how this effect, which is included in the estimates previously obtained by the methods of Engen *et al*. (2007) and Sæther *et al*. (2007), can be isolated and estimated by a simple statistical method that only requires computation of the dominant left and right eigenvectors of the estimated mean projection matrix. This use of reproductive value makes the results more easily interpreted than by the previous approaches of Tuljapurkar (1982). We also estimate the variance of future population sizes and show how the influence of environmental autocorrelation can be estimated from data and how to test its statistical significance. Our derivation leads to a univariate diffusion approximation for the population size with two parameters, the long-run growth rate of the population and the environmental variance. Finally, we demonstrate the application of this method by analysing data on four mammal species, Bighorn sheep (*Ovis canadensis*), Columbian ground squirrel (*Urocitellus columbianus*), Red deer (*Cervus elaphus*) and Yellow-bellied marmot (*Marmota flaviventris*). We show that the small effect of environmental autocorrelations on primate population dynamics (Morris *et al*. 2011) also applies to the species included in the present analyses. This provides a general approach to estimate how environmental autocorrelations affect the autocorrelation structure (e.g. noise colour) of fluctuations in size of age-structured populations.