## Introduction

Ecologists are increasingly recognizing the importance of studying the characteristics of environmental fluctuations (Inchausti & Halley 2003) and synchrony (Hastings 2010). Theoretical studies have shown that the autocorrelation of environmental fluctuations and synchrony have significant effects on population dynamics and extinction risks (Ripa & Lundberg 1996; Liebhold, Koenig & Bjørnstad 2004; Lögdberg & Wennergren in press). Theoretical studies have also demonstrated how autocorrelation and synchrony of ecological systems may be modelled by the spectral representation of time series (Halley 1996; Cuddington & Yodzis 1999). In this study, we demonstrate how this approach, combined with Bayesian inference, can be used for ecological time-series analysis. We show how model parameters may be estimated and how different hypotheses of fluctuation may be compared. Consequently we use analysis of autocorrelation and synchrony to make inference about processes that influence the dynamics observable in ecological datasets.

Ecological time-series data are usually positively autocorrelated (Steele 1985; Inchausti & Halley 2002), denoted ‘reddened’ because the spectral representation is dominated by low frequencies. Autocorrelation is commonly also present over multiple scales (Pimm & Redfearn 1988) and consequently first-order autoregressive (AR) models may neglect important aspects of autocorrelation. Consequently, environmental fluctuations are commonly analysed as Flicker noise, also known as ‘one over f’ (1/f) noise (Halley 1996). This assumes a fractal dynamic with self-similarity in the autocorrelation at different temporal scales and is described by a single-scale free colour parameter *γ* given by a powerlaw relationship of the power spectral density function (PSDF) of a time series with . Time series with *γ* = 0 have no autocorrelation (denoted ‘white noise’) whereas time series exhibiting random walk properties have *γ* = 2. Figure 1 (panels a and b) illustrates the difference between white and autocorrelated dynamics. Environmental fluctuations are generally estimated at *γ* ≈ 1, that is, ‘pink noise’ (Halley 1996). Pimm & Redfearn (1988) argued that ecological systems with similar dynamics are likely driven by the underlying environmental fluctuations.

However, internal processes may also cause autocorrelated dynamics of ecological systems. If studying individuals as part of a population, the growth rate of the individual may be autocorrelated by factors at the individual level (e.g. pathogens, pests, microclimatic factors). Similarly, local processes may cause autocorrelated dynamics if considering subpopulations as part of a metapopulation. To conclude that reddened dynamics of ecological processes are caused by some global factor (e.g. environmental fluctuation), it is therefore important to consider synchrony between time series. Synchrony is often defined as the co-fluctuation of two or more time series. Figure 1 (panels c and d) illustrates the difference between synchronized and non-synchronized dynamics. The synchrony of local populations is commonly attributed to either a classical ‘Moran effect’ where nearby patches are all influenced by a severe event, which makes their inherent local oscillations coordinated, or by a less strict interpretation of the ‘Moran effect’ where the populations are directly and continuously driven by fluctuations of environmental factors such as climatic variables (Heino *et al*. 1997). Furthermore, the synchrony of local populations may also arise as a result of between-patch dispersal (Abbott 2011) or by trophic interactions with other species already being synchronized (Liebhold, Koenig & Bjørnstad 2004). Individuals of a population may also be more or less synchronized in, for example, their growth. This synchrony is mainly attributed to the less strict interpretation of the ‘Moran effect’ where the growth rates are driven by environmental factors. If considering individuals as part of a population, synchrony may be caused by, for example, outbreaks of pathogens or pests within the population and common environmental factors. Different processes may be of different importance at different time-scales and synchrony may be more apparent at time-scales where global external factors are more influential. Further, when comparing two or more internally synchronous groups, the groups may be considered to actually co-fluctuate as one synchronous group (Fig. 1e) or as two different groups (Fig. 1f).

In this study, we present a novel method to analyse fluctuations and synchrony from ecological time-series data. We implement the method on dendrochronological data of the pedunculate oak (*Quercus robur*) and demonstrate how it may be used to highlight the importance of factors at the local, individual (population) level or by global factors. The pedunculate oak is one of the most important tree species in Europe for invertebrates associated with veteran trees (Jansson *et al*. 2009). The annual growth of oaks is known to be a result of a complex interaction of abiotic and biotic factors, for example, precipitation, temperature and defoliating insects (Thomas, Blank & Hartmann 2002). Climatic factors are expected to be both temporally autocorrelated and have a synchronous effect on tree growth. Insect outbreak may also cause synchronization (Bjørnstad *et al*. 2002) but the temporal autocorrelation caused by this is likely to be acting on shorter time-scale and potentially be cyclical rather than fractal. Hence, if we find that the fluctuation in annual growth is synchronous as well as autocorrelated similarly to climatic variables (i.e. Flicker noise with *γ* ≈ 1), we may expect that climatic variations are the main drivers.

We may also expect that years of particularly unfavourable conditions are followed by a recovery period of low annual growth. Such periods of low growth cause a reddened spectrum as well as synchronization if the unfavourable conditions are affecting the whole population. If trees cannot recover, such bad years may lead to a prolonged decline in annual growth and ultimately synchronized mortality of individual oaks (Andersson, Milberg & Bergman 2011). A number of synchronized mortality events for European oaks (*Quercus*) have been documented throughout Europe, the earliest dating back to 1739–1748 in north-eastern Germany (Thomas 2008). The causes of the synchronized mortality events seem to vary, from large-scale factors as climate to defoliation by insect larvae. One such event took place in Sweden in *c*. 2002–2007. To investigate if the trees that died and those that did not were historically influenced by different factors, we compare annual growth of trees that died with trees that managed to recover. Using hierarchical Bayesian modelling, we compare the posterior distributions of the estimated autocorrelation parameters. This approach is beneficial in that we include the uncertainty at the individual level (i.e. in *γ* of individual trees) when we estimate the population-level parameters that are most relevant for comparison. If the oaks that died exhibited a higher degree of autocorrelation in annual growth, we may expect that these trees historically have had longer periods of recovering after years of bad conditions or been particularly sensitive to different environmental factors. We compare two competing models: *M*_{1} – all trees come from the same synchronous group, and *M*_{2} – live and dead trees are different and co-fluctuate differently. If *M*_{1} is more probable, we may conclude that live and dead trees have historically been influenced by the same environmental factors. If instead *M*_{2} is more probable, we may conclude that they have been sensitive to different factors. Recent studies suggest that phase-shift analysis is a promising approach to model synchrony (Vasseur 2007; Keitt 2008; Hastings 2010; Lögdberg & Wennergren in press). In systems with cyclic behaviour, the phases may be easily identified (Hastings 2010). Yet for many ecological systems, including tree growth, we do not expect cyclic dynamics. We will in our study show how phase-shift analysis can be combined with Bayesian inference and used for analysis of synchrony in systems without apparent oscillations.

The purpose of this paper is to present a methodology for analysis of fluctuations and synchrony based on spectral analysis and Bayesian inference. We apply these methods on data of oak tree growth and demonstrate how we may extract valuable insight into the dynamic behaviour by studying spectral colour of time series and synchrony between them. We further demonstrate how this approach may be used to distinguish the degree of synchrony at different time-scales. We employ hierarchical Bayesian modelling, an approach that is being increasingly used within ecological studies with the benefit that it allows for parameter uncertainty at different levels of the model (Clark 2005).