## Introduction

Vertebrate populations often show synchronous population fluctuations in space. It has been well known for several decades (Elton 1924) that the populations of some species such as the snowshoe hare (*Lepus americanus*) or the arctic lemming (*Lemmus norvegicus*) in many areas show similar fluctuations at sites separated by hundreds of kilometres (Stenseth *et al*. 1999). Recently, comparative analysis involving several species living in the same area have in general demonstrated large interspecific variation, even among closely related species, in the spatial scaling of the synchrony in population fluctuations (Lindström, Ranta & Lindén 1996; Paradis *et al*. 1999, 2000; Cattadori, Merler & Hudson 2000). We still do not understand the mechanisms generating such interspecific differences in synchrony.

Analyses of linear or log-linear models have shown that the autocorrelation in the fluctuations in log size of two populations with no migration between them equals the autocorrelation in the environmental noise (Moran 1953; Lande, Engen & Sæther 1999). This also applies for between-years changes in log population sizes. Some evidence suggests that such a Moran effect can synchronize population fluctuations (Grenfell *et al*. 1998). Lande *et al*. (1999), Bjørnstad & Bolker (2000), Kendall *et al*. (2000), Ripa (2000), Engen (2001) and Engen, Lande & Sæther (2002a,b) extended these results to also include migration and showed that even short-distance migration may contribute substantially to an increase in the scale of population synchrony, especially if population-density regulation is weak. Extensive simulation studies of discrete time models have supported these results (Ranta, Kaitala & Lundberg 1997b; Ranta, Lindström & Helle 1997a). Thus, these analyses demonstrate that factors affecting environmental variation in space and time as well as processes acting within the populations themselves (e.g. strength of density regulation) influence the strength and scaling of synchrony in population fluctuations. Geographical patterns in population fluctuations often are studied using the Mantel test (Koenig 1999). Two matrices are computed: a distance matrix of the distances between all possible pairs of sites and a correlation matrix where the elements are the correlation coefficients of annual variation in population sizes at all possible pairs of study sites. A standard way to present the data is to plot the correlation coefficient computed from pairs of time-series observed during the same period against the distance between the different pairs of study plots. How rapidly the correlation decreases with distance indicates the spatial scaling of the synchrony in the population fluctuations. Although several modifications of this method have been suggested (Bjørnstad, Stenseth & Saitoh 1999a; Peltonen *et al*. 2002), a major problem with these non-parametric approaches is to link theory with data (Bjørnstad, Ims & Xavier 1999b). Another problem with the kind of non-parametric approach described above is that demographic stochasticity in population fluctuations is not taken into account. Demographic stochasticity is dominating compared to environmental stochasticity when local populations are small (Lande, Engen & Sæther 2003), and has by definition (Engen, Bakke & Islam 1998) no spatial scaling. Consequently population synchrony, which can only be affected by environmental stochasticity, is expected to increase with population size.

Here we provide a model that utilizes knowledge about the demographic variance and incorporates the possibility of a decrease of synchrony with population size due to demographic stochasticity. Our approach is especially useful for fully censused populations in which we can ignore sampling errors in population estimates and work with the full likelihood function for the problem. One advantage of this, in addition to the possibility of including demographic stochasticity, is that we can include any observed population size in the analysis and do not need to pay attention to the fact that the time-series only partly overlap in time. Rather than using raw estimates of correlations we work directly with the likelihood function using the complete set of residuals obtained after fitting models to the local populations.

The correlations then enter the model through the parameters of a spatial autocorrelation function. A third advantage with our approach is that we can assume that the local populations possibly have different dynamics. We do this by fitting stochastic population models to each local population, and include in the spatial analysis only the environmental component of the residuals, correcting for expected change in population size for different values of population sizes the previous year. Because theoretical as well as empirical analyses have focused previously on the synchrony of (log) population sizes or between years change in (log of) these (see Ranta *et al*. 1997b and Bjørnstad *et al*. 1999), we also include a theoretical analysis of the spatial scaling of residuals in a continuous homogeneous model in space and time with migration.

We apply this method to estimate the pattern of synchrony in the fluctuations of breeding populations of the Continental Great Cormorant *Phalacrocorax carbo sinensis* in Europe. We focus especially on how a large-scale climate phenomenon, the North Atlantic Oscillation (Hurrell *et al*. 2003), affects the spatial scaling of population synchrony.