## Introduction

The regional and local spatio-temporal variation of the environment affects mean levels as well as variability of population demographic processes, such as survival, recruitment or reproduction (Stenseth 1999; Stenseth *et al.* 2002; Ågren, Ehrlén & Solbreck 2008). An understanding of the links between population dynamics and environmental variability, combined with information on how these factors change over time, is necessary to understand and predict population dynamics and viability in changing environments (Gotelli & Ellison 2006) and to scale up from local processes to large spatial scales (Jeltsch *et al.* 2008).

Weather variation has been identified as a main driver of variation in demographic processes (Lima *et al.* 2001; Stenseth *et al.* 2002; Jönsson *et al.* 2009), and the population size of some species tracks weather variability (Jongejans *et al.* 2010). Moreover, while processes driving inter-annual variation in abundance may act at regional scales, organisms also interact with each other, and with their substrate at local scales (Ovaskainen & Cornell 2006). It has even been shown that variation in local conditions can buffer the population response to weather variation (Davison *et al.* 2010). A current challenge is to disentangle the relative importance of environmental drivers at different spatial scales on the inter-annual variation in population abundance.

Different approaches have been used to model population dynamics driven by regional and local processes. Autoregressive models are widely used to analyse population time series by statistically linking the population autocorrelation with environmental variability (Royama 1992; Inchausti & Halley 2003; Stenseth *et al.* 2004). This approach, however, makes it difficult to separate a lagged density-dependent structure of the population from a lag in the response to the environmental variability (Turchin & Berryman 2000). Another approach is by mechanistic demographic matrix modelling (Caswell 2001), which has recently become complemented with submodels for the relationship between the demographic parameters and environmental variables (Keith *et al.* 2008; Davison *et al.* 2010; Hunter *et al.* 2010; Toräng, Ehrlén & Ågren 2010). Coulson *et al.* (2008) even combined these approaches by averaging age-specific survival and recruitment rates in the estimate of the population growth rate, and allowing this relationship to vary more realistically over time. We used an approach which is applicable for inconspicuous species such as our study species: a model simple enough to be fitted to observational time series data, but sophisticated enough to contain parameters that have a biological meaning, such as the population growth rate (Klein *et al.* 2003). This approach has been used to model the effect of environmental variability on the rates of population or metapopulation dynamics of beetles (*Callosobruchus* spp.) and of a rodent (*Cynomys ludovicianus*) (Bull & Bonsall 2008; Snäll *et al.* 2008).

Inferences and predictions about population dynamics require proper accommodation of environmental variability and data uncertainty on different spatial scales (Clark & Bjørnstad 2004). The hierarchical Bayesian framework allows accounting for different sources of variation across different spatial scales in population models (Clark 2005). It has recently been applied to the demographic modelling approach to estimate demographic parameters, e.g. the reproductive rate (Evans, Holsinger & Menges 2010), and to model population or metapopulation dynamics (Bull & Bonsall 2008; Snäll *et al.* 2008).

Population models can inform us about the trend and long-term viability of populations. A suitable estimator of species trend and long-term viability is the stochastic growth rate, log λ_{S} (Lewontin & Cohen 1969; Tuljapurkar & Orzack 1980). The parameter accounts for the effect of inter-annual variation in population size on population viability, as the mathematical definition includes variance in year-specific growth rate for the study period. If log λ_{S} is < 0, the population is bound to decline, while if it is ≥ 0, the population is viable. The confidence in statements about the long-term viability of populations (log λ_{S}) largely depends on the variability accounted for when estimating the parameter. Methods exist to estimate the effect of variability in demographic processes on the stochastic growth rate and its uncertainty (Caswell 2001; Evans, Holsinger & Menges 2010; Hunter *et al.* 2010). However, we are not aware of estimates of the full probability distribution of log λ_{S} in which the variability across spatial scales is accounted for, and where this distribution is contrasted with estimates based on other methods.

We present and evaluate a hierarchical Bayesian model for the population dynamics of the inconspicuous epixylic moss *Buxbaumia viridis*. We hypothesize that the dynamics are affected 1) by regional processes, in this case, inter-annual variation in autumn frosts and spring precipitation and temperature, by regulating the growth rate of the species; and 2) by the local variable preceding year’s population abundance, as it constitutes the parent generation, and by the variable dead wood, as it sets the resource amount. The hypotheses are based on previous studies of the study species (Wiklund 2002, 2003; Wiklund & Rydin 2004) and of a congeneric species (Hancock & Brassard 1974). We also provide an assessment of the local long-term viability of the species based on the full probability distribution of log λ_{S}, and we provide an estimate of the confidence in this assessment. Finally, we compare this full probability distribution with estimates of log λ_{S} based on other methods.