Hierarchical Bayesian estimation of the population viability of an epixylic moss

Authors


Correspondence author. E-mail: alejandro.ruete@slu.se

Summary

1. Understanding the variation in population abundances requires accounting for the environmental variability and uncertainty on different scales. We developed and evaluated a Bayesian hierarchical model for the inter-annual variation in population abundance of the epixylic bryophyte Buxbaumia viridis. The model accounts for spatio-temporal variability on two spatial scales. We used data on population abundance and on the weather variables at regional level collected between 1996 and 2003, and data on dead wood amount collected between 1996 and 2008. We also provide a Bayesian estimate of the population viability, specifically the population stochastic growth rate (log λS), which accounts for natural variability and uncertainty.

2. Previous estimates of population viability did not account for uncertainties in a satisfactory way. First, point estimates of log λS cannot, by definition, express variation. Second, the commonly used approach to estimate log λS and its confidence interval underestimates uncertainties. The approach aims to estimate the mean of log λS, with the confidence interval representing the uncertainty in the estimate of this mean. The interval does not reflect the natural variation and uncertainty.

3. We estimated a probability distribution of log λS, where the probability distributions of the year-specific growth rates (log λy) are accounted for. The species is likely to decline under current environmental conditions. Based on the probability distribution of log λS, we estimated this risk to be 81%.

4. We found support for the hypotheses that the population dynamics are driven by autumn frosts, by spring precipitation and temperature (regional variables), and by the preceding year’s population abundance (local variable).

5.Synthesis. Statements about the viability of populations should not be based on point estimates of log λS. Instead, the full probability distribution of log λS should be used, which explicitly accounts for the hierarchically structured natural variability and uncertainty. This distribution allows estimating the risk for a population decline, or providing an estimate of the confidence in a statement about a decline. This quantitative information can be weighed against other interests. We expect this Bayesian approach to be especially useful in the viability analysis of natural populations experiencing environmental variability.

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 λS (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 λS 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 λS) 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 λS 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 λS, and we provide an estimate of the confidence in this assessment. Finally, we compare this full probability distribution with estimates of log λS based on other methods.

Materials and methods

Study species

Buxbaumia viridis (DC.) Moung. & Nestl. (Buxbaumiaceae) is a dioecious moss that grows on strongly decayed dead wood patches in coniferous or deciduous forest (Nyholm 1979). The gametophyte is minute and difficult to find in the field, but the sporophyte varies in length from 7 to 25 mm (Möller 1923) and can hold 6 million spores (Wiklund 2002) with a mean diameter of 12 μm (Boros & Járai-Komóldi 1975). Little is known about the life cycle of B. viridis. In eastern-central Sweden, sporophytes usually emerge in October, and most spores are released in middle June the following year (K. Wiklund personal observations). Based on what is known about the life cycle of the congeneric B. aphylla in North America (Hancock & Brassard 1974), we believe that the life cycle of B. viridis is annual or paucennial—capable of completing the cycle in one year or surviving a few years with reduced or arrested growth, depending on weather conditions. A population persists on a patch of dead wood in the form of either perennial protonema (the first stage in the bryophyte life cycle) (Hancock & Brassard 1974), or brood bodies (groups of cells which are specialized for survival under harsh conditions), or by immigration of new individuals by spores. Brood bodies might rescue populations from local extinction during dry periods (Goode, Stead & Duckett 1993). However, as the dead wood patches are temporary (Söderström 1988), landscape-scale persistence relies on recurrent spore dispersal and establishment (Wiklund 2003). Buxbaumia viridis has a circumboreal distribution, but the population size is small wherever it occurs (Hallingbäck 2002). The species is classified as ‘Vulnerable’ in Europe (European Committee for Conservation of Bryophytes 1995).

Study area

The 7-ha study area is situated in eastern-central Sweden (Vipängen, Uppsala, 59° 49′N, 17°39′ E), 30 m a.s.l. The boreo-nemoral forest is dominated by Picea abies followed by Populus tremula, Pinus sylvestris, Sorbus aucuparia, Betula pendula and Salix caprea, and is surrounded by mainly arable lands. The annual precipitation is 600–700 mm, and the length of the growing season is approximately 190 days (Sjörs 1999).

Empirical data

We first surveyed the study area for dead wood patches occupied by B. viridis in October–November 1995. Next, we established six 25 × 25 m permanent plots with the aim of representing the variation in density of occupied dead wood patches in the forest. In each plot, we mapped all patches of dead wood > 10 cm in diameter, and those with 5–10 cm diameter having a length of > 50 cm. For each dead wood patch, we recorded the stage of decay in eight classes, based on the hardness and outline of the wood patch (Söderström 1988; Appendix S1 in Supporting Information), and wood taxon (coniferous or deciduous). Every June in the following eight years, 1996–2003, and again in 2008, we counted mature sporophytes on each occupied dead wood patch in all plots. In June 2008, we mapped all the dead wood patches a second time using the same methods as in 1995, whereby we had complete data on dead wood patches from two surveys. The data from this last survey were utilized in the model evaluation. Data on sporophyte abundance was summarized per plot (the local scale modelled).

We obtained daily precipitation and temperature data for the period 1995–2008 from Ultuna weather station, located 100 m north of the forest. The weather data were summarized as the number of days with temperatures below 0 °C for the period October to November, sum of precipitation (mm), and mean of monthly temperature means (°C) for the period March to June (autumn frosts, spring precipitation and spring temperature, respectively). These variables were believed to be critical in the species life cycle, as sudden freezing spells in autumn may cause high mortality of immature sporophytes (Hancock & Brassard 1974), and water stress in spring may reduce the survival of the sporophytes in the maturation phase (Wiklund & Rydin 2004).

Statistical modelling

The time series on counts of mature sporophytes allowed us to develop a model for the yearly abundance of B. viridis sporophytes as a function of local and regional conditions. We next detail the full start model, but the final model presented is the result of a selection procedure. All variables and parameters are also described in Table S1.

The deterministic skeleton of population regulation took the form of an underlying exponential population growth model, which is reasonable for bryophytes (During 1979),

image(eqn 1)

We linearized the model and applied the hierarchical generalized linear modelling framework (Gelman & Hill 2007): we assumed that the local-scale (i.e. plot-level) abundance of sporophytes followed a Poisson distribution with mean μy,p, specifically,

image(eqn 2)

where

image(eqn 3)

where Ny,p is the observed abundance of sporophytes in year = 2,…, 8 (1997–2003) in plot = 1,…, 6, and λy is the regional, year-specific population growth rate, which was modelled further as a function of weather (eqn 5). Ny−1,p is the observed abundance of sporophytes in the preceding year. The ‘effect-size’ parameter β1 determines the proportion of spores produced by the sporophytes in the preceding year that gives rise to sporophytes in the focal year. We added 10−3 to avoid the term being equal to zero for technical and biological reasons. Technically, the logarithm of Ny−1,p = 0 is not defined whereby log(μy,p) and Ny,p would not be defined. Biologically, Ny,p may very well be > 0, although Ny−1,p = 0 as a result of immigration. The constant chosen is a number as close as possible to zero in order not to bias the sporophyte count and not to affect the estimate of the parameter β1. The overdispersion term εy,p follows a normal distribution with mean 0 and SD σ (‘∼Normal (0, σ)’ henceforth). The term models excess variation compared with that assumed by the Poisson distribution (mean = variance). It also scales the abundance of the preceding year relative to all other sources of spores that may be the origin of the sporophytes in the focal plot and year. For example, if εy,p > 0, there are more spores which develop into sporophytes than expected, given the assumed Poisson distribution and given the amount of spores produced by sporophytes in the focal plot in the preceding year (Ny−1,p). These additional spores may be spores older than 1 year in the spore bank or spores immigrating from outside the plot. The term also includes effects of unknown local processes, e.g. intra- or interspecific competition, or measurement error (residual variation). The variable Dy,p is the number of suitable dead wood patches. The values of the variable for the years 1996–2003 were estimated by a submodel for dead wood dynamics (see Appendix S1). The reason was that we only had data from two complete counts of dead wood patches (1995 and 2008). As this submodel was fitted jointly with the sporophyte model, the uncertainty in estimated dead wood amount affected the uncertainty of the parameter estimates of the sporophyte model. However, for technical and biological reasons, we cut the link in the opposite direction: sporophyte abundance does not affect the amount or dynamics of dead wood. The ‘effect-size’ parameter β2 regulates how many sporophytes are to be found per dead wood patch. We subtracted Dy,p with its mean for years = 1,…, 8, inline image, to reduce parameter correlations in the estimation process (Gelman et al. 2004). We defined all dead patches of decay classes 4–8 (Appendix S1, Table S1.1) as suitable because the sporophyte rarely occurred on less-decayed wood (Wiklund & Rydin 2004).

Moreover, we modelled the logarithm of the regional, year-specific growth rate, log λy, as a linear function of number of autumn frosts (F), spring precipitation (P) and spring temperature (T). Specifically,

image(eqn 4)

where σλ quantifies the interplot variation of log λy, unknown regional processes, such as unexpected interaction effects between weather variables, or measurement error (residual variation), and where

image(eqn 5)

where ϕ0 is an ‘intercept’ parameter and ϕi | i = 1, 3 are ‘effect-size’ parameters. The weather variables were subtracted with their means (inline image = 12.3 days, inline image = 166.8 mm, and inline image = 7.6 °C) to reduce parameter correlations in the estimation process (Gelman et al. 2004), and divided by their SD (σF = 6.8 days, σP = 33.9 mm and σT = 0.97 °C) to allow comparison of the relative importance of the ‘effect-sizes’ parameters ϕi | i = 1, 2, 3. The parameters to be estimated were βi, εy,p, ϕi, σ and σλ, of which the last two are so-called hyperparameters (Gelman & Hill 2007).

We applied the Bayesian modelling approach because it is convenient for fitting complex hierarchical models with a formal mathematical treatment of the natural variability and data uncertainty (Ellison 2004; Gelman et al. 2004). One of the features of a Bayesian model is that it provides the full probability distribution of the model parameters, the posterior distributions, given the data and a priori knowledge about the parameters. The modeller may have vague prior knowledge about the parameters, and technically this is implemented as flat prior distributions for the parameters (e.g. Table S1). The estimated posterior distributions can be characterized by different statistics, e.g. the most likely value for the parameter can be identified (the mode). Also, the posterior probability distribution allows to estimate how likely it is that a parameter is above or below a certain value, e.g. the probability that an effect-size parameter is larger than 0 (Ellison 2004). In contrast, the frequentist approach aims to estimate the true value of parameters, and the p-value quantifies the probability for this ‘true’ estimate not to be different from zero.

The final model presented is the result of a selection procedure and model averaging based on the deviance information criterion (DIC henceforth), an information-theoretic criterion which is appropriate for Bayesian hierarchical modelling (Spiegelhalter et al. 2002). The DIC is equal to the posterior mean of the deviance plus two times the estimated number of effective parameters (Table 1; Spiegelhalter et al. 2002). The lower the DIC, the better the model is able to predict a new data set, and thus, the DIC penalizes for increasing model complexity just as the commonly used Akaike’s information criterion (AIC; Burnham & Anderson 2002). If no model is clearly superior to other models in a set of plausible models, model averaging is an appropriate way to describe the biology of the system, and to reduce the potential bias of a specific final model (Burnham & Anderson 2002). One model averaging approach is based on DIC weights (Brooks in the Discussion of Spiegelhalter et al. 2002), and corresponds to the Akaike weights suggested by Burnham & Anderson (2002). This means weighing the models proportional to their DIC, and presenting a weighted average of the parameters of the plausible models (Burnham & Anderson 2002; Jackson, Sharples & Thompson 2010).

Table 1.   Diagnostics for models for the population dynamics of Buxbaumia viridis
ModelLocal variablesRegional variablespDDevianceDICΔDIC w i
  1. *NULL model, does not include any explanatory variable.

  2. Variables included in the models are N, preceding year’s abundance; F, frosts; P, precipitation; T, temperature; D, dead wood amount. For each model, the estimated number of effective parameters as measure of model complexity (pD; Spiegelhalter et al. 2002), the posterior mean of the deviance (Deviance), the deviance information criterion (DIC; Spiegelhalter et al. 2002), and the difference between the model with the lowest DIC and the focal model (ΔDIC) are given. For a set of plausible models (models 1 and 2), DIC weights (wi) were used for model averaging (Brooks in the Discussion of Spiegelhalter et al. 2002).

 0– (NULL)*– (NULL)29.3138.8168.12.4
 1NF + P + T29.6134.5164.10.00.53
 2F + P + T29.3135.0164.30.20.47
 3NF + P29.5135.4164.90.8
 4NP + T30.2135.0165.21.1
 5NF + T29.5135.5164.90.8
 6NF30.1135.3165.41.3
 7NP29.5136.7166.22.1
 8NT30.3135.9166.22.1
 9N30.1136.3166.42.3
10D + NF + P + T41.5138.9216.951.2

Model fit and evaluation

For a description of parameters, symbols and the non-informative prior distributions used, see Table S1. We ran two Monte Carlo Markov chains of 200 000 iterations. Visual inspection suggested convergence of the chains after 10 000 iterations, which were discarded as ‘burn-in’. The models were fitted using the software OpenBUGS 3.0.7 (Thomas et al. 2006). For effect-size parameters, we report the probability that they are above or below 0. We also report highest posterior density intervals (HPDI henceforth), which are obtained using the hdrcde library (Hyndman 2010) for r 2.11.0 (R Development Core Team. 2010). To facilitate the dissemination of this approach by other ecologists, we have included the OpenBugs code in Appendix S3.

We evaluated the final averaged model in two ways. First, we assessed the goodness-of-fit by checking whether the data collected in 1997–2003 were included in the 50% HPDI of the abundance estimated by the model. Second, we evaluated the ability of the model to reconstruct the yearly sporophyte abundance observed in 1997–2003 and 2008 by means of projection simulation. We started the projections with data observed in year 1996, and projected the dynamics for 1997–2008 utilizing the projection estimate of the preceding year’s abundance and observed weather data as explanatory, driving variables. The simulations were iterated 20 000 times, each with an independent set of draws from the joint posterior distribution of the parameters. We then checked whether the observed data on sporophyte abundance collected in 1997–2003 and in 2008 were included in the 50% and 75% HPDIs of the simulated yearly abundance (Gelman & Hill 2007).

Stochastic growth rates

We estimated the stochastic growth rate, log λS, for the period 1996–2003, according to Lewontin & Cohen (1969):

image(eqn 6)

where inline image is the geometric mean of λy (eqn 4). This estimate accounts for the inter-annual variation in λy and for its full posterior distribution in every year, in any possible order of succession. We compared the posterior distribution of log λS with two of the most commonly used estimates of stochastic growth rate. The first was log λSP, a point estimate of log λS as in eqn 6, but only using the estimated modes of λy. The subscript P stands for ‘point estimate’. The second estimate was derived from structured population models and can be applied on any population time series (eqn 14.59 in Caswell 2001; and references therein),

image(eqn 7)

where t are iteratively simulated years that tend to T—a large-enough number of iteration for convergence to the expected value (Caswell 2001), here T = 10 000 iterations. Nt is the tth element of a uniform random sequence of all the abundances simulated for the period 1997–2003. The estimate used was based on 100 independent estimates, each with a different random sequence of abundances. We henceforth refer to this estimator as the ‘limit approximation’, shortened log λSL. Finally, we estimate approximate 95% confidence intervals for λSL (Caswell 2001),

image(eqn 8)

We report two estimates of the confidence interval, one using T = 10 000 and one using T = 100 000.

Results

The B. viridis population showed peaks in abundance in 1996, 1998, 2002 and 2008, and low levels in 2000 and 2001 (Fig. 1). According to the final averaged model (a weighted average of the Models 1 and 2, Table 1), the regional-scale weather variables had larger effects than the local-scale abundance in the preceding year on the abundance of sporophytes in a focal year—the standardized effect-size parameters for frosts, precipitation and temperature were larger than the effect-size parameter for abundance in the preceding year (Fig. 2). The variable amount of highly decomposed dead wood was not included in the final model, as including it gave a higher DIC-value (Model 10, Table 1).

Figure 1.

 Time series on observed Buxbaumia viridis abundance of sporophytes (dots). Vertical bars show the 50% highest posterior density interval (HPDI) for the estimated abundance; that is, the ability of the model to estimate the focal year abundance based on data on abundance in the preceding year and on weather data in the focal year. The dark and light areas show the 50% and 75% HPDI, respectively, of the projected sporophyte abundance. The simulations utilize the abundance observed in 1996 and weather data in the focal year. Dotted, dotted-dashed and dashed lines represent autumn frosts (F), spring precipitation (P) and spring temperature (T), respectively. The sums of the abundances among the six sample plots are shown. For plot-specific abundances, see Appendix S3.

Figure 2.

 Parameter estimates for the final model for the population dynamics of Buxbaumia viridis. The parameters are associated with the abundance in the preceding year (β1), autumn frosts (ϕ1), spring precipitation (ϕ2) and spring temperature (ϕ3). The modes (short vertical lines), 50% (thick horizontal lines) and 95% (thin horizontal lines) highest posterior density intervals are shown. To the right are the proportions of the posterior distributions being higher or lower than 0. The dashed (0) vertical line is a visual aid.

The year-specific population growth rate (λy) was negatively correlated with the number of autumn frosts (ϕ1 = −0.28; 95% HPDI: −0.77–0.10; Pr(ϕ1 < 0) = 0.91) and with spring temperature (ϕ3 = −0.45; 95% HPDI: −0.88–0.08; Pr(ϕ3 < 0) = 0.96), and it was positively correlated with spring precipitation (ϕ2 = 0.74; 95% HPDI: 0.26–1.30; Pr(ϕ2 > 0) > 0.99; Figs 2 and 3). The SD of the estimated year-specific growth rate (σλ = 0.1; 95% HPDI: 0.01–0.87; Figs 2 and 3) accounted for measurement error or unknown regional processes.

Figure 3.

 Estimated mean (contours) and SD (σλ; background colouring) of the natural logarithm of the year-specific growth rate (log λy) of Buxbaumia viridis, as a function of autumn frosts, spring temperature and spring precipitation. The growth rate (contours) is calculated using the mean level of the weather variable not included in each panel (mean temperature and mean frosts, respectively). The blue dots show the data for the 1996–2008 studied period, and the blue rectangle delimits the data range used for fitting the model. The past 100 years’ weather data (orange squares for 1896–1959, and green triangles for 1960–1995) are shown for comparison.

The local-scale variable abundance in the preceding year had a small, yet positive, effect on the abundance in the focal year (β1 = 0.03; 95% HPDI: −0.05–0.12; Pr(β1 > 0) = 0.81). The difference in DIC between the models including and excluding the abundance in the preceding year was small, but we decided to keep it because it makes sense biologically. The parameter εy,p and its variability (σ = 1.33; 95% HPDI: 1–1.79; Fig. 2) scaled the local recruitment from sporophytes in the preceding year to unknown spore sources, and to measurement error.

The model suggests that the population will decline in the long term under environmental conditions that are similar to current conditions; the mode for the stochastic growth rate was < 0, log λS = −0.14 (95% HPDI: −9.52–1.2), and its posterior distribution had a long left tail (Fig. 4). The risk for this to take place, or the confidence in the statement of population decline, is 81%—the proportion of the posterior distribution of log λS, which is < 0.

Figure 4.

 (a) Estimated posterior distribution of the stochastic growth rate, log λS, for Buxbaumia viridis based on the estimated posterior distributions of the year-specific growth rates. The shaded area and the percentage are the proportion of the posterior distribution which is < 0. The thick solid vertical line shows the mode of the posterior distribution (−0.21). The dashed vertical line represents the point estimate of the stochastic growth rate, log λSP = −0.59. (b) The inset shows a detail of the plot near log λS = 0, where the thin solid vertical line shows the limit approximation of the stochastic growth rate, log λSL = −0.0003, with the whisker showing its 95% confidence interval using T = 10 000, eqn 8.

The estimates of the commonly used measures of the stochastic growth rate were included in the 95% HPDI of the posterior distribution of log λS (−9.52–1.2), log λSP = −0.59 and log λSL = −0.0003. However, the confidence intervals of the limit approximation of the stochastic growth rate did not include the point estimate log λSP or the mode of the posterior distribution of log λS (Fig. 4). Also, the width of the confidence interval varied with T: −0.056–0.057 if T = 10 000, and −0.015–0.015 if T = 100 000 (Appendix S2, Fig. S2.2).

The averaged model fitted the data satisfactorily. First, the observed abundances for years 1996–2003 were included in the 50% HPDI of the abundance estimated by the model (Fig. 1, vertical bars). Second, the projections of the dynamics starting with the conditions observed in 1996 reconstructed adequately the inter-annual variation in sporophyte abundance in 1997–2003 and 2008. The observed data were included in the 50% HPDI of the simulated yearly abundances, with exceptions in the years 1999, 2000, 2002 when they were included in the 75% HPDI. The data for 2008 were influenced by an extreme event in sample plot 6 (see plot-specific abundances in Appendix S2), and were therefore not included in these HPDIs.

Discussion

We have developed a hierarchical Bayesian model based on a population time series for species living in temporally and spatially varying environments. The Bayesian approach adds a fundamental aspect to the population viability estimate: natural variability and sampling uncertainty across regional and local scales (Clark & Bjørnstad 2004; Clark 2005). This approach allows the stochastic growth rate to be estimated under all possible combination of observed environmental variability. Therefore, we can provide an estimate of confidence in the statement about the long-term viability of the population. The approach also allows disentangling the relative importance of different drivers of the inter-annual variation in population abundance. We expect this approach to be especially useful in the viability analysis of natural populations experiencing environmental variability.

Previous estimates of the stochastic growth rate, a widely used measure of population growth in fluctuating environments (Caswell 2001), did not account for uncertainties in a coherent and satisfying way. First, point estimates (single values) such as log λSP or log λSL cannot, by definition, express any variation. Second, the estimate of the confidence interval based on the limit approximation approach (Caswell 2001) underestimates uncertainties. The main reason is that the approach aims to estimate the mean of the growth rate, with the confidence interval representing the uncertainty in the estimate of this mean. That is, the confidence interval does not reflect the natural variation and uncertainty in population growth rate, which is accounted for in the hierarchical model. It should also be noted that the width of the interval decreases the longer the analyst runs the projections, i.e. how large T is chosen to be in eqns 7 and 8. The full posterior distribution of log λS presented in the current study explicitly utilizes the full posterior distributions of the year-specific growth rates (i.e. their natural variability and uncertainty), and thereby accounts for all possible combinations of year-specific population growth rates and their relative effects. It can also be noted that the estimate of log λSL (eqn 7) is higher than the point estimate of log λSP or the mode of the probability distribution (eqn 6). The reason is probably that eqn 6 includes the geometric mean (first term), which give high weight to values (of λy) close to zero, and that it includes the variation in yearly growth rate (second term). Lewontin & Cohen (1969), who presented this estimator, compared the problem of viability of small populations with the problem of growth of a repeatedly gambled capital.

We think that statements about the viability of a population should be based on the full probability distribution of the stochastic growth rate, and not only on its mode. Also, when reporting the confidence in estimates of population viabilities, attention should be paid to the proportion of the posterior distribution below or above a critical value, not only to the confidence interval around the mean. The estimate of the confidence interval of log λSL using the limit approximation approach suggested no population decline as the interval included zero. This categorical conclusion could have negative consequences if used in conservation strategies. In contrast, the full probability distribution of the estimate of log λS provides information on how likely a population decline is, or the confidence in a statement about a decline, Pr(log λS < 0) = 81%, which is more informative and can be weighed against other interests.

We are aware of the risk of underestimating the environmental variability when using time series data (Beissinger & McCullough 2002), as the approach uses only observed environmental variability. However, the weather data represent the conditions of the last decades of the century (Fig. 3). Moreover, because both the autumn and spring weather have changed during the last century in the study region (see orange squares and green triangles in Fig. 3; IPCC 2007), statements about population viability based on environmental conditions observed long ago, when they were different (e.g. before 1960, Fig. 3), would be inaccurate. We are therefore confident in our estimate of population viability, as long as the weather conditions remain approximately the same.

This is, to our knowledge, the first estimate of a decline of a bryophyte species based on time series data. Bryophytes have a complex life history, and modelling their dynamics is not straightforward (During 2006). Yet, we are aware of an example that has also disentangled the mechanisms controlling their population dynamics: an increase in the abundance of a common ground-floor bryophyte was a result of improved growth conditions (Økland et al. 2009).

Species population dynamics and life cycle

Our analysis of B. viridis data shows the insights into the population dynamics of the species that can be gained from a hierarchical Bayesian model. The population dynamics are mainly driven by regional weather fluctuations, which have also been shown to have a key influence on the demographic processes of some vascular plants (e.g. Ågren, Ehrlén & Solbreck 2008; Evju et al. 2010; Jongejans et al. 2010; Toräng, Ehrlén & Ågren 2010). The first likely regional environmental mechanism driving the population dynamics is sudden spells of freezing before the winter, causing high mortality among immature sporophytes, as observed for the congeneric B. aphylla (Hancock & Brassard 1974). The second is water stress resulting from high temperature and low precipitation conditions, which decreases survival of the sporophytes in the spring (Wiklund & Rydin 2004). The effect sizes of the three weather variables were similar, although there was a tendency for spring precipitation to be more important (larger effect-size parameter). The local-scale variable dead wood did not explain the inter-annual variation in B. viridis abundance. We believe that the reason is the low inter-annual variation in simulated dead wood amount among years during the study period. Also, B. viridis occurs on dead wood of a specific stage of decay (Appendix S1), but only 8% and 7% of the highly decomposed logs were occupied in the two complete surveys. Therefore, other factors at lower scales, e.g. wood nutrient or moisture conditions, may also explain the abundance of the species and why dead wood did not explain the variation.

Our hierarchical model provides further insights into the life cycle of the species. Our results support the hypothesis that B. viridis’ spore germination and fertilization can occur within 1 year after the spore release. This is shown by the positive effect of the preceding year’s abundance on the abundance in the focal year. Annual or paucennial life cycles are common among bryophytes (and vascular plants) living in habitats with high environmental variation (During 1979)—here temporary dead wood patches and variable weather. The study species has also been suggested to have perennial protonema, brood bodies (Wiklund 2003) or a short-lived diaspore bank (Wiklund 2002). However, the longevities and relative importance of the protonema, the brood bodies and the diaspore bank are unknown. We also think that there may be a considerable spore rain from outside the study plots, and that, hence, the preceding year’s abundance in a sampling plot is not the only spore source.

Uncertainties are an important part of the Bayesian framework (Clark 2005). In addition to modelling known regional and local processes, we modelled unknown processes on the same scales. The parameter σλ quantifies the inter-plot variation in population growth rate. This may quantify the potential buffering ability of certain local populations to temporally unsuitable regional conditions. For example, in a year with bad weather, certain local populations may not suffer as much as most populations do (Davison et al. 2010). The parameter also quantifies unknown processes that drive the regional-scale inter-annual variation in population growth rate. Moreover, the parameter εy,p and its variability σ account for unknown local recruitment processes, e.g. germination of spores from the spore bank, or measurement error.

We now know that the persistence of this species is determined by a combination of regional processes (weather variability), local population dynamics and resource levels (variation in dead wood amount). A change in the balance between these processes may lead to a further decline of the species. The regional-scale weather variables are likely to be the major drivers of the population dynamics, as judged by the estimated standardized effect sizes. However, the relative importance of the preceding year’s abundance, unknown local recruitment processes, and not-included variables should not be underestimated in future conservation strategies. Dead wood was not included in the final model, but its future importance is most likely to be high. Its availability can be regulated by forestry activities (Hallingbäck 2002), but its natural dynamics (e.g. decay rates) may also change in the future because of climate change (Weslien et al. 2009). Moreover, both the mean and the variability in all seasons’ temperature and spring precipitation increased during the last century and are predicted to increase in Scandinavia in the next century (IPCC 2007). These variables have opposite effects on the population growth, and it is therefore difficult to anticipate the future trend of the B. viridis population without making simulations where the population dynamics are driven by predictions of future temperature and precipitation (e.g. Snäll, Benestad & Stenseth 2009). However, short-lived species, such as B. viridis, may be more sensitive to increments in environmental variability than long-lived species (Morris et al. 2008).

Acknowledgements

We are grateful to Pär Forslund, Tomas Pärt and Christer Solbreck and three anonymous reviewers for constructive comments on the manuscript, and to Niklas Lönnell and M. Celina Abraham for support in the field. We are also grateful to Jane Morrell for improving the English, to Per Nyman for providing us with the weather data, and to Tobias Jeppsson for comments on Appendix S3. The work was funded by grants 2005-933 and 2006–2104 from FORMAS to T.S.

Ancillary

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