The distribution pattern of many species reflects the past rather than the current structure of landscapes. Consequently, species are most often not in equilibrium with the current landscape structure. Yet this is a well-known fact, there is no appropriate approach to estimate the colonization rate of non-equilibrium species based on only data on the species occurrence pattern in the landscape.
We present an approach to estimate the colonization rate of non-equilibrium metapopulations. The approach requires only data on species presence/absence among its patches (occurrence pattern), data on patch ages and data on the historic distribution of the patches in the landscape. By estimating the past occurrence patterns and colonization events leading to the current pattern of occupied and non-occupied patches, we estimate the colonization rate, including the dispersal kernel. We also show how to estimate effects of local patch conditions and how to include an independent estimate of the local extinction rate based on other data. We use nine epiphytic lichen species confined to beech trees to illustrate the method.
Five species had restricted dispersal range, between 200 and 4700 m, and their colonization rate decreased with increasing fragmentation. Species colonization rates were related to niche width. Among the demographic parameters, the force of colonization was more important than the dispersal range in explaining the colonization rates. Local patch conditions did not explain the colonization probability of any species. In metapopulation projections that did not account for restricted dispersal range, higher future metapopulation sizes were projected.
Synthesis. The presented approach uses data on only species occurrence, patch age and landscape history to estimate the species colonization rate and dispersal kernel. It can also utilize independent data on local extinction rate. Rather than identifying factors explaining the occurrence pattern, the model estimates the rate of change in the occurrence pattern. This dynamic modelling allows testing general and applied questions on the dynamics or viability of metapopulations of sessile species. The approach is applicable for species whose distribution pattern reflects the past rather than the current landscape structure, for example, certain epiphytes and ground-floor plants.
Many species live in patchy landscapes, and their dynamics can be generalized by the metapopulation theory (Hanski & Gaggiotti 2004), which has a basis in island biogeography (MacArthur & Wilson 1967). Species with high colonization and extinction rates, for example, certain insects or annual ground-floor plants (Hanski 1994; Dornier, Pons & Cheptou 2011), can be expected to be in equilibrium with the landscape structure. Thus, the metapopulation distribution pattern and proportion of occupied patches (occupancy henceforth) reflect today's spatial patch configuration and patch areas. Patches that are close to occupied ones are more likely to be occupied than isolated ones because of restricted species dispersal range. Moreover, small patches are less likely to be occupied than large ones because the local extinction risk is larger in small ones. For species in metapopulation equilibrium, we can estimate the colonization and extinction rates, including the dispersal kernel, simply based on occurrence pattern data using the incidence function model (IFM; Hanski 1994). This is useful because we rarely have access to long-term data on metapopulation dynamics (Hanski & Gaggiotti 2004).
Also metapopulations of species with slow colonization and extinction rates may be in equilibrium with the landscape structure. In addition, their dynamics may be influenced by the very dynamics of their patches (e.g. Thomas 1994). This was acknowledged by Verheyen et al. (2004), who extended the IFM to be applicable for dynamic landscapes with varying patch ages. That is, they modified Hanski's (1994) IFM to allow estimating the rates of colonization and extinction, including the dispersal kernel, of species whose metapopulation dynamics are influenced by landscape dynamics. Johansson, Ranius and Snäll (2012) extended this model to also estimate the time at which patches become suitable. Hodgson, Moilanen and Thomas (2009) instead modified the IFM to account for local patch succession, that is for changing patch qualities. However, these modifications of the IFM still assume that the species is in equilibrium with the dynamic landscape.
There are limitations with the IFM and its extensions. The main one is that the distribution pattern of many species reflects the past rather than the current landscapes structure – the species are not in equilibrium with the current landscape structure (e.g. Tilman et al. 1994; Snäll et al. 2004; Helm, Hanski & Pärtel 2006; Kuussaari et al. 2009; Johansson, Ranius & Snäll 2013). Reasons may be low extinction rates of local populations, or slow colonization of newly created patches (Hanski 1999; Snäll, Ehrlén & Rydin 2005; Herben et al. 2006; Vellend et al. 2006).
An approach to fit models for the dynamics of non-equilibrium metapopulations living in dynamic landscape based on only occurrence pattern data would be very useful. This can be exemplified by both basic ecological and conservation questions answered by applying the approaches earlier developed. Verheyen et al. (2004) suggested that life-history trade-offs explain colonization–extinction trade-offs, which affect persistence in static and dynamic landscapes. For example, a species with efficiently dispersed seeds had a higher persistence than a species with restricted dispersal in a dynamic landscape, while in a static landscape, the persistence of the two species was opposite. Hodgson, Moilanen and Thomas (2009) found that habitat turnover can obscure the connectivity–occupancy relationship in occurrence pattern data, whereby we may underestimate the importance of connectivity for species persistence. Johansson, Ranius and Snäll (2012) showed that epiphyte colonization rates were related to species traits; specifically, that they are higher for species with wide niches and small dispersal propagules than for species with narrow niches or large propagules. Johansson, Ranius and Snäll (2013) showed that low tree regeneration rates increased the epiphyte extinction risks, but that species declines were slow. Conservation actions that increased regeneration after 100 years of low regeneration decreased the extinction risks to very low levels.
We are aware of only one attempt to parameterize non-equilibrium metapopulation models for species whose dynamics are affected by landscape dynamics just using occurrence pattern data. Snäll et al. (2005) aimed at estimating the parameters of both a forest landscape model and a metapopulation model exclusively based on occurrence pattern data on an epiphyte and its host trees. They concluded that the forest landscape model could not be parameterized based on only occurrence pattern of the host trees. However, given a specific spatiotemporal landscape history, the parameterization of the metapopulation model was promising. In the present study, we utilize data on forest stand ages and on the historic forest distribution to fit non-equilibrium metapopulation models. The model system is lichens that are confined to beech in the landscape. Beech forest stands constitute habitat patches.
We present an approach to estimate colonization rates, including the dispersal kernel, of non-equilibrium metapopulations that are influenced by the dynamics of the landscape. The approach requires data on only species presence/absence in focal patches (species occurrence henceforth), on patch ages and on the historic distribution of the patches in the landscape. It consists in reconstructing the most likely time series on the occurrence of the species in focal patches and using this estimated time series to fit the colonization model. The approach includes estimating the effects of local patch conditions on the colonization rate. We also show how to include local extinction rates from independent sources.
Materials and methods
Study Area and Focal Forest Stands
We studied a 1750 km2 landscape in southern Sweden (Appendix S1, Fig. S1 in Supporting Information). Our focal spatial unit is the forest stand which (i) constitutes an easily defined patch for many species, (ii) is an appropriate spatial unit for studying metapopulation processes at the landscape scale, (iii) is the spatial unit of forestry and (iv) is the typical habitat patch in a fragmented forest landscape. We utilize an extensive published data set (Fritz, Gustafsson & Larsson 2008) which includes 150 beech-dominated stands according to the broad-leaved forest inventory in the municipality (Forestry Agency, unpublished data) fulfilling the following criteria: (i) stand area ≥0.5 ha and ≤5 ha and (ii) average estimated age ≥95 years. However, the sample size varied among species because it depends on the minimum age at which a stand becomes suitable for each species (Fritz, Niklasson & Churski 2009; Table 1). In these stands, the average beech volume was 80%. Rough estimates of forest stand ages were available, but we improved these by coring the four trees that were judged to be the oldest in 2009. For further details on the criteria for selecting the focal trees and stands, for a sensitivity analysis of these criteria and for dendrochronological procedures, see Appendix S1.
Table 1. Lichen study species, shortest spore dimension (μm) (Foucard 2001; Smith 2009), minimum suitable stand age, occupancy in 2000, sample size (N; number of focal stands used for model fitting) and estimated mean dispersal range. All species are epiphytic, except C Chaenotheca brachypoda which is epixylic. The species nomenclature follows Santesson et al. (2004)
For each focal stand, we used presence/absence data from Fritz, Gustafsson and Larsson (2008) on the nine epiphytic crustose lichens (Table 1) that are confined to beech in the study region in 1998 (2000 henceforth for ease of interpretation). The study species grow on old beech bark and are usually found in old–growth forest (Fritz, Niklasson & Churski 2009). One of them is epixylic (Table 1), but we hereafter use ‘epiphytic’ for simplicity. We choose species that are confined to beech which made it easy to define suitable habitat patches. The occurrence of the study species was recorded up to 2 m height above the ground on each standing, living and dead beech. Many epiphytes have their main distribution on the lower part of the trunk (Snäll et al. 2004; Johansson et al. 2010), including our study species (Fritz 2009).
The Colonization Model
For each focal species, the estimation of the colonization rate is based on estimation of the patch- and time-specific probabilities of species occurrence in each focal stand between the year it became suitable for the focal species and 1990, and on observed occurrence pattern data from 2000. We assume that the observed occurrence pattern is the result of past colonization events. Since we only utilized observed data from 2000, the decadal occurrence data between the year when the stand became suitable for the focal species and 1990 were missing. All these missing data were estimated by data imputation (see below). We also inform the dynamic model with temporal data, specifically stand ages determining how long each stand has been available for colonization. We assume that a stand could become colonized after the year at which it became suitable for the focal species, and this year was determined by the minimum suitable stand age for the species (Table 1). As described below, the approach also allows estimating effects of local patch conditions on species colonization and estimating a dispersal range parameter. Finally, we describe how to include an independent estimate of the local extinction rate based on data from another study.
Technically, we modelled the species occurrence (Fi,t) in each focal stand i every ten-year time steps t between the year at which the stand became suitable and year 2000. The response variable Fi,t takes the values 1 – species present, 0 – species absent or NA – missing data. We assume Fi,t = 0 for stands younger than the minimum suitable age at time t. Since occurrence data were only available from 2000, all missing data on species occurrence between the year at which the stand became suitable and year 1990 (specified as NAs) were estimated by data imputation (Gelman et al. 2004). Specifically, when estimating the parameters of the metapopulation model (eqns (eqn 1), (eqn 2), ( eqn 3) below), we jointly estimate the missing data on Fi,t between the year at which the stand becomes suitable and year 1990. All parameters (Fi,t and parameters in eqns (eqn 1), (eqn 2), ( eqn 3)) are estimated jointly, so the uncertainty in the estimates of species occurrence in each focal stand in each decade affects the uncertainty of the metapopulation model parameters, and vice versa.
In fitting the colonization model, we assumed that the number of propagules arriving to a focal stand follows a Poisson distribution, so that the probability of at least one successful propagule establishment in an unoccupied stand is , where λi,t is the ‘force of colonization’ parameter. More specifically,
where ηi,t is the patch- and time-specific colonization probability and the mean of a Bernoulli distribution. It was modelled further as
where cloglog denotes the complementary log-log link function, φ is the force of colonization, which regulates the decadal rate of colonization resulting from dispersal from surrounding beech stands, Ti is a measure of stand suitability defined as log(Ti) = βX, where X is a matrix of explanatory stand variables, β is a vector of associated effect size parameters and Si,t is the connectivity. In the following, we describe Ti and Si,t.
The connectivity to potential dispersal sources in the surrounding landscape was quantified by Si,t, specifically
( eqn 3)
where α regulates the exponentially decaying influence of the surrounding landscape with increasing distance, Di,j in 100 m units. We set the maximum Di,j to 10 km, and this is hence the upper limit of the spatial scale of our study. No focal stand was located < 10 km from the border of the study area. We set this maximum Di,j because of the outer border of the study area. We could have chosen a higher value but then, in order to avoid inaccurate quantification of the connectivity for stands located close to the border of the study area (closer than this new Di,j), we would have had to remove focal stands from the data set lending to smaller sample size. The indicator variable pi,t equals 1 if broad-leaved forest was present in the cell j in year t. Since the landscape changed drastically during the last 150 years, we assumed that the connectivity of a stand i in year t (Si,t) was best described by the distance to, and amount of surrounding forest at that time, that is at time t. We estimated the connectivity for the times 1850–1880, 1890–1970 and 1980–2000 based on the broad-leaved forest present in 1850, 1920 and 1995 according to historical maps (Appendix S1). Specifically, we divided the maps into a 300 by 300 m grid, and measured the distance (Di,j) from the centroid of the focal stand i to the centroid of the surrounding grid squares j. To ensure that all forest patches considered as potential sources were at least as old as the minimum suitable stand age for each species (Table 1) in year t, we included only forest grid squares whose centroids intersected with stands at least as old as the minimum suitable stand age in the year t. For example, for estimating the probability of colonization in 1900 for a species with a minimum suitable stand age of 120 years, we only included source stands present on both maps from 1850 and 1650. Source stands that only occurred on the 1850 map were excluded since they had potentially not reached the age of being suitable and constituting dispersal sources.
Adding an Extinction Model
The modelling approach does not allow estimating both colonization and extinction rates. At first, we therefore assumed no stochastic extinctions from stands, so once a stand became colonized it remained occupied. Indeed, the extinction rate from trees that remain standing is negligible for some (e.g. Snäll, Ehrlén & Rydin 2005; Johansson, Ranius & Snäll 2012) but not all epiphytes (e.g. Öckinger & Nilsson 2010; Fedrowitz, Kuusinen & Snäll 2012). However, we can make use of independent estimates of the stochastic extinction rate based on data from another study, and this is technically described in Appendix S2. We illustrate the approach using Öckinger and Nilsson's (2010) estimate for a foliose epiphytic lichen, 13% per forest stand per decade. This is the only stand-specific estimate of a lichen extinction rate that we are aware of. The estimate may be high, but it is nevertheless useful for illustrating how to include an independent estimate of the local extinction rate utilizing data from other studies, and for illustrating the effect of adding an extinction rate to the estimate of the colonization rate.
Model Fitting, Selection and Evaluation
We applied the Bayesian modelling approach because it is convenient for fitting complex models with a formal mathematical treatment of the natural variability and data uncertainty (Ellison 2004; Gelman et al. 2004). Using the Bayesian approach, it is relatively easy to estimate missing observations (data imputation, Gelman et al. 2004), such as the probability of species occurrence in each patch and time step in this case. Another feature of a Bayesian model is that it provides the full probability distribution of the model parameters, the posterior distributions, given the data and á priori knowledge about the parameters. For the uninformative prior parameter distributions used, and model fitting details, see Appendix S3.
We report the Bayesian 95% credible intervals for the posterior distribution of the estimated parameters. To facilitate the dissemination of the approach, we provide the JAGS code used (Appendix S2). The code provided includes an adaptation to account for false absence records, given access to such data (MacKenzie et al. 2005).
The species-specific final models were the result of a selection procedure based on the posterior distribution of key parameters, our understanding of the biology of the system (in accordance with Gelman & Hill 2007) and the posterior mean deviance of the model, which is a measure of fit or ‘adequacy’ (Spiegelhalter et al. 2002). The lower the deviance, the better the model fits to the data. The difference in mean deviance between nested models follows a χ2 distribution. According to the likelihood ratio test, a model M1 with the parameters of model M0 plus an additional one (d.f. = 1) is superior if the deviance of M1 is 3.84 units lower than the deviance of M0 (5% significance level). If two parameters are added to M0, the critical value would be 5.99 units (d.f. = 2, 5% significance level). Currently, the deviance information criterion (DIC; Spiegelhalter et al. 2002) is preferred over difference in mean deviance for model selection (Gelman & Hill 2007). The reason why we did not use the DIC is that it not defined for models including mixture distributions, here for example, the modelled probability of occurrence in t which depends on the modelled probability of occurrence in t−1. However, we adopt the fact that DIC estimates for multilevel (hierarchical) models are suggestive rather than definitive – in accordance with Gelman & Hill (2007), we accepted models with a difference in deviance somewhat smaller than 3.84 if the estimated model parameters were biologically reasonable.
We first compared a null model which only included the force of colonization (φ in eqn (eqn 2)) with a model including the dispersal kernel (eqns (eqn 2) and ( eqn 3) including φ and α; ‘spatial model’ henceforth). Next, we added the explanatory variables (eqn (eqn 2) including Ti).
To investigate the performance of alternative final models, we compared the fitted time series 1850–2000 and the projected time series 2000–2100 between the spatial models and the null models. We expected that the spatial models would project lower future occupancies in the fragmented landscape than the null models. We also extended the final colonization models for each species with the extinction model, re-fitted the parameters in eqns (eqn 1), (eqn 2), ( eqn 3) and made projections for 2000–2100.
The Colonization Model
Based on data on species occurrence in sampled focal beech forest stands from one point in time, data on the age of the focal beech stands, and on the historic distribution of beech stands, we fitted dynamic colonization models with the force of colonization parameter φ for all nine study species (Fig. 1). In five of the species, the colonization rate increased with increasing connectivity to potential dispersal sources in the surrounding landscape, and we could fit the dispersal range parameter α (‘spatial models’, Fig. 1 and Table 2). In species for which we could not fit spatial models, the estimate of φ is lower, but since these species colonize all patches with equal probabilities, the resulting colonization rate is not necessarily lower (Fig. 1). In none of the nine species, did stand-specific variables explain the colonization probability (Table 2).
Table 2. Decrease in deviance (Δ Deviance) when adding connectivity or stand-specific variables to null colonization models (Δ deviance = 0) for the lichen species. Final colonization models that included the connectivity variable (‘spatial models’) are indicated using bold font. ‘·’ indicates interaction. ‘Prod’ means forest site productivity index
Slope · Aspect
The two species with the highest colonization rates (Lecanora glabrata and Pirenula nitida) also had the lowest minimum stand age (Table 1 and Fig. 1). Lecanora glabrata had both lower force of colonization and shorter dispersal range resulting in lower landscape scale occupancy. The other three species for which we could fit spatial colonization models (Chaenotheca brachypoda, Opegrapha viridis and Pachyphiale carneola) had higher minimum stand age and lower force of colonization (Table 1, Fig. 1). Moreover, O. viridis had the shortest dispersal range and largest spore size, and P. nitida had the longest dispersal range and smallest diaspore size (Table 1, Fig. 3). Among the four species for which spatial models did not improve the model fit, all had high minimum suitable age and three of them had low occupancies. Pertusaria multipuncta had higher occupancy and also considerably larger spores than any of the nine species (Table 1 and Fig. 2).
The developed colonization models are based on estimated time series on patch- and time-specific probabilities of species occurrence. The resulting colonization rates were estimated to be very low during the first decades when only a small proportion of the focal stands had become suitable for the study species (Fig. 2). However, although the median rates were zero, the high upper 95% credible limits show that colonization events indeed were estimated to occur (Fig. 2). In species with short dispersal range, the spatial models predict decreased colonization rates when the amount of surrounding suitable forest decreased, see for example, decreasing colonization rate after 1980 when the amount of surrounding forest decreased (Figs 2 and S1).
The estimated past time series and the projected future time series on occupancy varied between models (Fig. 2, right column). Occupancies prior to 2000 estimated by the spatial models were higher than, or equal to occupancies estimated by null colonization models. However, the occupancies projected 100 years into the future by spatial models were lower than the ones projected by null models, and this difference is noticeable after approximately five decades. Since these models assume that extinctions from stands do not occur, we projected increasing occupancy after 2000, both with the null and with the spatial models.
The spatial scale at which the potential dispersal sources in the surrounding landscape explained colonization probabilities varied among species. The mean dispersal (α−1) ranged from 218 to 4695 m among species (Table 1; Fig. 3). The modes of the joint posterior distribution of the dispersal parameter α and the force of colonization parameter φ (black dot in Fig. 3) are the basis for the mean dispersal range, instead of the mode of the marginal posterior distribution of α shown in Fig. 1. Note also the correlation between the parameters φ and α (Fig. 3). This parameter correlation and uncertainty are expressed in the confidence intervals in metapopulation projections (Figs 2 and 4). The probabilities of the surrounding landscape beyond 10 km having an effect on the colonization probabilities ranged between <0.001 and 0.1 among species (Fig. 3).
Adding an Extinction Model
When adding a model for stochastic extinctions based on data on a different epiphytic lichens from another study, there were more colonization events early during the time series, and the estimated colonization rates increased for all species (compare Fig. 4 with Fig. 2). However, the colonization rates for species with a restricted dispersal range (spatial models) still decreased when the amount of surrounding suitable forest decreased (Fig. 4).
The distribution pattern of species typically reflects the past rather than the current structure of landscapes (Tilman et al. 1994; Hanski 1999; Ovaskainen & Hanski 2002). For these species, metapopulation equilibrium is an assumption that should not be made in metapopulation modelling as it, for example, may overestimate the viability of the metapopulation (Hanski, Moilanen & Gyllenberg 1996; Moilanen 2000). We present the first approach to fit a spatial colonization model for non-equilibrium metapopulations that requires only occurrence pattern data, data on patch ages and on the historic distribution of the patches in the landscape. Such data are relatively easy to obtain for many sessile species. The metapopulation model is fitted to estimated patch- and time-specific probabilities of occurrence of the species in the focal patches, that is a time series on species occurrence for each patch. We also show how to make use of independent estimates of the local stochastic extinction rate, which can be obtained by re-surveying local populations.
We show that for species that were estimated to have restricted dispersal range in the fragmented landscape, increasing fragmentation slowed down the colonization rate. Preceding studies suggest that the spatial scale at which the surrounding landscape explains colonization events of sessile species depends on the spatial scale of the study, and on how connectivity is measured. Studies of colonizations of individual trees over areas ranging from 24 to 210 ha report mean dispersal ranges of 16 to 50 m (Öckinger, Niklasson & Nilsson 2005; Snäll, Ehrlén & Rydin 2005; Werth et al. 2006; Johansson, Ranius & Snäll 2012). In the present study, we focus on dispersal among forest stands within an area of 175 000 ha, and our estimated mean dispersal ranges from potential dispersal sources per decade ranged between 218 and 4695 m among species. The spatially explicit effect of the surrounding landscape up to 10 km found here may correspond to what Johansson, Ranius and Snäll (2012) described as background deposition in their study of a smaller spatial and temporal scale, that is, long dispersal range at the landscape scale between decades may be interpreted as a yearly, background propagule rain when studying metapopulation dynamics among individual trees between single years. Correspondingly, the lack of an effect of the surrounding landscape found for some species here may be explained by a restricted dispersal range at a much larger scale (Muñoz et al. 2004). We should also acknowledge the parameter uncertainty and correlation in this dispersal modelling (Fig. 3). Parameter correlations are typical for nonlinear models and mean that changing one parameter will change the likely value of another parameter. One strength in applying Bayesian parameter estimation in these situations is that it does not focus on point estimates, but rather looks at the whole posterior distribution, including any correlation. In the projections of the metapopulation dynamics, these correlations are accounted for and expressed in the projection probability distributions. For further details, see (Snäll, O'Hara & Arjas 2007).
We found the highest mean colonization rates in the two species with the lowest minimum suitable age. These have the widest niches and should therefore have the largest amount of dispersal source area and diaspore output into the landscape. The same has been found in, for example birds (Lack 1976) and other epiphytic lichens (Johansson, Ranius & Snäll 2012), and agrees with metapopulation theory in that increasing connectivity increases the colonization rate (Hanski 1999). The two species with the longest and shortest dispersal range indeed had the smallest and largest diaspores, in accordance with Verheyen et al. (2003) and Löbel & Rydin (2009). Nevertheless, we neither found clear relationships between the force of colonization and spore size, nor between spore size and colonization rate as suggested before (e.g. Verheyen et al. 2003, 2004; Johansson, Ranius & Snäll 2012). We argue that estimates of colonization rates of sessile species result from combinations of multiple interacting traits (e.g. Verheyen et al. 2003) and the past changing landscape. There may be different explanations why spatial models did not improve the model fit for four species. One may be that their dispersal range is short in comparison with the distance between our focal stands and the surrounding landscape assumed to be dispersal source. For example, Pertusaria multipuncta had a high colonization rate and may mainly disperse (efficiently) over very short distances with its large spores. Another explanation is that the focal stands represent a poor regeneration niche for the focal species – they are specialized on habitat conditions that are uncommon among our focal stands (e.g. Fritz, Brunet & Caldiz 2009; Fritz, Niklasson & Churski 2009). Nevertheless, we did not find support for any local environmental variable explaining the colonization rates. The weak explanatory power of slope, aspect and productivity did not surprise us, as they were in accordance with Fritz, Gustafsson and Larsson (2008), but we had strong belief in the hypothesis that stand area increases the colonization rate. However, other studies of epiphyte colonization of individual trees have also failed to detect positive relationships between tree size and colonization rate (Snäll, Ehrlén & Rydin 2005; Fedrowitz, Kuusinen & Snäll 2012).
We find it is reasonable to assume a negligible rate of stochastic extinction at the stand level for epiphytes in managed forest landscapes, whereby the extinction rate is deterministically set by the rate of patch destruction (patch-tracking metapopulation dynamics; Snäll, Ribeiro & Rydin 2003). In several epiphytic species, the stochastic extinction rate has namely been very low at the tree level (Snäll, Ehrlén & Rydin 2005; Gjerde et al. 2012; Johansson, Ranius & Snäll 2012). Nevertheless, significant stochastic extinction rates have been observed at the tree level in certain foliose epiphytic lichens (Fedrowitz, Kuusinen & Snäll 2012), in an epiphyllous bryophyte (Zartman et al. 2012) and in epixylic lichens (Caruso, Thor & Snäll 2010). Among ground-floor vascular plants, the extinction rate at stand level varies (Verheyen et al. 2004). These significant local stochastic extinction rates are the reason why we also show how to fit the model given available estimates of the extinction rate. Such estimates are relatively easy to obtain by repeatedly surveying patches that are occupied by the focal species (but see Kéry et al. 2006). The appropriate survey frequency of course depends on the species extinction rate, but decadal surveys may be appropriate and make it straightforward to simulate the metapopulation dynamics at decadal time steps.
Limitations and Potential of the Approach
Our modelling approach also has limitations that must be acknowledged. First, predictions using simple models based on species occurrences are more imprecise than predictions based on more complex, individual-based models that are available for certain model species (e.g. Harrison, Hanski & Ovaskainen 2011). However, even simple models may provide qualitatively correct predictions about future metapopulation persistence among alternative land use scenarios (Hanski 1994). The current models also have more specific limitations. First, using all suitable forest stands as potential dispersal sources, instead of occupied stands, we may have overestimated the connectivity between the patches. Therefore, the projected future occupancies are likely optimistic. Moreover, different choices of which stands to include in the modelling can be made. We have included all stands that were estimated to be less than 150 years. Some of these stand polygons intersected with forest on the map from 1850, and should therefore be older than 150 years. However, the mapping conducted in 1850 is rough, so the forest boundaries are not perfectly correct. We investigated the sensitivity of the parameter estimates to excluding from the data set stands recorded to be younger than 150 but occurring on the 1850 map. This led to a greatly reduced data set (62–66% reduction), lower model support and somewhat lower forces of colonization. However, the confidence intervals clearly overlapped the intervals obtained when using the full data set. For further details, see Appendix S1.
Nevertheless, the presented approach to estimate the metapopulation colonization rate advances the potential use of presence/absence data from non-equilibrium metapopulations that have been collected at one point in time. This advancement is clear when comparing with the inferences previously made using similar data sets. These preceding studies make inferences about which factors explain the occurrence of species. For example, Snäll et al. (2004) used similar data to make the qualitative conclusions on which local stand conditions (e.g. host tree numbers) explained the species occurrence pattern. Moreover, they found that the probability of species occurrence increased with increasing connectivity to the species in the surrounding landscape and that the probability increased with increasing connectivity to the suitable forest in the surrounding landscape that was present in the past. Yet another example study is Fritz, Gustafsson and Larsson (2008) who collected the data used in the present study. They made the qualitative conclusion that forest continuity and environmental stand conditions (e.g. substrate amount and quality) explain species abundance.
The current approach of fitting a dynamic model provides an estimate of the rate of change in species occurrence. The model thus estimates the rate of the dynamic process that led to the current occurrence pattern, rather than just identifying the factors that explain the pattern. As already pointed out by Hanski (1994), the applications of dynamic metapopulation models are countless. They can for example be used to answer general questions such as how the rate of spatial dynamics of species varies depending on species traits (e.g. Verheyen et al. 2004; Johansson, Ranius & Snäll 2012). They can also be used to investigate whether the metapopulation persistence is different given different scenarios of land use, that is, in metapopulation viability analysis (Beissinger & McCullough 2002). For example, Fedrowitz, Kuusinen and Snäll (2012) concluded that infrequent, sexually dispersed lichens, but not frequent, asexually dispersed ones are likely to be lost from small woodland set-asides in intensively managed landscapes. Roberge et al. (2011) showed that increased host tree (= patch) destruction resulting from edge effects or ash dieback increases the metapopulation extinction risk. Johansson, Ranius and Snäll (2013) showed how different conservation actions affect the future metapopulation persistence depending on the niche breadth of the species. Finally, metapopulation models for species whose dynamics are driven by the landscape dynamics can be applied to test the numerous theoretic predictions on these coupled metapopulation-landscape dynamics (e.g. Fahrig 1992; Keymer et al. 2000; Johst et al. 2011).
In summary, the approach presented is directly applicable for patch-tracking metapopulations with negligible local stochastic extinctions (Snäll, Ribeiro & Rydin 2003). We have also shown that the approach is applicable for species with significant local stochastic extinction rates given access to independent estimates of the extinction rate.
We thank Mauricio Fuentes, Krister Larsson and Mats Niklasson for sharing their expertise and for help in the field, Lena Gustafsson for enriching discussions, Mari Jönsson for providing comments on the manuscript and Stefan Anderson (The Swedish Forestry Agency) for providing data from the regional survey of broad-leaved forests and GIS data. Finally, we are grateful to two anonymous referees for suggestions that improved the manuscript. The work was funded by grants 2005-933 and 2006-2104 from FORMAS to TS.
Data available from the Dryad Digital Repository, including complete observation data set and forest stand age estimates, and broad-leaved forest cover over the study area and focal forest stand location (Ruete, Fritz & Snäll 2014).