Toward improved species niche modelling: Arnica montana in the Alps as a case study

Summary

• 1
  Under the effects of rapid environmental change, such as climate change and land degradation, assessment of plant species potential distribution is becoming increasingly important for conservation purposes. Moreover, land administrators need reliable predictions of species suitability for planning a wide range of management activities.
• 2
  In this study, we used the recent Maxent algorithm for modelling the niche of Arnica montana within a Site of Community Importance in the Alps, with the ultimate aim of providing a rigorous evidence base for management of this locally threatened species. We built a final suitability map taking into account (i) the minimization of spatial autocorrelation through the use of a constrained random split of sampled data; (ii) the use of a stepwise selection of predictors in order to obtain a reduced model containing only meaningful variables; (iii) the comparison of the predictive power of three sets of environmental predictors; (iv) the identification of the most suitable areas by overlaying predictions of three competing models; (v) the use of divergence maps as a complement to conventional performance comparison assessments.
• 3
  Maxent improved accuracy both on training and test data sets. Elevation, geomorphology and hosting habitats performed as effective primary predictors. A reduced model based on the outcomes of a preliminary stepwise selection analysis of predictors gave the best accuracy score on test data. Two parts of the study area have been selected for management as a result of areas of agreement between the three competing models.
• 4
  Synthesis and applications. There remain important methodological issues that need to be improved in order to increase confidence in niche modelling and ensure that reintroduction and management activities for threatened or rare plant species are based on reliable distribution models. Modellers can improve predictions of plant distribution by addressing methodological topics that are often overlooked, as demonstrated for A. montana in this study.

Introduction

Hutchinson (1957) used a mathematical definition of the niche as a hyper-volume, called the fundamental niche, located in an n-dimensional ‘hyper-space’ enclosing conditions that allow a species to survive and reproduce. In an ideal environment, a species is expected to live in a geographical area that strictly corresponds to the projection of its fundamental niche, thus occurring everywhere environmental conditions are suitable (Pulliam 2000). In reality, the fundamental niche is unlikely to be observed and we usually detect the realized niche (Guisan & Thuiller 2005). Interspecific competition may exclude individuals from some parts of their fundamental niche. Moreover, when dispersal across the landscape is limited by a hostile matrix (e.g. geographical barriers or human artefacts), parts of the fundamental niche may remain uninhabited whatever their local suitability (Pulliam 2000). Selection of best areas within the fundamental niche may also limit the extent of the realized niche (Hutchinson 1978). A species might be absent from suitable habitats because of local extinction events or limited dispersal ability, or it might occur in a sink habitat where its population growth rate is less than 1, and thus where it would disappear without constant immigration from source habitats (Guisan & Thuiller 2005).

Niche modelling, also known as species or habitat potential distribution modelling (Guisan & Thuiller 2005), is used to inductively interpolate or extrapolate fundamental niche outside the locations where a species is present (i.e. realized niche), by relating species presence to environmental predictors (Franklin 1995). Niche modelling is important for a range of land management activities. Examples include predicting the distribution of rare and threatened species and plant communities (Engler, Guisan & Rechsteiner 2004), risk assessment of invasive species in new environments (Peterson 2003), and estimation of the likely intensity of biological responses to climate change (Guisan & Theurillat 2000; Thuiller 2004). Reintroduction and augmentation of small natural populations of threatened or rare plant species (Sutherland & Hill 1995; IUCN 1998) should rely on reliable distribution models (Sergio et al. 2007) for management and other applied ecology purposes (Seddon, Anderson & Schapire 2007).

Various methods have been developed for niche modelling. Some of these methods, including generalized linear models (GLM; McCullagh & Nelder 1989) and generalized additive models (GAM; Hastie & Tibshirani 1986), require presence/absence data in order to generate statistical or discriminant functions rules. However, there is growing interest in making use of presence-only data, consisting of occurrences but with no reliable data on where the species was truly absent. In fact, the large majority of available data consist of presence-only data sets, coming from atlases, museum and herbarium records, observational data bases and in situ field surveys (Pearce & Boyce 2006). Therefore, a second group of methods, including for instance the genetic algorithms (GARP; Stockwell & Peters 1999) and Bioclim (Busby 1991), is gaining more consideration. The recently proposed Maximum Entropy (Maxent) algorithm (Phillips et al. 2006) allows the use of presence-only data and categorical predictors. In addition, Maxent has been shown to perform better than other algorithms. For example, Elith et al. (2006) demonstrated that Maxent performed very well when compared to more established methods such as Bioclim, GARP, GAM and GLM. In addition, Barry & Elith (2006) noted similarities among Maxent, GLM, and GAM, specifically in their ability at fitting nonlinear response surfaces that are frequently observed in biological data. Hernandez et al. (2006) tested four modelling methods and showed that Maxent had the strongest performance among the tested methods, since it performed well and remained reasonably stable in prediction accuracy across all sample size categories and produces maximal accuracy levels for the smallest sample size categories. Finally, Sergio et al. (2007) showed that Maxent outperformed GARP when applied to presence-only herbarium collection data. In effect, recent studies on potential distribution assessment are mainly focused on the comparison among methodological tools. Although it is a critical topic, many modelling features remain largely overlooked and several authors (Araujo & Guisan 2006; Barry & Elith 2006; Guisan et al. 2006) have recently noted frequent algorithmic uncertainties and ambiguities in predictive distribution modelling, such as for instance scarce attention to: (j) the choice of environmental predictors, (ii) the problem of autocorrelation, and (iii) the contribution of each predictor to the model accuracy.

Accordingly, in this study besides using the Maxent algorithm, we conceptualize and test whether handling methodological topics that are often overlooked by modellers may improve the prediction of plant distributions by: (i) the use of a constrained random split of sampled data, in order to minimize biases due to spatial autocorrelation; (ii) the use of a stepwise selection of predictor variables for the evaluation of the contribution of each predictor to model accuracy, in order to obtain a less overfitted reduced model containing only meaningful variables; (iii) the comparison among three different sets of environmental predictors (i.e. an initial set with seven predictors, a set employing only topoclimatic variables, and a pruned set resulting from the stepwise predictor selection applied to the initial set); (iv) the choice of the most suitable areas for species augmentation as a result of the degree of concordance among three competing models, each based on a different set of predictors; and (v) the use of divergence maps as a complement to conventional performance comparison assessments.

We applied the proposed approach to presence-only data of Arnica montana L. (Asteraceae), aiming at a further rigorous reinforcement of this threatened species within a Site of Community Importance (hereafter SCI) in the Alps. This approach should result in improvements in the modelling of species distributions.

Methods

species description and study area

Arnica montana is a long-lived perennial species, whose distribution is restricted to Europe (Hultén & Fries 1986). Formerly, A. montana was a common plant of nutrient-poor grasslands and dry heathlands (Kahmen & Poschlod 2000). Recently, habitat fragmentation, abandonment of pasturing and collection for herbal use have led to its rapid decline. Consequently, A. montana is considered one of the most threatened grassland species in the Netherlands and in Central Europe (Pegtel 1994). In the Alps, A. montana is locally present in mountain areas with low-intensity land use practices, and its collection has been regulated since 1932 at the national level and since 1977 regionally due to its frequent use as a medicinal plant. Following the European Union (EU) Habitats Directive 92/43/EEC (Annex V), A. montana has been designated as a plant species of community interest, whose exploitation may be managed and whose conservation should be encouraged.

The Site of Community Importance ‘Val Viola Bormina-Ghiacciaio di Cima dei Piazzi’ (IT 2040012) belongs to the EU Network ‘Natura 2000’ resulting from the Habitats Directive, and is located in the Eastern Alps, within the Lombardy Region (Italy). The SCI extends over 59·66 km² (centroid coordinates: 10°14′34″E, 46°25′42″N) ranging from 1710 to 3441 m above sea level in altitude. The climate of the SCI is central European montane to subalpine (Ellenberg 1988). The study area is predominantly covered by natural and semi-natural grasslands, in particular siliceous alpine and boreal grasslands (EU habitat 6150, with reference to Directive 92/43/EEC; 25·92% of the study area), siliceous screes of the montane to nival belts (EU habitat 8110; 19·6%) and alpine and boreal heaths (EU habitat 4060; 10·2%). The geological substrate is mainly siliceous rocks (Staub 1946).

field sampling and basic spatial statistics

Field surveys were conducted from 2004–2006 aimed at identifying all sites of A. montana presence within the study area. The coordinates of each location were measured with a global positioning system (GPS) using differential correction techniques to improve the accuracy of data location (error < 1 m). Basic descriptive spatial statistics (minimum, maximum and mean distances among growing sites) and the convex hull (the area of the minimum convex polygon containing the sampled points; Graham 1972) around the set of points were estimated.

choice of the environmental predictors

Predictive modelling of species distribution involves the use of environmental data describing factors that are known to have either a direct (proximal) or indirect (distal) impact on a species. Proximal variables directly affect the distribution of the species, while distal variables are correlated to varying degrees with the causal ones (Austin 2002). The niche of plants is typically defined from biophysical variables such as air temperature, soil water content and solar radiation (Dymond & Johnson 2002). Elevation, slope angle and slope aspect are often considered as effective distal variables to predict these biophysical variables (Moore, Norton & Williams 1993; Horsch 2003). For instance, (i) elevation is highly correlated with temperature and humidity, (ii) slope angle regulates soil wetness, erosion and wind impacts, (iii) slope aspect influences solar input and snow persistence (Hoersch, Braun & Schmidt 2002). Soil wetness and incoming solar radiation are regarded as substantial proximal variables (Moore, Grayson & Ladson 1991; Dymond & Johnson 2002). Moreover, when working on limited areas, climatic factors may be overridden by local factors such as geomorphology and land-cover. Some authors have suggested that land-cover exerts prevailing control of species’ niche at a finer spatial resolution than climate (Pearson, Dawson & Liu 2004). Geomorphology is a distal variable with impacts on nutrient and water availability for plant growth (Nichols, Killingbeck & August 1998). Therefore, to model the relationship between the occurrence of A. montana and the physical environment, we used seven predictors (Table 1), encompassing both proximal and distal variables.

Table 1.  Environmental predictors employed for Arnica montana niche modelling

Predictor variable	Source	Data type	Unit	Scale	Used for model
Elevation	Digital elevation model (DEM)	Continuous	Metres	1:10 000	Full, topoclimatic, pruned
Slope angle	Derived from DEM	Continuous	Degrees	1:10 000	Full, topoclimatic
Slope aspect	Derived from DEM	Categorical	9 categories	1:10 000	Full, topoclimatic
Topographic wetness index	Derived from DEM	Continuous	m^2 m^−1	1:10 000	Full, topoclimatic
Potential solar radiation	Derived from DEM and latitude	Continuous	MJ cm^−2 year^−1	1:10 000	Full, topoclimatic
Geomorphology	Geomorphologic map	Categorical	23 categories	1:10 000	Full, pruned
Habitats	Habitat map	Categorical	15 categories	1:10 000	Full, pruned

A digital elevation model (DEM) for the area was obtained by digitizing available topographic maps of the Lombardy Region. The DEM at 1:10 000 scale was then used to derive slope angle, slope aspect, wetness index and potential solar radiation. Slope aspect and slope angle were calculated for each position in the DEM by taking local derivatives of elevation in the x and y direction. Slope aspect values were then grouped into nine categories (flat areas; N: 337·5°–22·5°, NE: 22·5°–67·5°, E: 67·5°–112·5°, SE: 112·5°–157·5°, S: 157·5°–202·5°, SW: 202·5°–247·5°, W: 247·5°–292·5°, NW: 292·5°–337·5°).

Topographic wetness index (TWI hereafter) estimates the accumulation of overland water flow across the catchment and site slope steepness. For vegetation studies, it is used to represent the spatial variability of soil wetness. It was calculated as follows (Moore, Grayson & Ladson 1991):

• (eqn 1)

where TWI_i is the wetness index at location i, C_iis the catchment area (the area draining into a pixel, expressed in m² m⁻¹) and S_i is the slope angle (in degrees) at location i. For slope, the classic average maximum slope calculation was used (Van Niel, Laffan & Lees 2004). This method fits a plane to its first order neighbours to determine slope for the centre cell. It is the most commonly applied calculation, and it is the default option in most GIS software packages. Although TWI only represents the topographic conditions, and therefore omits other aspects, such as the rainfall/runoff coefficient, this index has been used successfully in numerous studies for the assessment of catchment wetness conditions (e.g. Store & Kangas 2001). More detailed models including soil texture and variation in the water flux within the drainage area (O’Loughlin 1986) were not used since the required data were not available.

The amount of solar radiation that reaches each portion of the study area was determined by the potential annual direct incident radiation (SOLAR from now on). The term ‘potential’ accounts for the limitations of this index, since SOLAR does not consider cloud cover and shading by adjacent topography. SOLAR was estimated following McCune & Keon (2002):

• (eqn 2)

where:

• f(latitude, slope, aspect) = −1·467 + 1·582 * cos(latitude) *  cos(slope) – 1·5 * cos(aspect) * sin(slope) *  sin(latitude) – 0·262 *  sin(latitude) * sin(slope) + 0·607 * sin(aspect) * sin(slope)   (eqn 3)

Land cover was accounted for by digitizing the EU habitat map of the study area, obtained through field surveys in 2006 and digital orthophotos. As a result, a 15-class categorical map at 1:10 000 scale was realized. A geomorphology layer at 1:10 000 scale with 23 classes was provided by Lombardy Region (2004).

maxent methodology

Maxent is a machine-learning technique based on the principle of maximum entropy (Jaynes 1957), i.e. when approximating an unknown probability distribution. Maxent searches for the approximation that satisfies a set of constraints on the unknown distribution and that, subject to those constraints, maximizes the entropy of the resulting distribution (Phillips, Anderson & Schapire 2006). Given m sample points x₁ ... x_m (i.e. occurrence data), a study area composed of k pixels and a set of features f₁ ... f_n (i.e. environmental predictors), each feature f_iassigns a real value f_i (x_j) to each point x_j (e.g. the altitude of sample point x_j). The empirical average of each feature f_iis thus defined as:

• (eqn 4)

Maxent searches for the probability distribution that maximizes

• (eqn 5)

under the constraints that for each feature f_i

• (eqn 6)

for some constants β_i (also known as regularization value) and

• (eqn 7)

where p_k(x_j) is the unknown quantity (i.e. the probability assigned to each pixel). Without the regularization value (an empirically tuned value depending on the sample size), the distribution being computed is likely to undergo overfitting. In the regularized case, it is used as a relaxed constraint where feature expectations are only close to empirical average over sample locations rather than exactly equal to them (Phillips, Anderson & Schapire 2006). Because these probabilities must sum to 1, each probability is typically extremely small. Hence, Maxent presents the probability distribution in a cumulative representation (i.e. the value assigned to a pixel is the sum of the probabilities of that pixel and all other pixels with equal or lower probability, multiplied by 100 to give a percentage). As a result, Maxent assigns a non-negative probability to each pixel in the study area with results ranging from 0 to 100, providing a map showing the probability gradient for the species’ potential distribution (Phillips, Anderson & Schapire 2006). Pixels with values close to 100 are the most suitable, while cells close to 0 are the least suitable within the study area. Maxent was applied using the software detailed in Phillips Anderson & Schapire (2006).

model calibration

Models were calibrated using a random 70% of the data as a training sample (60 points) and evaluated with the remaining 30% (test data; 25 points) (split-sample approach; Fielding & Bell 1997). In spatial analyses, the simple count of sample units is not an adequate estimator of effective sample size. The amount of pseudoreplication of a variable depends on the distance between sample points (i.e. a set of closely spaced observations effectively provides less information than the same number of observations more widely separated in space). Such spatial dependency is termed spatial autocorrelation (Cliff & Ord 1973) and, although often overlooked (Drake, Randin & Guisan 2006), may bias model accuracy (Segurado, Araujo & Kunin 2006). In order to minimize this shortcoming, we used a constrained random split of sampled locations forcing all pairs of points less than a threshold distance to split dichotomously into the training and the test sets. Such distance was determined through a simple algorithm: (i) within a GIS, for each sampled point, we calculated the distance from all other locations; (ii) we imposed the 25 closest pairs of locations to be split dichotomously into the training and the test sets; (iii) the remaining 35 points were added to the training data. Hence, the threshold distance was an empirically tuned value depending on the sample and equal to the maximum distance among the distances of the 25 closest pairs of locations (49·76 m in this study). These two sets of locations have been used as training (60 points) and test (25 points) sets.

competing suitability models

Our modelling strategy was based on a stepwise algorithm. Firstly, a model using all the potentially useful predictor variables likely to be predictive for A. montana was employed (full model from now on; Table 1). The full model may be oversized (i.e. one or more predictors may have little predictive power), overfitted (i.e. the number of predictors may be too high compared to the number of observations) or redundant (i.e. some predictors may be correlated in a significant manner, hence resulting in multicollinearity). Since some variables are categorical, the estimation of unimportant predictors could not be achieved using classic correlation analysis. The outputs from exploratory tools, such as stepwise selection of predictors, are useful to make an optimized selection of predictors as input to predictive models. Hence, we used a leave-one-out stepwise selection to build a significant subset of the initial set of predictors. The stepwise selection was applied by systematically dropping predictors one at a time and assessing the resulting variation in model accuracy on the test data. Besides the use of the leave-one-out stepwise selection, we employed a further analysis of predictor importance by assessing the accuracy scores of niche models when predictor variables were used in isolation (i.e. one-predictor niche models). This second step allowed the creation of a reduced (pruned) model by removing the least important predictors. Thirdly, a topoclimatic model with five variables (DEM, slope, aspect, SOLAR and TWI) was used. This climate-driven model assumes that any direct influence factor on the spatial distribution of a certain species can be parameterized by landform and topography-derived climate parameters (Davis & Goetz 1990); the major advantage of the topoclimatic model is that georeferenced topographic data are available for large areas, thus providing a reliable and easily available predictor data set.

accuracy evaluation

To evaluate the accuracy of the three competing models, we used the Receiver Operating Characteristic (ROC) curve (Hanley & McNeil 1982) both on training and test data. The accuracy evaluation on test data is useful for assessing if the resulting suitability model tends to overfit the training data; hence, losing its predictive ability. The ROC curve represents the relationship between the percentage of presences correctly predicted (sensitivity) and 1 minus the percentage of the absences correctly predicted (specificity). The area under the curve (AUC) measures the ability of the model to classify correctly a species as present or absent. AUC values can be interpreted as the probability that, when a site with the species present and a site with the species absent are drawn at random, the former will have a higher predicted value than the latter. Following Araujo & Guisan (2006), a rough guide for classifying the model accuracy is: 0·50–0·60 = insufficient; 0·60–0·70 = poor; 0·70–0·80 = average; 0·80–0·90 = good; 0·90–1 = excellent.

model comparison

The suitability maps resulting from the application of the three competing suitability models were compared using three approaches. Firstly, a global comparison using Spearman's rho correlation coefficient was calculated as:

• (eqn 8)

where n is the number of pixels, d_i is the rank of the i^th pixel in the first map minus the rank of the i^th pixel in the second map. It should be noted that the Spearman's rho correlation coefficient is more appropriate than Pearson's r correlation coefficient since Maxent provides ranking scores (cumulative probabilities).

Secondly, the three suitability maps were cross-tabulated to discover the portions of the study area to which all models ascribe suitability scores greater or equal to 75% (BEST₇₅ henceforth), 90% (BEST₉₀) and 95% (BEST₉₅), respectively.

Lastly, a comparison among the three suitability models was calculated on a per-pixel basis as:

• (eqn 9)

• (eqn 10)

where:

• – 
  | | is the absolute value of the difference between two suitability models;
• – 
  min and max operators represent the minimum and maximum respectively of the differences between the suitability models;
• – 
  a, b and c refer to full model, pruned model and topoclimatic model in the order given.

Hence, DIVERG_min is a map where each pixel gives the minimal divergence (i.e. maximum agreement) among the three models while DIVERG_max is a map where each pixel uncovers the maximal divergence (i.e. minimum agreement) among models.

Results

Eighty-five growing sites of A. montana were identified in field surveys (Fig. 1). Minimum, maximum and mean distances among locations were 14·27 m, 4360·07 m and 744·55 m, respectively. The convex hull covered 2566·18 ha; thus, our niche models are interpolative with regard to 43·05% of the study area and extrapolative for the remaining 56·95%.

Figure 1. Position of the 85 identified field locations of Arnica montana within the study area. The dashed line delimits the convex hull around sampled locations.

The stepwise selection of predictors applied to the full model (Fig. 2) revealed that TWI, SOLAR, slope angle and slope aspects had the least predictive power when applied to the test data (25 locations). The overall AUC score of the full model on the test data was 0·864. The predictor variable with the highest AUC value when used in isolation (i.e. one-predictor niche model) was habitat (AUC = 0·848) followed by elevation (AUC = 0·8394) and geomorphology (AUC = 0·811). In addition, SOLAR (AUC = 0·666), TWI (AUC = 0·670), slope aspect (AUC = 0·714) and slope angle (AUC = 0·766) had little predictive power when used in isolation. The environmental variable that decreased AUC score the most when omitted (i.e. leave-one-out niche model) was elevation (AUC = 0·852), followed by habitat (AUC = 0·855) and geomorphology (AUC = 0·860). Omitting slope aspect (AUC = 0·877), TWI (AUC = 0·866) and SOLAR (AUC = 0·870) increased the AUC, while excluding slope angle (AUC = 0·864) had a negligible effect on AUC. Hence, the stepwise selection suggested a pruned model based on the three predictors: elevation, habitat type and geomorphology.

Figure 2. Results of the stepwise selection of predictors applied to the full model (TWI, topographic wetness index; SOLAR, potential annual direct incident radiation).

The accuracy of the three niche models measured through the ROC curve (Fig. 3) demonstrated that the topoclimatic model was outperformed by both the full and pruned models. The full model was the most accurate on the training data (AUC = 0·941), while the pruned model was most accurate on the test data (AUC = 0·888). All three models performed significantly better than random prediction (AUC = 0·5).

Figure 3. Accuracy assessment of the three competing models (full, topoclimatic and pruned) through ROC and AUC analyses.

The three niche models resulted in similar suitability rankings for the pixels constituting the study area. The full and pruned models were highly correlated (Spearman's rho = 0·918) and both had lower correlations with the topoclimatic model (correlation coefficients equal to 0·774 and 0·635, respectively).

The three resulting suitability maps have similar spatial patterns (Fig. 4). The full (Fig. 4a) and pruned (Fig. 4c) models clearly identify three valleys with high suitability scores within the study area (Val Viola, Val Dosdè and Val Verva). The topoclimatic model (Fig. 4b) suggests the same areas, but is spatially less restrictive and smoother. BEST₇₅ (Fig. 4d) and BEST₉₀ (Fig. 4e) covers 29·13 ha and 7·79 ha, respectively. BEST₇₅ encloses two areas that are 2300 m distant, being the first adjacent to the Val Viola lake (zone 1) and the second close to the northern limit of the study area (zone 2). Zone 1 is a 1450-m long strip with a maximum width of about 150 m, while zone 2 is a rectangular area approximately 400 × 250 m. BEST₉₀ restricts the suitable niche to zone 1, while BEST₉₅ (Fig. 4f) identifies a minimal subset of zone 1 with an extent of only 1·15 ha.

Figure 4. Maps of predicted potential distribution for Arnica montana using: (a) full model, (b) topoclimatic model, (c) pruned model; portions of the study area indicated by all models as having a Maxent score greater or equal to (d) 75%, (e) 90% and (f) 95%, respectively.

The spatially explicit comparison among the niche models (Fig. 5) identifies few areas where the three suitability maps disagree. DIVERG_min highlights a minor valley (i.e. Val Cantone) in the upper left portion of the SCI to which the full model assigns average suitability values, while the pruned model ascribes high scores. DIVERG_max depicts three portions of the study area (Val Cantone and two regions close to zone 1 and zone 2, respectively) where the disagreement among the niche models is entirely due to the difference between the topoclimatic model and the other two niche models.

Figure 5. Local disagreement among the three competing suitability models, estimated through (a) minimal (DIVERG_min) and (b) maximal (DIVERG_max) divergences.

Discussion

competing suitability models

The use of multiple competing niche models reflecting various ecological hypotheses may contribute to a deeper knowledge on species–environment relationships and lead to a more stringent assessment of potential distribution. To predict potential distribution of A. montana within the study area, we compared and combined the outcomes from three models, each based on a different set of environmental predictors. Some predictor variables may be absent from the full model as a result of a lack of knowledge regarding all the environmental factors that determine the distribution of A. montana. However, the use of distal (indirect topographic-derived climate predictors) rather than proximal predictors is likely to have minimized this problem. Nevertheless, an adequate set of predictors may be defined as that which provides improved predictive accuracies especially on test data, with respect to a specific context and purpose. From this perspective, the high accuracy level of the full model on the test set supports our initial choice of environmental predictors for A. montana. Nonetheless, the stepwise selection of predictors showed that the full model was overlarge, since the omissions of slope aspect, TWI and SOLAR, increased the AUC on test data, while the exclusion of slope angle left AUC unaltered. Despite being overlarge, the full model was not overfitted: following Harrell's rule of thumb (Harrell 1984), the risk of model overfitting starts when less than 10 cases per independent variable are used. However, the model accuracy on test data (AUC = 0·864) indicates that the full model did not lose its predictive ability outside the training set; also thanks to the regularization value that is used by Maxent to prevent overfitting.

The topoclimatic model was of particular interest to us, since it only requires a cartographic source to generate a DEM, with the other predictors (slope angle, slope aspect, TWI and SOLAR) being derived from the DEM. This property not only reduces the need for environmental data for the area under study, but also facilitates the transferability of the niche model to areas neighbouring the SCI, since only DEM of such new areas is needed. The topoclimatic model provided a good, but not excellent, level of accuracy. We hypothesize different reasons for its performance. First, it makes use of two first-order (slope angle and slope aspect) and two second-order derived data (TWI and SOLAR). The more-derived a variable is from the source data (i.e. DEM), the greater is its uncertainty due to potential error propagation (Van Niel, Laffan & Lees 2004). Secondly, the degree of information assumed by TWI and SOLAR is partially correlated, even if not redundant, with topographic predictors since they were built as spatially explicit functions of such variables. Thirdly, as suggested by various authors (e.g. Pearson, Dawson & Liu 2004), climate variables may be weak predictors at a local scale where they could be overridden by other factors such as land cover (Blasi, Capotorti & Frondoni 2005). Lastly, an ecological explanation is also conceivable, i.e. topoclimatic predictors only weakly influence spatial distribution of A. montana within the study area.

The pruned model addressed the need for a limited in size, not overfitted and not redundant model, exclusively based on significant predictors. These properties are also likely to result in a final model with more straightforward ecological interpretation. The pruned model was the most effective, following two criteria: first, it gave the best AUC score on test data and, secondly, it met the requirements of the quest for parsimony (i.e. accuracies being approximately equal, the best model is the simplest one). Parsimonious models are also more transferable from a predictive viewpoint to external areas in the close neighbourhood with respect to larger models (Araujo & Guisan 2006).

spatial autocorrelation

Spatially autocorrelated data might affect the explanatory power and predictive accuracy of suitability models (Randin et al. 2006). Therefore, we used a constrained random split of sampled locations. This procedure has an advantage, particularly for small sample sizes, when compared to the approach of discarding points that are too close to others in that no points are left out. This step of the proposed procedure may be further refined, for instance basing on spatial autocorrelation measures (e.g. Moran's I or Geary's c) made on the same data, and then using the correlogram to define at which distance no more autocorrelation is observed. Nonetheless, we rejected this approach since it still requires some data that are closer than the autocorrelation distance to be discarded. On the other hand, the split of data into test and training data on the basis of spatially close pairs means that a point in the test data set has a counterpart in the training data set, and this might increase the performance (greater AUC) on the test data because the model is fitted to nearby points in the training data set.

divergence maps

The use of divergence maps allowed for a spatially explicit comparison between the niche models. Spearman's rho correlation coefficients among suitability maps gave an overall measure of the good agreement among models, but it was not able to depict portions of the study area making exception to this trend. Areas of particular interests are those where DIVERG_min bears high values and DIVERG_max low ones, respectively. The former reveals portions of the study area where the disagreement among the suitability models is high; the latter discloses areas where the agreement is nearly absolute. This is a step forward with respect to the quantification of uncertainty in predictions by looking at prediction maps, as suggested by Barry & Elith (2006).

final suitability map

The choice of the most suitable areas as a result of synthesizing the outputs from the three suitability models may seem restrictive. A more relaxed modelling approach could be applied by cross-tabulating the results from the two most predictive models (i.e. the pruned and full models). Nonetheless, under the prerequisite that the overlayed suitability models bear good or excellent AUC scores, the use of a further confirmation aims to a more stringent potential distribution assessment. However, in only identifying areas that are confirmed by several models, this approach has the potential disadvantage of limiting the total area that is identified as suitable for the study species. This could be an issue if the study area is limited in size. Nevertheless, management activities for species are often limited to a few hectares, although it would also be possible to bring down requirements by using lower thresholds (e.g. BEST₇₀). The parts of the study area that were identified as most suitable for A. montana (i.e. BEST₇₅ and BEST₉₀) are large enough for proactive conservation management, while BEST₉₅ resulted in an area too small for this kind of activity. In addition, both zone 1 and zone 2 lie within the convex hull, thus representing an interpolative rather than extrapolative outcome of the niche modelling procedure. Belonging to the spatial envelope of the sampled locations is a further verification of the goodness-of-the-niche prediction, since it provides high probability that zone 1 and zone 2 are located within the environmental envelope of A. montana as well, thus satisfying a further suitability criterion (not requested by Maxent) on which some methodologies (e.g. Bioclim) are based. As a result, these two areas will soon undergo conservation management as an action of the management plan of the SCI, using seeds collected inside the SCI during summer 2006 and now stored in the Lombardy Seed Bank (University of Pavia) and in the Millennium Seed Bank (Royal Botanic Gardens, Kew) within the activities of the EU-project ENSCONET.

Acknowledgements

This project was supported by the University of Pavia (Italy), the local Government of Lombardy Region and the provincial administration of Sondrio. Comments from three anonymous reviewers greatly improved this manuscript, while Dr. Matthew I. Daws (Royal Botanic Gardens, Kew, UK) helped improve the English.