Testing the efficacy of downscaling in species distribution modelling : a comparison between MaxEnt and favourability function models

Testing the efficacy of downscaling in species distribution modelling: a comparison between MaxEnt and favourability function models.— Statistical downscaling is used to improve the knowledge of spatial distributions from broad–scale to fine–scale maps with higher potential for conservation planning. We assessed the effective� ness of downscaling in two commonly used species distribution models: Maximum Entropy (MaxEnt) and the favourability function (FF). We used atlas data (10 x 10 km) of the fire salamander, Salamandra salamandra, distribution in southern Spain to derive models at a 1 x 1 km resolution. Downscaled models were assessed using an independent dataset of the species' distribution at 1 x 1 km. The FF model showed better downscaling performance than MaxEnt, and the models that were based on linear combinations of environmental variables performed better than models allowing higher flexibility. The FF model minimized model overfitting compared to the MaxEnt.


Introduction
Atlases are available for a large variety of taxa world� wide and represent species occurrence in the form of maps (Harrison, 1989).These maps often cover extensive geographic regions and their production usually involved large numbers of volunteers collecting data.Finer-scale maps of the distribution of species, required for conservation planning, can be derived from atlas maps by using statistical downscaling techniques (Kunin et al., 2000;Araújo et al., 2005;Barbosa et al., 2010).Such finer resolution maps have been valuable to understand how environmen� tal variables and different spatial resolutions affect species distributions (Barbosa et al., 2010) and to determine the impact of climate change on species ranges (Araújo & Rahbek, 2006) and the links between human nutrition and biodiversity protection (Fa et al., 2014(Fa et al., , 2015)).Downscaling assumes that a property within larger units is the arithmetic average of the property within smaller units (Bierkens et al., 2000).
The ability of a model to produce fine-scale species distributions from a coarser resolution has been described as model generality or model transferability (Vanreusel et al., 2007;Gray et al., 2008).Comparisons of model transferability using different Species Distribution Models (SDMs) have been undertaken by several authors (Peterson et al., 2007;Gray et al., 2008).In this paper, we test for the first time the performance of the favourabil� ity function (FF) model and the Maximum Entropy (MaxEnt) model.The FF model was designed to derive species distribution maps by obtaining, from logistic regression, an environmental favourability function from a species occurrence whose results are not affected by an uneven proportion of pres� ences and absences (Real et al., 2006;Acevedo & Real, 2012); this property permits direct comparison between models when different species are involved in the analysis, and allows for model combinations (e.g., Fa et al., 2014).MaxEnt is a presence-back� MaxEnt is a presence-back� ground profile method that has been successfully applied in a number of fields (Phillips et al., 2006).Considered consistently competitive with the highest performing methods (Elith et al., 2006(Elith et al., , 2010)), this is the most widely used SDM algorithm (Fourcade et al., 2014).MaxEnt uses input from a set of lay� MaxEnt uses input from a set of lay� ers or environmental variables as well as a set of georeferenced occurrence locations, to produce a model of relative suitability across the study area.
Here, we evaluate the performance of MaxEnt and FF in downscaling (from a spatial resolution of 10 x 10 km squares to 1 x 1 km squares) by using the known distribution of a terrestrial species, the fire sa� lamander, Salamandra salamandra (Linnaeus, 1758) (Urodela, Salamandridae) in Andalusia (southern Spain).We test how well each model performs by assessing whether it: (1) displays sufficient discrimi� nation power at finer spatial resolutions; (2) accurately predicts observed distributions at finer resolutions; and (3) significantly resembles predictions obtained with models trained using independent data at a finer resolution.

Species and study area
The fire salamander, Salamandra salamandra, occurs throughout most of the western Palaearctic region.This species is not included in the worldwide IUCN Red List, but is considered vulnerable in Spain (Pleguezuelos et al., 2004).The Andalusian region (fig.1A) harbours three of the nine subspecies present in the Iberian pe� ninsula: the entire distribution of S. s. longirostris in the south, most of the range of S. s. morenica in the north, and some populations of S. s. crespoi in the west.The presence of S. salamandra is closely related to spe� cific environments: wet and shaded zones with high rainfall located on medium to high mountainous areas; forests with ponds or streams; and wet grasslands surrounded by hedges or stone walls (García-París et al., 2004;Pleguezuelos et al., 2004).The specific habitat requirements of the selected species result in a clear environmentally-defined distribution that makes it suitable for model evaluation.
To test the downscaling performance of the various models we first used presence-absence atlas data (10 x 10 km UTM square grids) of S. salamandra in Andalusia (fi gs.1A, 1B).Data from the species dis� gs.1A, 1B).Data from the species dis� ).Data from the species dis� tribution map published in Pleguezuelos et al. (2004) were used.The species was present in 328 (37%) 10 x 10 km UTM squares.Additionally, we used an independent dataset of 1,090 presences of the spe� cies (1 x 1 km resolution) to evaluate the results of the downscaled models.These records were direct observations of the species throughout Andalusia, obtained from field surveys undertaken between 1980 and 2003 (Tejedo et al., 2003;fig. 1C).

Environmental variables
We considered a total of 34 environmental varia� bles for our modelling procedures (table 1).These variables included climate, topography, land cover, and human activities.Climate may affect the distri� bution of species, mainly at global and meso-scales, whereas topography and land cover act at mesoand micro-scales (Mackey & Lindenmayer, 2001).A set of climatic variables was selected to provide information about average annual values, average seasonal values, and intra-annual variation.To� pographical variables were selected because they represent a level of strong integration between multiple factors that are biogeographically important for species (Hof et al., 2012), for example, tempera� ture, air pressure, humidity, precipitation, availability of area with certain environmental conditions, soil erosion, risk of extinction, and refuges for Pleisto� cene species.Vegetation cover could be a suitable descriptor for the distribution of S. salamandra since the most favourable areas for the species in southern Spain are ecosystems associated with forests but not with pastures and crops (Miñano et al., 2003;Romero et al., 2012); the Normalized Difference Vegetation Index (NDVI; Tucker, 1979) was used as a descriptor of vegetation because it is considered a good indicator of photosynthetically active biomass (Sellers, 1985;Khan et al., 2010).Distance to inland waters was selected because S. salamandra uses river courses and water bodies for reproduction (Miñano et al., 2003).Finally, anthro� pogenic factors (or human activities), such as roads or settlements, as well as land cover change due to agricultural activities, may have serious impacts on the habitats of S. salamandra in southern Spain (Pleguezuelos et al., 2004).
All variables were available at a 1 x 1 km resolution, or were resampled at this resolution from finer ones.We computed the means of all variables within each 10 x 10 km square.Before inclusion in the modelling procedure, all variables were tested for multi-col� linearity.Variables with high multi-collinearity were removed until all remaining variables had a variance inflation factor (VIF) of < 10 (Marquardt, 1970;Mont� gomery & Peck, 1982).The remaining variables were included in the modelling routines, so that all models operated with the same variable set. Figure 2 shows a schematic description of the subsequent method� ological procedure.

Modelling techniques
The FF model assesses variation in the probability of occurrence of a species, under certain conditions, with respect to the overall prevalence of the event (Real et al., 2006;Acevedo & Real, 2012).Favourability values are, thus, independent from the species prevalence.This property enables direct comparison between models when several species are involved in the analytical design, and allows for model combinations through fuzzy logic (Barbosa & Real, 2012).For this model, we used a forward-backward stepwise logistic regression to obtain a linear combination of variables (y).Favourability values were then computed using the following equation: F = e y / [(n 1 / n 0 ) + e y ] (equation 1) where e is the basis of the natural logarithm, and n 1 and n 0 are the number of presences (= 328) and of absences (= 638), respectively.Absences were considered to be those squares not included in the presences subset.The use of a large number of   Pleguezuelos et al. (2004); C. Localización de las cuadrículas de 1 x 1 km con presencia según Tejedo et al. (2003).To avoid this, we controlled the false discovery rate (FDR) due to multiple tests using the procedure proposed by Benjamini & Hochberg (1995).
To estimate the weight of variables in the model, we used the estimated Wald test (Hosmer & Lemeshow, 2000, p. 16).
MaxEnt estimates the most uniform probability distribution (i.e., maximum entropy) of each environ� mental variable within the study area with the cons� traint that the expected value of each variable under this estimated distribution matches the mean values for the set of occurrence data (Phillips et al., 2006).MaxEnt is based on distinguishing known occurrence sites for a species from the 'background', that is the sum of presences and absences.For the occurren� ce of a given species, MaxEnt defines a probability distribution q l (x) according to this equation: where f j (x) is the value of a set of features (f 1,…, f n ), which are derived from environmental variables, at each x site, l = (l 1 ,...,l n ) is a vector of feature weights, and Z l is a normalizing constant which ensures that q l (x) sum to 1 over the study area.The results of the analyses were presented in a logistic output format (Q) so that large differences in output values better matched large differences in suitability: where z is a vector of environmental variables, q l (x(z)) is the probability distribution in sites x with environmen� tal conditions z, and H is the entropy of q l .We ran 500 iterations using the Maximum Entropy Species Distribution Modelling v3.3.3 software.Two alternative procedures were considered for feature classes and they determined two degrees of flexibility for the model fit: (1) auto features -where linear, quadratic, product, threshold, and hinge features were combined-with the application's auto-option for datasets having more than 80 training samples (Phillips & Dudík, 2008); and (2) linear features, where only linear features are permitted -as performed in the FF model.We used the regularization parameter settings (aimed at minimizing overfitting) proposed by Phillips & Dudík (2008) for more than 100 occurrence records, but we did not apply regularization multipliers (Radosavljevic & Anderson, 2014).Through the choice of these options in MaxEnt, we aimed to analyse the performance of downscaling at the edges of a whole range of degrees in model flexibility.
Maps resulting from SDMs were downscaled from a 10 x 10 km resolution to 1 x 1 km squares.The models were thus projected to a 1 x 1 km resolution grid across the study area by applying the model equations (i.e., equation 1 for the FF model, and equation 3 for the MaxEnt models) to predictor variables at this resolu� ) to predictor variables at this resolu� tion (see examples in Araújo et al., 2005;Barbosa et al., 2010).This method was chosen for simplicity.It is known as the direct downscaling approach (Bombi & d'Amen, 2012), and classified by Bierkens et al. (2000) as 'downscaling based on mechanistic models through a deterministic function (i.e., either equation 1 or equation 3)'.

Comparison of model transferability
We assessed the capacity of the models to discrimi� nate between presence and absence the receiving operating characteristic (ROC) plot.The area under the ROC curve (AUC) provides a single-number discrimination measure across all possible thresholds (Fielding & Bell, 1997).For the initial models, the ROC was assessed for the 10 x 10 km training data, and re-calculated after downscaling to the 1 x 1 km dataset.A higher AUC can indicate better perform� ance on condition that models are compared for the same species in the same study area (Lobo et al., 2008), because AUC can be influenced by the spe� cies prevalence or relative occurrence area (Chefaoui et al., 2011;Khaoruba et al., 2013).Sensitivity and specificity were also calculated before and after downscaling, considering thresholds throughout the whole suitability range.
Suitability values predicted for the study species in presence grids were analysed for the FF and MaxEnt models.We used the Mann-Whitney U-test to eva� luate changes in suitability resulting from downscaling to a finer spatial resolution.
We performed a threshold-dependent assessment to evaluate the model's capacity to describe and predict.A threshold defining 'highly suitable' areas is needed to classify the distribution of a species.Suitability values obtained with the FF model were divided into three classes: highly suitable, i.e. = > 0.8; intermediate, 0.2-0.8;and 'highly unsuitable', = < 0.2.This is equivalent to defining a prediction with odds higher than 4:1 for 'highly suitable' and 'highly unsuita� ble' sites (Muñoz & Real, 2006).For the MaxEnt mo� dels, we used the 'equalized predicted area' criterion for model comparison, which selects a threshold so that the compared models have the same predicted area (Phillips et al., 2006).Thus, for every MaxEnt model, thresholds were chosen in such a way that the sum of the grids in the results map -classified as 'highly suitable' and 'highly unsuitable'-equated those in the FF model.Thresholds calculated for the 10 x 10 km resolution outputs were also used in the 1 x 1 km outputs.
We determined the descriptive capacity of the models before downscaling (at 10 x 10 km) by com� puting the percentage of presence grids in the 'highly suitable' and 'highly unsuitable' areas.To assess the models' predictive capacity after downscaling, we used presences at a 1 x 1 km resolution from Tejedo et al. (2003) as independent data (see also Araújo et al., 2005;Barbosa et al., 2010).To calculate if the differences between percentage presences were significant we used the arcsine test of equality of percentages (Sokal & Rohlf, 1979, p. 663).
Finally, we also compared the downscaled models with alternative models trained using the 1 x 1 km dataset derived from Tejedo et al. (2003).Absences of S. salamandra were recorded if the species was not cited within the grid.The Pearson correlation coefficient was used for model comparison.

Model description
Only 16 of the original 34 variables were retained after the multicollinearity analysis.The set of varia� bles related to actual evapotranspiration was deleted, because these variables were strongly correlated with the NDVI (correlations were often higher than 0.8) and with temperature, radiation, and rainfall (correlations were sometimes higher than 0.7).All the average annual variables were also deleted because of their high VIF.If the values were reasonable, we preserved the variables that provided information on mean seasonal values and intra-annual variations of temperature, rainfall, and NDVI.Therefore, the annual temperature range and the average monthly spring rainfall were retained, despite the VIF being higher than 10 in the MaxEnt models (12.8 and 11.3, res� pectively).The 16 remaining variables were included in the three modelling routines, although only eight of these variables were selected by the stepwise procedure for the FF model (table 1).
Both the 'linear-feature' and 'auto-feature' MaxEnt models showed great similarities in ranking the impor� tance of the variables (table 1).The FF model also selected variables with high relevance in the MaxEnt models; average winter NDVI and average summer solar radiation were the most important variables in the three models.In general, suitability for S. salamandra was defined by mountain areas close to inland water, with high vegetation cover and climatically conditioned by variables describing energy and water availability throughout the year (see table 1), in agreement with the known habitat requirements for the species.

Model discrimination, sensitivity, and specificity
The FF and MaxEnt models produced visually similar geographical outputs and predictions (fig.3A).Higher suitability values were obtained for areas where S. salamandra was present (compare with fig.1).The discriminatory capacity of the three models at the lower resolution was > 0.9 for the AUC, i.e., 'outs� tanding' (Hosmer & Lemeshow, 2000, p. 162), and around 3% higher when comparing the 'auto-feature' MaxEnt model with the models combining variables linearly (see table 2).After downscaling, the capacity of the three models to discriminate between 1 x 1 km squares with and without presences was between 0.8 and 0.9 i.e., 'excellent'; the AUC was around 2.5% higher for the FF model.
For all three models, the minimised difference bet� ween sensitivity and specificity was close to the 0.4 suitability value for the higher spatial resolution (fig.4A).The sensitivity of the FF model was significantly higher than for the MaxEnt models where suitability was > 0.4.After downscaling (fig.4B), the minimised differen� ce between sensitivity and specificity was at the 0.8 suitability value for the FF model, but remained close to the 0.4 suitability value using MaxEnt.Sensitivity increased considerably after downscaling in the FF model, whereas it decreased minimally in MaxEnt; the specificity decreased in all three models, though more strongly in the FF.

Suitability values
Average suitability for presence grids increased signifi� cantly after downscaling in the FF and 'linear-feature' MaxEnt models (Mann-Whitney U-test, p < 0.001); the increase was steeper in the FF model (see table 3, fig.5), but suitability values remained stable in the 'auto-feature' MaxEnt model (p > 0.1).

Threshold-dependent evaluation
After downscaling, the percentage of 'highly suitable' squares increased (almost doubling) and the percen� tage of 'highly unsuitable' grids decreased slightly when using the FF model.Both percentages remained almost stable when using MaxEnt (fig.3B; table 2).
Compared to the results of the MaxEnt models, the FF model showed significantly more presence grids in the 'highly suitable' area after downscaling (arcsine test of equality of percentages, p < 0.01) and significantly fewer presence grids in the 'highly unsuitable' area (p < 0.01).The results computed using the 'linearfeature' MaxEnt showed significantly more presence grids in the 'highly suitable' area than the results computed using 'auto-feature' MaxEnt (p < 0.01).
The percentage of recorded presence grids in 'highly suitable' areas significantly increased after downscaling when using the FF model and also when using 'linear-feature' MaxEnt (arcsine test of equality of percentages, p < 0.01; the increase was steeper in the former case, see table 2, fig.5).Within the 'highly unsuitable' areas, this percentage of presence grids increased significantly using 'auto-feature' MaxEnt (p < 0.01), and decreased significantly using the FF model (p < 0.01).

Downscaled models vs. models trained with high resolution data
The AUC of the models trained with independent 1 x 1 km data (fig.3C) were higher than 0.8 (table 2) whiche� ver modelling technique was used; this was only slightly higher than the corresponding downscaled models.Correlation coefficients between models derived from downscaling and models trained with 1 x 1 km data performed using the same algorithm (main diagonal in table 4) were significant (p < 0.01), being higher than 0.7 using 'auto-feature' MaxEnt, higher than 0.8 using 'linear-feature' MaxEnt, and higher than 0.9 using the FF model.The three downscaled models had their highest correlation with the model trained with 1 x 1 km data using the FF model (left column in table 4), and correlations with the downscaled FF model (upper line in table 4) equalled the intra-technique correlation with the models trained at 1 x 1 km (main diagonal).

Discussion
Model downscaling: valuable for conservation planning?Species distribution maps from biological atlases are often too coarse for real-world conservation planning (Kunin et al., 2000;Araújo et al., 2005;Barbosa et al., 2010;Bombi & D'Amen, 2012).National atlases, such as those for mammals in Spain, are at best at resolutions no Fig. 2. Schematic description of the methodological procedure: * Downscaled suitability values in 1 x 1 km squares are also compared to suitability values predicted by models trained with 1 x 1 km data.less than 10 x 10 km (Palomo & Gisbert, 2008), but those for larger areas, such as those for continental European mammals, can be coarser (50 x 50 km) (Mitchell-Jones et al., 1999).For management purposes, downscaling to finer resolutions is useful.Although some authors argue that it is not possible to achieve a higher level of detail beyond that contained in the initial coarser-resolution maps (Stockwell & Peterson, 2003), others suggest that the properties that define a species distribution are the same at whatever resolution (Kunin, 1998).Fig. 3. Geographical representation of the three distribution models performed for the fire salamander, Salamandra salamandra, in Andalusia: A. Suitability based on the models performed with 10 x 10 km resolution data, before and after downscaling to a 1 x 1 km resolution; B. Subdivision of Andalusia in highly suitable, intermediate and highly unsuitable areas (the threshold criterion for the FF model was defining a prediction with odds higher than 4:1 for 'highly suitable' grids, and lower than 1:4 for 'highly unsuitable' grids; the equalized predicted area criterion was chosen with MaxEnt, i.e., equating the 'highly suitable' and the 'highly unsuitable' surface areas to those in the FF); C. Suitability based on the models performed with 1 x 1 km resolution data; 1. Favourability; 2. Linear-feature MaxEnt; 3. Auto-feature MaxEnt.

Fig. 2. Descripción esquemática del procedimiento metodológico: * Los valores de idoneidad al aumentar la resolución en cuadrículas de 1 x 1 km también se compararon con los valores de idoneidad predichos por los modelos calibrados con datos de 1 x 1 km.
Fig. 3. Representación geográfica de los tres modelos de distribución realizados para la salamandra común, Salamandra salamandra, en Andalucía: A. Idoneidad basada en los modelos realizados con datos con resolución de 10 x 10 km, antes y después del aumento de resolución a 1 x 1 km; B. Subdivisión de Andalucía en áreas altamente idóneas, de idoneidad intermedia y altamente inadecuadas (el criterio para establecer umbrales en la función de favorabilidad consistió en definir como "altamente idóneas" las cuadrículas con un pronóstico de presencia mayor que 4:1, y como "altamente inadecuadas" las cuadrículas con un pronóstico de presencia menor que 1:4; con MaxEnt se ha utilizado el criterio de área predicha igualada, que consiste en igualar la superficie de las áreas "altamente idóneas" y "altamente inadecuadas" a las observadas con la función de favorabilidad); C. Idoneidad basada en los modelos realizados con datos con resolución de 1 x 1 km; 1. Favorabilidad; 2. Ajuste lineal de MaxEnt; 3. Ajuste automático de MaxEnt.In our study, we showed that downscaling from 10 x 10 km maps to a 1 x 1 km resolution caused only a slight loss of discriminatory capacity.Overall, both methods had outstanding discriminatory capacities at the coarser resolution, even though the species' prevalence in Andalusia (37%) was not remarkably low; these capacities remained excellent after downscaling.Some loss of discriminatory capacity after downscal� ing is not surprising.As spatial resolution becomes finer, local effects become more important (Hewitson & Crane, 1996).Thus, a reduction in accuracy is ex� pected because the importance of climate in influencing species distributions decreases at more local scales, whereas the influence of micro-environmental factors such as topography increases (Pearson et al., 2002; Table 2. Model assessment: FF.Favourability function; LMx.Maxent using only linear features; AMx.MaxEnt using the auto-feature option; a State variables were based on the distribution of S. salamandra according to Pleguezuelos et al. (2004) (10 x 10 km resolution) and to Tejedo et al. (2003) (1 x 1 km resolution); b Models performed with 10 x 10 km resolution data; c Models downscaled to a 1 x 1 km resolution; d Models performed with 1 x 1 km resolution data; e Asterisks indicate significant differences between percentages (arcsine test of equality of percentages; p < 0.01) compared to FF; f Asterisks indicate significant differences between percentages (arcsine test of equality of percentages; p < 0.01) compared to those before downscaling.
Tabla 2. Evaluación del modelo: FF.Función de favorabilidad; LMx.MaxEnt utilizando solo el ajuste lineal; AMx.MaxEnt utilizando el ajuste automático; a Las variables se basaron en la distribución de S. salamandra según Pleguezuelos et al. (2004) (resolución de 10 x 10 km) y Tejedo et al. (2003) (resolución de 1 x 1 km); b Modelos realizados con datos con resolución de 10 x 10 km; c Modelos con resolución aumentada a 1 x 1 km; d Modelos realizados con datos con resolución de 1 x 1 km; e Los asteriscos indican diferencias significativas entre porcentajes (transformación arcoseno de igualdad de porcentajes; p < 0,01) en comparación con FF; f Los asteriscos indican diferencias significativas entre porcentajes (transformación arcoseno de igualdad de porcentajes; p < 0,01) en comparación con los porcentajes antes del aumento de resolución.Trivedi et al., 2008).Araújo et al. (2005) suggested that interpolation is possible if governing processes affecting species' distributions at coarser resolutions are also important in driving distributions at finer resolutions.In our case study, a combination of environmental fac� tors that could affect the distribution of S. salamandra at different spatial resolutions was used to prevent excessive loss in the biological meaning of the model.According to Guisan et al. (2007) and Bombi & D'Amen (2012), a 10-fold shortening of the grain size -referring to pixel side length-as is our case, should not severely affect predictions of species distributions.However, this meant that the number of finer-scale observations and predictions was roughly 100 times higher, and that the proportion between observed pres� Fig. 4. Sensitivity and specificity of the three models before (A) and after (B) their interpolation to a finer resolution.The fire salamander, Salamandra salamandra, distribution data provided by Pleguezuelos et al. (2004) were used at a 10 x 10 km resolution, and by Tejedo et al. (2003) at a 1 x 1 km resolution (see fig. 1).
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 ence and absence was much lower than that at the coarser-scale.Consequently, in many suitable 1x1 km sites, the species was not observed, reducing the discriminative capacity of the model.In this situation, the downscaled model cannot -and should not-be very specific, and sensitivity should be considered as a better measurement of performance than specificity.This is why AUC, which weighs sensitivity and specifi� city equally, is expected to be lower after downscaling as a result of a much lower presence/absence ratio in the 1 x 1 km resolution dataset than in the training data.Bombi & D'Amen (2012) also observed that a more general effect of downscaling is a reduction of specificity.
Our finding that our downscaled models had excellent discriminatory capacity, with a high capacity to predict species occurrences, suggests that this approach can be valuable for decision-making in conservation.None� theless, model downscaling will be more successful for wide-ranging taxa than for taxa with smaller home ranges that often have especially aggregated distribu� tions, and low dispersal (Kaliontzopoulou et al., 2008;Barbosa et al., 2010).This is because the former are more dependent on macro-environmental factors than spatially restricted species.

Threshold independent comparison
Using a visual inspection of the geographic output, Araújo et al. (2005) found that downscaled maps were not only able to recover original spatial patterns of rich� ness observed at the coarser resolution, but were also able to identify finer gradients that were invisible in the original resolution.In our study, the 'auto-feature' MaxEnt model produced a more fragmented 1 x 1 km geographi� cal pattern than the other models (fig.3A); this could be related to the degree of flexibility of this model being higher than the other two.The 'hinge' feature class in the auto-features option makes MaxEnt resemble the procedure for generalized additive models (GAMs) that find a flexible geographical relationship between species and the environment (Elith et al., 2010).This leads to a high model fit to observed species distributions (see fi g. 6), which in our case provided a clear match between predicted highly suitable areas for S. salamandra and mountain rivers (see fig. 3A).Barbosa et al. (2010) also observed that the courses of rivers emerged as suitable areas for Lutra lutra and for Galemys pyrenaicus in Spain after downscaling from a 10 x 10 km resolution model, despite the fact that river locations were not explicitly included as predictor variables in their models.
At a more quantitative level, AUC comparisons suggested that model flexibility may have enhanced the discrimination capacity of the original output, at the cost of hindering its capacity to discriminate after downscaling.This is demonstrated by the fact that the 'auto-feature' MaxEnt model was the best discrimina� tor before downscaling, whereas the FF model was the best discriminator after downscaling (table 2).As observed in our results, previous MaxEnt-GLM com� parisons showed AUC values that were on average 2%-3% higher in MaxEnt models than in logistic-re� MaxEnt models than in logistic-re� gression models (e.g., Elith et al., 2006 [n = 10]; Gibson et al., 2007 [n = 1]; Marini et al., 2010 [n = 12]).Here, this only happened with the 'auto-feature' MaxEnt model, which permitted the highest flexibility, and also yielded the lowest AUC values when downscaled at the 1 x 1 km resolution.

Threshold-dependent comparison
The FF model was more sensitive to both 10 x 10 km and 1 x 1 km presence observations of S. salamandra than the MaxEnt models, especially when the suitability threshold was high (fig.4).Once a threshold for 'highly suitable' areas was defined, the FF model produced the highest proportion of presence observations in 'highly suitable' areas at the 1 x 1 km resolution (table 2).However, below the threshold for very low unsuitability, the FF model had the lowest proportion of presences, that is, the lowest over-prediction rate at this threshold (Barbosa et al., 2013).In contrast, the lowest proportion of presence grids in 'highly suitable' areas was obtained in the 'auto-feature' MaxEnt model (table 2).A possible explanation for this is that the detailed definition of water courses in the downscaled 'auto-feature' MaxEnt model   .Promedio de la idoneidad: de F para la función de favorabilidad (ecuación 1), y de Q para MaxEnt (ecuación 3) en las cuadrículas con presencia de salamandra común (Salamandra salamandra), y proporción de cuadrículas con presencia en áreas consideradas muy idóneas desde el punto de vista ambiental según los modelos, antes y después del aumento de resolución espacial.Los valores de MaxEnt se han representado de acuerdo con ambas opciones de ajuste, "lineal" y "automático", entre presencias y variables ambientales.

Average suitability value
Presence proportion in highly suitable areas 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3  from the 'background', selecting 'wrong' backgrounds may cause erroneous outputs that could be incorrectly interpreted as failures of transferability.In this study, the two MaxEnt models used the same 'background' for comparisons, i.e., the entire set of 10 x 10 km squares in the study area.When a very flexible combination of variables was accepted, our results suggested the same relationship between overfitting and transferability cost in MaxEnt as that described in Peterson et al. (2007).Constraining MaxEnt to linear combinations of variables largely solved the problem of overfitting, but the transferred Favourability model still showed a better fit to the finer-resolution data than the MaxEnt models.

Fig. 1 .
Fig. 1.Geographical context of the study area and distribution of the fire salamander, Salamandra salamandra, in Andalusia: A. Spain is coloured grey, with Andalusia (southern Spain) shaded dark grey; B. Distribution in 10 x 10 km UTM squares according to Pleguezuelos et al. (2004); C. Location of 1 x 1 km grids with presence according to Tejedo et al. (2003).

Fig. 5 .
Fig. 5. Average of suitability: F for favourability function (equation 1), and Q for MaxEnt (equation 3) in fire salamander, Salamandra salamandra, presence grids, and the proportion of presence grids in areas considered environmentally highly suitable by the models, before and after downscaling.MaxEnt values are represented according to both the 'linear-feature' and the 'auto-feature' options for the relationship between presences and environmental variables.

Fig. 6 .
Fig. 6.Response curves showing how four environmental variables (x-axes) affected the model predictions (i.e., suitability for the fire salamander, Salamandra salamandra, in the y-axes, that is either the favourability function value F in equation 1 or the MaxEnt output Q in equation 3) while keeping all other variables constant in their average sample value: Black.Favourability function; Dark grey.Linear feature in MaxEnt; Light grey.Auto feature in MaxEnt.

Table 1 .
List of variables considered for the environmental models of Salamandra salamandra in Andalusia.The relative importance of each variable entered in the models was measured with the Wald parameter for the favourability function (FF), and with the percentage contribution for the two MaxEnt models: AMx.
b Average spring evapotranspiration (mm) b Average summer evapotranspiration (mm) b Average autumn evapotranspiration (mm) b Average winter evapotranspiration (mm) b Average annual solar radiation (Wh/m 2 ) c,d Average spring solar radiation (Wh/m 2 ) c,d

Table 3 .
Mann-Whitney U-test to compare the rank of suitability values for Salamandra salamandra before and after downscaling: (see fig.3Band discussion of the threshold-independent comparison) may be an artefact; it excluded locations, often far from water courses, in which individuals of this species are generally found outside the breeding season.