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\paragraph{Species distribution modeling}  The evaluation approach of the models  showed that both AUC and adjusted explained deviance values increased with larger background radius (Fig. 2). 55 models(each model being a combinations of an equation and a background radius)  were above the selection thresholds for both AUC (0.65) and adjusted explained deviance (0.15). The average AUC values for the best models was 0.78 (0.05 standard deviation), while average explained deviance was 0.23 (0.07 standard deviation). The final habitat suitability model (Fig. 3) showed high habitat suitability values and low standard deviation (high agreement among models) in central and southern France, Italian Peninsula, Mediterranean Islands, Atlantic coast of the Iberian Peninsula, southern coast of the Black Sea and the Oriental coasts of the Mediterranean Sea. Medium habitat suitability values were found at the continental areas of the Iberian Peninsula, British islands, Dinaric Alps, Balcanicn Mountains, continental areas of Anatolia and southern coasts of the Baltic Sea. Very low habitat suitability values consistent with non suitable habitats were dominant across Alps, Pyrenees, North European Plains (continental areas of Poland, Belarus and Ukraine), across the Scandinavian Shield, and the current Arabian Desert.  The lack of agreement among the best models was high in the north of the British islands, Scandinavian Mountains, margins of the Alps and Pyrenees, Carpathian mountains, central plains and north western coast of the Iberian Peninsula, and the northern limit of the Arabian desert.  The habitat suitability values of the presence recordsused to fit the models  (Fig. 4) was generally high (average 0.69, standard deviation 0.16), but all sites at the north and eastern distribution limits (Germany, Slovakia and Turkey) showed low habitat suitability values. The average standard deviation for the Neanderthal sites was 0.12 (standard deviation 0.04), but two sites showed higher deviation values than the others: Karain Cave (KARA, Turkey) and Hôrka-Ondrej (HORK, Slovakia). \paragraph{Importance of environmental factors}  The analysis of variable importance performed with Random Forest was robust  (93.61 explained deviance) deviance), and  showed that minimum winter temperature (64.48 \% increment in mean squared error), annual rainfall (60.10 \%IncMSE), and slope (59.23 \%IncMSE) were the most important  factors shaping habitat suitability at the continental scale. The temperature of the warmest month showed an intermediate importance (45.19 \%IncMSE), while summer rainfall (27.66 \%IncMSE) and topographic diversity (25.00 \%IncMSE) were the least important variables. The response curves of the most important predictors (Fig. 5) showed that the optimum habitat suitability values happened when the temperature of the coldest month was higher than 5 Cº), Cº,  the annual rainfall was between 700 and 1200 mm, and the topographic slope was between 3 and 7 degrees. The recursive partition analysis of the 44 localities revealed the different environmental processes shaping habitat suitability across Europe. The model based on variable values (Fig. 6) showed that a minimum temperature of the coldest month around -3.7 ºC separated localities with low and high habitat suitability, and localities with a maximum temperatures of the warmest month between 29 and 34 ºC showed the highest habitat suitability, and were mostly concentrated along the Mediterranean coast. A few Mediterranean spots did not follow this pattern due to extremely hot summers (bio5 > 34ºC): South-eastern coast of Iberia (27), the region of Marmara (2) and the Black Sea region in Anatolia (3), and Chipre (17).   The model based on the local R-squared of the predictors (Fig. 7) showed terminal nodes with a wide range of habitat suitability values, specially for nodes 3, 5 and 6, and a general low reliability (high difference between actual and predicted suitability according to the recursive partition model). Node 3 highlighted localities in which both slope and temperature of the warmest month were important in defining low habitat suitability values, and correspond to areas in which both variables show extreme values, like Pyrenees (ID=28, steep terrain and cold climate) or Central Anatolia (ID=5, flat terrain and hot climate). According to this model, the highest habitat suitability (node 7) is reached when the local R-squared of slope is lower than 0.38, and no other predictor seems to be important.We will discuss this counterintuitive result in the Discussion section.  Finally, the model based on the local coefficients of the predictors showed that higher habitat suitability values happen when the coefficient of the slope is between 0.07 0.070  and -0.006 (zero indicates optimum slope) and the coefficient of the average temperature of the coldest month is lower than 0.004 (approaching zero and indicating optimum value for the predictor). A very high coefficient of the slope (>0.07) indicated a low habitat suitability due to a  lack of rugosity in the terrain, and therefore low suitability, terrain complexity,  as happens  in the Po Valley (ID=32), (ID=32)  or the Pannonian Plain (ID=8).