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In this paper we present the first model of Neanderthals potential distribution across Europe during the Last Interglacial (Eemian, MIS 5e, ~130 ka BP). We have applied a robust state-of-the-art modeling approach to take advantage of a sparse set of archaeological records and a palaeoclimatic simulation to produce the model. We analyzed the model using regression-tree based statistical methods to assess how environmental variables shaped habitat suitability at continental and local scales, and assessed to what extent our preliminary hypothesis on the abiotic drivers of Neanderthals distribution matched the results of the model.  The discussion about the distribution of Neanderthals across Europe during the Eemian has been the focus of just a few papers during the last years \cite{Richter2005, Richter2006, Wenzel_2007, Gaudzinski2011}. \cite{Krause_2007} neanderthals in asia and siberia \cite{Beeton_2013}  Check the results of this paper!: http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0003972 10.1371/journal.pone.0003972  \textbf{How do the model results fit the hypothesis presented in the introduction?}  According to our proposed hypothesis, the northern edge of Neanderthals distribution should be controlled by cold winters due to low availability of small and big game and cold stress. Our model confirms that minimum temperature of the coldest month influences habitat suitability at the continental scale, but the local scale analysis shows that the low habitat suitability of places within and beyond the northern edge (for example: Southern Baltic coast, Jutland peninsula, Southern Sweden, Southern Finland) can be explained by a combination of cold winters and plain slopes (LARGE FORESTED PLAINS WITH LOW PRODUCTIVITY??). We hypothesized that hot and dry summers may have shaped the southern limit of Neanderthals distribution due to heat stress and low summer productivity resulted from drought, but also that this effect could have been buffered by the nearness of the sea in the Mediterranean coastlines. The analysis of our model showed that southern localities with summer temperatures higher than 34 ºC (2, 3, 17, 26, 27) consistently presented low habitat suitability, while in the Mediterranean coastlines (and southern coast of the Black Sea), with summer temperatures between 29 and 34ºC, habitat suitability was generally very high.  but we failed to detect a consistent negative effect of summer rainfall.  \textbf{Implications for Neanderthal dispersal} 
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Our model shows at least six suitable geographical regions more or less isolated from each other: the Iberian Peninsula, limited by the Pyrenees; France and Western Germany, limited by the Alps, the north European plains and the sea; British Islands, but Neanderthals presence during the Eemian has not been confirmed there \cite{Penkman_2011}; Italian peninsula, limited by the Po Valley and the Alps; Peloponese and Aegean Islands, limited by the sea (but see \cite{Ferentinos_2012} and \cite{Broodbank_2006} for a discussion about the possibility of seafaring by Neanderthals), but without any reliable Neanderthal material attributed to MIS 5e; and the coasts of Anatolia, Lebanon and Israel.  In the long term, the compartimentation of Neanderthal populations could have lead to genetic differentiation, as shown in \cite{Fabre_2009}. In this paper the authors studied genetic variation within twelve samples of mitochondrial DNA obtained from Neanderthal samples dated between 100 and 29 ka BP (but according to \cite{Higham_2014} the latter date may be more probably around 40 ka BP). They defined two genetic demes within our study area (see Fig. 2 in \cite{Fabre_2009}): \cite{Fabre_2009}:  a northern region that comprises the North of the Iberian Peninsula, Eastern France, Germany and the rest of central Europe; a southern region comprising the East of the Iberian Peninsula, South of France, Italian Peninsula, and the Balcans. These demes do not completely fit with our habitat suitability model: there seem to be connectivity between Eastern and Western Iberian Peninsula due to a corridor of suitable habitat at the south of Pyrenees, and between Western and Eastern France, that show a large extension of well connected suitable habitat. But both models agree in the important effect of the Alps as the main geographic accident shaping the separation between ecologically suitable areas and genetic demes. A detailed analysis of habitat suitability maps along with cultural and genetic evidences can help to understand the causes for gradients of cultural and genetic change across large territories. \textbf{Given our conclusions, what is our new understanding of the distribution of Neanderthals and the abiotic factors shaping it?}