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Our knowledge about Neanderthals has greatly increased over the last two decades with more than two thousand research papers published in areas as diverse as chronology \cite{Hardy_2010, Gaudzinski_Windheuser_2014, Higham_2014}, ecology \cite{Stewart_2005, Finlayson_2007, Henry_2010}, population dynamics \cite{Trinkaus_1995, Premo_2008, Fabre_2009, S_rensen_2011, Bocquet_Appel_2013}, adaptive traits \cite{Sorensen_2001, S_rensen_2009, Rae_2011}, diet \cite{Stringer_2008, Richards_2009, Henry_2010, Hardy_2011}, technology and cognition \cite{Wynn2004467, Hardy201323, Fern_ndez_Peris_2012, Riel_Salvatore_2010}, genetics \cite{Pr_fer_2013, Briggs_2009, Sanchez_Quinto_2014}, and their relationship with anatomically modern humans \cite{Gibbons_2014, Sankararaman_2012,Banks_2008, d_Errico_1998}. Still, our knowledge about their geographical distribution across time and the environmental factors limiting it remains sparse and shallow, and only a few authors have tried to go beyond local or regional presence patterns. For example, \citet{Wenzel_2007} describes the distribution of Neanderthals during MIS 5e across central Europe, and discusses briefly the influence of climatic conditions and habitat features over the general occupancy pattern, but does not establish any quantitative link between presence and environment. \citet{Banks_2008} analyzed the distribution of Neanderthals between 43.3 and 36.5 ka BP applying pSDMs, assessed niche conservatism, and analyzed the importance of climatic predictors, concluding that temperature was the most important driver shaping the distribution of this hominin.  In this paper we propose a hypothesis about the abiotic drivers of Neanderthals distribution deeply rooted on the Grinnellian niche concept \cite{Grinnell_1917}, the hierarchical framework to explain the importance of species-distribution drivers at different scales proposed by \citet{Pearson_2003}, and our current knowledge on the ecology of Neanderthals (see references above). According to \citet{Pearson_2003}, climate influences species distribution at global to regional scales, while the effect of topography is restricted to scales ranging from regional to local.\fixme{Check that my edits that split the parameters relevant in the north from those relevant in the south are actually correct.} local.  We hypothesize that the northern edges of the Neanderthals range were limited by cold winter temperatures and the southern edges of their range by a combination of high temperatures and low water availability during the summer (whereas a high topographic diversity combined with moderate slopes could have favored occupation at the local scale). Winter temperatures at the northern edge could have exert a negative impact over Neanderthals populations in two related ways: 1) lower availability of small and big game than in temperate areas \cite{Badgley_2000} would have compromised the high caloric intake required by this species \cite{Steegmann_2002, S_rensen_2009}; 2) increased cold stress worsened by a low-caloric diet would have lead to an even more reduced fertility \cite{Bocquet_Appel_2013} and a higher mortality rate \cite{Steegmann_2002}, compromising population survival. Summer temperatures linked to continentality and the higher solar radiation of lower latitudes could have prevented the occupation of southern plains in the Mediterranean peninsulas due to increased heat stress, specially considering the low surface area/volume ratio of this species \cite{Churchill2006}, which hampers heat dissipation. Also, summer drought could have probably reduced access to plant resources, making these areas unsuitable during the warm season. Mediterranean coastal areas could have been suitable because of the buffering effect of the sea over temperature and the permanent availability of resources like shellfish \cite{Hardy_2011}. At the local scale, high topographic diversity, which fosters biodiversity by an increased availability of ecological niches \cite{Tews_2003}, provided the required abundance and diversity of preys and shelter \cite{Daujeard_2012}, but moderate to low slopes may have been important to reduce the high energetic expenditure of mobility in steep terrain. This limitation, combined with the lower abundance of animals at higher elevations \cite{Brown_2001}, could have prevented Neanderthals presence in the higher elevations of the European mountain ranges. To test our hypothesis, we have modeled and analyzed the distribution of Neanderthals during the full Eemian (MIS 5e, ~130 ka BP) using a palaeo species distribution modeling approach (PSDMs hereafter) \cite{Franklin20151, Svenning20112930, Varela2011451}. PSDMs rely on the same principles of species distribution modeling \cite{Guisan2000147, Guisan_2005}. PSDMs require presence coordinates representing the location of fossil remains, a set of predictors (usually palaeo-climatic simulations), and an algorithm to define the relationship between presence and predictors. The result of a PSDM can be defined as a \emph{habitat suitability map}, a raster map in which each cell is ranked according how well they resemble the ecological conditions of those locations at which the focal species occur \cite{Soberon_2005}. PSDMs rely on the assumption that the modeled species is in equilibrium with its environment, and therefore occurs in all the localities with suitable habitat \cite{Guisan_2005}. Aside from providing a graphical description of the potential distribution of the species, PSDMs can be used to analyze the influence of particular drivers over the species distribution and gain ecological knowledge, as in \citet{Varela_2009} or \citet{Rodr_guez_S_nchez_2008}. PSDMs have been used before to study the distribution of Neanderthals and Anatomically Modern Humans \cite{Burke201435, Banks_2008, Banks2008481, Beeton_2013}, but in this paper we also offer a new approach based on recursive partition trees to assess the importance of the predictors at the local scale, and offer a novel insight into the drivers of Neanderthals distribution. 
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In summary, in this paper we propose a hypothesis about how abiotic drivers (climate and topography) could have shaped Neanderthals distribution throughout Europe, and we test it by fitting and analyzing a PSDM describing their distribution during the Last Interglacial.