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Our hypothesis and objectives.  We hypothesize that the northern and southern edges of Neanderthals range were respectively limited by cold winter temperatures, and a combination of high temperatures and low water availability during the summer, while 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) reduction of primary productivity and low availability of small and big game would have compromised the high caloric intake required by this species \cite{Steegmann_2002}; 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. High summer Summer  temperatures linked to the higher  solar radiation of lower latitudes  and continentality 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 (Churchill 2006), that hampers heat dissipation. But Mediterranean coastal areas could have been suitable because of the buffering effect of the sea over temperatures temperatures,  and the permanent availability of resources like shellfish \cite{Hardy_2011}. At the local scale, high topographic diversity linked with diversity, that fosters biodiversity by  an increased availability of ecological niches (CITATION) could have \cite{Tews_2003},  provided the required abundance and diversity of preys, preys \cite{Daujeard_2012},  but moderate to low slopes may have been important to reduce the energetic cost of local mobility. Our hypothesis does not