About 39 thousand years ago all sign of the presence of Neanderthal bands in their extensive range across western Eurasia disappears. Their demise occurred during a period of relative warmth (Marine-Isotope Stage-3) following a cold period at its worst around 65 ka (MIS-4). They had previously thrived since their first appearance in Eurasia at about 250 ka, surviving at least two full glacial cycles. Their demise occurred around 5 thousand years after they were joined in western Eurasia by anatomically modern humans (AMH). During their long period of habitation they had adapted well to a range of climatic zones from woodland to tundra. During their overlap both groups shared much the same food resources, dominated by large herbivores whose numbers burgeoned during the warm period, with the difference that Neanderthals seemed to have depended on ranges centred on fixed sites of habitation while AMH maintained a nomadic lifestyle. Having shared a common African ancestry about 400 thousand years ago, DNA studies have revealed that the two populations interbred regularly, probably in the earlier period of overlap in west Asia from around 120 thousand years ago and possibly in Europe too after 44 ka. Considering their previous tenacity, how the Neanderthals met their end is something of a mystery. It may have been a result of competition for resources with AMH, which could be countered by the increase in food resources. Maybe physical conflict was involved, or perhaps disease imported with AMH from warmer climes. Genetic absorption through interbreeding of a small population with a larger one of AMH is a possibility, although DNA evidence is lacking. An inability to adapt to climate change contradicts the Neanderthals long record and their disappearance during MIS-3. Previous population estimates of changing Neanderthal populations in the Iberian Peninsula (see Fig. 2 in Roberts, M.F. & Bricher, S.E 2018. Modeling the disappearance of the Neanderthals using principles of population dynamics and ecology. Journal of Archaeological Science, v. 100, p.16-31; DOI: 10.1016/j.jas.2018.09.012) show decline from about 70,000 to 20,000 before MIS-4, then recovery to about 40,000 before the arrival of AMH at 44 ka followed by a decline to extinction thereafter. Roberts and Bricher developed a model for investigating demographics from archaeological evidence that is neutral as regards any particular hypothesis for Neanderthal extinction.