The authors mention two competing hypotheses, namely
- the Neanderthals were unable to adapt to the changing environment, and
- competitive exclusion by anatomically modern humans (AMH, aka Cro-Magnon, aka Homo sapiens) drove them to extinction.
- Neanderthals and AMH interbred and the distinction between them disappeared,
- they perished because they
didn't publishweren't able to evolve resistance to some pathogen or other, and of course
- that they were wiped out by AMH in bloody feuds that so seem to define the winners to this day.
The idea behind the competition hypothesis is that for longer periods of time, only one species can exist in a single niche. A niche is a nice way to describe something which is very difficult to define, but it largely means "way of life," and is comprised of all the resources that the species utilize in order to survive. Figuring out exactly what those resources are for a given species is not easy at all, because it concerns not only the foods that are eaten, but, in the words of David Tilman
any substance or factor which can lead to increased growth rates as its availability in the environment is increased, and which is consumed by an organism. Substances that can be a resource include (but may not be limited to) chemicals, other living organisms, and space (but note that if you read this paper you'll find they use 'niche' as meaning 'geographical range').
If two species, or populations, are sharing a niche (or overlap to a large extent), then they will compete for the resources, and one will necessarily drive the other to extinction given enough time (though not a lot is needed on evolutionary time-scales). The selective pressure to utilize different resources can even drive two populations of the same species away from each other, resulting in a speciation event. The two species can then coexist exactly because they no longer utilize the same resources, and thus do not occupy the same niche anymore. This happens to be one of the research projects that I am working on at the moment, so enough about that.
The study looked at the three climatic periods to evaluate whether climate change can have been the cause of the Neanderthal extinction. The three periods are
- "Pre-H4": 43.3–40.2 kyr with mild weather,
- "H4": 40.2–38.6 kyr with cold weather, and
- "GI8": 38.6–36.5 kyr with mild weather again.
The computer simulation they applied (GARP, download) used archeological data from Neanderthal and AMH archeological sites to pinpoint their geographical ranges, as well as data about the landscape and climatic dimensions potentially relevant to shaping the distribution of the species. The results are that the geographical range of the Neanderthals was much smaller in GI8 than what would be expected based on the climatic data. Thus, they conclude, the vanishing distribution of Neanderthals was not due to problems coping with the changing environment, and must then have been due to competitive exclusion by AMH. The last words in the paper's discussion are these:
The AMH expansion and Neanderthal contraction of niche characteristics were concurrent, and we suspect causally related. It follows that there was certainly contact between the two populations, which may have permitted both cultural and genetic exchanges. Our findings clearly contradict the idea that Neanderthal demise was mostly or uniquely due to climate change  and looks towards AMH expansion as the principal factor. Hence, we contend that AMH expansion resulted in competition with which the Neanderthal adaptive system was unable to cope.So what they have done is to rule out one hypothesis (hyp. #1), and, without further proof, sort of succumb to the only other they list (hyp. #2), but which they did not investigate at all. As I mentioned earlier hyp. #3 has been ruled out, and hyp. #5 is perhaps part of hyp. #2 (driven to extinction by competition for resources and/or by direct hostile contact with humans). But since the Neanderthal population was very small towards the end, I don't think we can totally rule out that some disease had something to do with their demise. Additionally, it would be really nice with some direct evidence in support of the competitive exclusion hypothesis, which for example could take the form of data indicating that the resources utilized by the two Homo species were identical. Perhaps that already exists, in which case it would have been nice if it had been mentioned in this paper.
 William E. Banks, Francesco d'Errico, A. Townsend Peterson, Masa Kageyama, Adriana Sima, Maria-Fernanda Sánchez-Goñi (2008). Neanderthal Extinction by Competitive Exclusion PLoS ONE, 3 (12) DOI: 10.1371/journal.pone.0003972
 R GREEN, A MALASPINAS, J KRAUSE, A BRIGGS, P JOHNSON, C UHLER, M MEYER, J GOOD, T MARICIC, U STENZEL (2008). A Complete Neandertal Mitochondrial Genome Sequence Determined by High-Throughput Sequencing Cell, 134 (3), 416-426 DOI: 10.1016/j.cell.2008.06.021
 David Tilman, Resource Competition and Community Structure, 1982, Princeton University Press.