Some key points:
1. Positively selected for adapatations caused by mutations move slowly. "During the course of the Holocene, a strongly selected mutation might move only across a radius of a thousand or so kilometers."
2. If there is more than one positively adaptive mutation and each affects the same trait, each expands along a "wave of advance" and if they are equally advantageous, "once the "waves of advance" meet, they stop moving, forming a stable boundary." If one is more advantageous than the other, the encount of the waves of advance slows the spread of the more advantageous mutation, while stopping the spread of the other mutation, with the amount of slowing related to the relatively selective advantage conferred by the competing mutations.
Hawks' three paragraph explanation of what is going on in the mathematical model called the Fisher equation, that is used to explain the diffusion of mutations in time and space that leads to this effect is particularly lucid and deserves a read from anyone who wants to understand the math without actually doing the math.
3. In real life, these models have limitations:
[A]lleles are unlikely to substitute perfectly for each other. In many cases, they may work synergistically -- individuals carrying two selected alleles that affect the same function may outperform those carrying only one such allele. At some point, new selected mutations may start to have diminishing returns, even on a trait like skin pigmentation where dozens of alleles may have been selected in widespread human populations. So the current distribution may to some extent be "frozen", but by a more complicated dynamic than the simple intersection of waves of advance.
4. There are lots of allels (version of particular genes) that are not in equilibrium in modern humans; "there are really only few genes that have approached local fixation in recent human evolution. The current spatial pattern of recently selected alleles doesn't look like a tesselation with many alleles near local fixation. Over most of the Old World, it looks like populations have a very large number of very new alleles, far from fixation, and few up over 70 percent in frequency."
5. Population sizes influence the likelihood that the same adapatation will occur independently in different populations, rather than spreading form a single source.
[M]ore people means more mutations. In their case, they focus on population density over space (population number, when you multiply them) as a constraint on the number of possible adaptive mutations. They apply this idea as a hypothesis to account for parallel adaptations that may have emerged in recent human evolution.
6. Limited population sizes mean that modern humans (and even more so our predecessors) are mutation-limited. Only a small fraction of mutations that would be adaptive if the happened actually do happen:
[S]mall human populations would have been mutation-limited -- that is, the number of new mutations is small, making it unlikely that adaptive mutations will happen in any given generation. In such populations, the rate of adaptation is limited by the availability of new mutations. In an extreme -- in the very small effective sizes of Pleistocene human populations -- the rate of adaptation may be extremely slow and regional populations may come to differ at many weakly selected loci, which spread very slowly.
As the population grows, strongly adaptive mutations become more and more likely to happen somewhere in the species' range. Yet they are still relatively rare -- meaning that they have an opportunity to spread fairly far before encountering another equally strongly selected mutation affecting the same trait.
This process can give rise to very large differences on a continental scale, even when the selection pressures in different regions do not differ. In humans, the dispersal of selected alleles across space may have been significantly accelerated by actual dispersals of populations. It is not a mere coincidence that very widespread alleles in Eurasia also tend to be much older than 20,000 years old -- long-distance dispersals prior to that time had a higher chance of leaving a lasting influence on subsequent populations.
But as the population gets bigger and bigger, parallel mutations are more and more likely to happen. . . . at the extreme of large population size and likely mutations, you shouldn't see any new mutations emerging and spreading over very large areas. Any of these mutations would be very likely to encounter other new mutations that do the same thing.
Is this likely in humans? Clearly some mutations have happened recurrently. Making a broken gene is easy -- there's a large mutational target, since a large fraction of nonsynonymous substitutions might do the job. So if there's a net selective advantage to breaking a gene, we ought to see that happen recurrently in human populations.
In contrast, if the mutational target is very small, then mutations will still be rare even in a very large population. If only one base change can have an adaptive effect, that precise change will happen less than once in [1,000,000,000] births (remember that not just any mutation at a site, but some particular mutation is what we may need). If a rare duplication or gene conversion is the necessary change, then it may be much rarer.
Looking across the last few million years, when human population numbers were much smaller than the Holocene, we can be pretty sure that some aspects of our evolution were mutation-limited. The changes that took hold in our ancestors were the ones that happened, and that survived the winnowing of genetic drift. Many changes that would have been adaptive didn't happen in our ancestors. They just weren't lucky enough.
But some of those changes would still be adaptive now, if we could get them.
7. The spread of an advantageous mutation is path dependent.
Some mutations have "first mover" advantages. Once they are common, other adaptive mutations may still occur -- even mutations that are better from the standpoint of fitness -- but be lost or grow very slowly because their net fitness advantage over the common mutant is slight.
This model of evolutionary impacts isn't strongly mechanism dependent. Most of the insights can be applied to the spread of cultural and economic phenomena, adjusting the time and space parameters appropriately to reflect the analog of generations and geographic locality to more appropriate units of time and interaction network notions of space. For example, one can use the basic ideas presented to explain the spread of Apple v. Wintel computing technology.
Alternately, in a context closer to the one John Hawks was looking at, the model could be tweaked to analyze the spread of different competing human communities with a memone of different cultural practices some adaptive and some not, as opposed to different specific gene mutations. These models, for example, seem to capture the dynamics that drive the human replacement of Neanderthals, the Neolithic population replacement of modern human hunter-gatherers, and the various waves of incomplete population replacement accompanied by cultural adaptation in important historical era mass population and culture shifts.