Reading through P. A. P. Moran’s book, The Statistical Processes of Evolutionary Theory, I found this passage (p. 12):
It should be pointed out that the above stochastic models [of density dependence] usually result in there being a non-zero probability that the population will die out altogether. In genetic problems this is an unmitigated nuisance. In population genetics we are concerned with the variation and distribution of gene frequencies and it is very difficult to make stochastic models in which both the gene frequency and the population size are random variables (see Feller (1951), p. 242 for a beginning in this direction, in which, however, the population may die out). Many genetic phenomena do depend on the population size and the models we shall consider later nearly all assume that this size is held constant. It is true that if we have, for example, a situation in which a new mutant gene takes over the whole population by reason of some selective advantage, the total population size, which is held in check by density-dependent forces, can usually be expected to increase somewhat, or at any rate to change slightly, but this is not likely to have an important effect.
That’s interesting for several reasons. Recently I’ve been investigating the connections between selection and demographic growth. In humans, there are a number of recently selected genes whose advantage comes from relaxing density dependence (that is, increasing carrying capacity), for example by allowing greater resource extraction from the environment. In those cases, the effect of a selective fixation on population size will not be negligible. Examples of that kind may not be rare in nature, although in many instances selection may increase population size only to result in added pressure to various prey species, which then reduce the carrying capacity.
Another reason why this is interesting is that it reveals a fairly unusual way of thinking about selection. From one point of view selection is just a condition of the demography of alleles. In particular, both selection and genetic drift (and for that matter, mutation) are described by the same equations that describe demography. Under genetic drift, these allelic demographies are in all cases of similar form to the demography of the population in which those alleles are embedded. Selection, on the other hand, is notable for showing the demography of alleles to be inconsistent with the demography of the population. The most commonly considered case is where one allele increases while the population remains the same size. But balancing selection, for example, can be reduced to density-dependence on an allele’s frequency.
One of the easiest ways for selection to set itself apart from stochastic changes in populations is to be deterministic. But the results of selection are nonetheless stochastic, and it is good to be reminded.