58 Sahotra Sarkar
and complete dominance, the three zygotes survive to breed in the proportions
u^2 nAA:2unAa:(1-k)aa.Ifk>0, the dominants have a selective advantage,
ifk<0, the recessives are similarly favored. The change in the population is now
described by the non-linear difference equation:
un+1=
un(un+1)
un+1−k.If|k|<<1, andu 0 = 1, Haldane obtained the approximate relation:
kn=un+ln un− 1.One of the numerical consequences of this equation eventually became a staple of
textbooks of population genetics. Barrett (1895–1902) had described the case of
the peppered moth (then known asAmpidasys betularia) the dominant melanic
form of which first appeared in the Manchester area in 1848. By 1901 it had
replaced the recessives almost completely. From Haldane’s calculations, the selec-
tive difference was at least 1.5. As far as Haldane was concerned, this was not a
“very intense degree of natural selection” (p. 26); for Fisher and others, however,
it was unrealistically high.^16 There the matter lay for a generation, until Ket-
tlewell [1956] finally measured the selection coefficient due to predation by birds
and found it to be even higher. Even Haldane [1956] was not convinced that the
coefficient could have always been quite that high. Nevertheless, this example,
almost uniformly interpreted as a resounding success of evolutionary theory, per-
colated into the folklore of biology.^17 What it really shows is how difficult it has
been to connect the models of population genetics to data collected from natural
populations.
Besides these two cases, Haldane analyzed eleven other sets of models in the
1924 paper. He treated both autosomal and sex-linked loci and even considered
the differences between models in which selection acted on only one or on both
the sexes. What, in retrospect, appears to be the most innovative of these models,
are those of various types of familial selection, for instance, between members of
the same litter. Evolution slowed down considerably.^18 Most other models gave
similar results. For instance, models of “certation” (gametic selection) allowed
changes to be potentially as fast as in diploids. Even in those models in which
selection potentially allowed faster changes, the rate was not much faster.
(^16) See Provine [1971, 170].
(^17) For a detailed account of the study of industrial melanism, and its incorporation into evo-
lutionary studies, see Kettlewell [1973]. However, recent work has challenged the validity of
Kettlewell’s conclusions — see Majerus [1998].
(^18) The standard approximate treatment (with|k|<<1andu 0 = 1) now gave, instead of
kn=un+ln un−1, the equation:
1
2 kn=un+ln un−1.
Thus, the “species changes its composition at half the rate at which it would change if selection
worked on the species as a whole, and not within families only” (p. 28). If the members of a
family shared only one parent, the slowdown factor was 3/4 instead of 1/2, but the potential
rate of change remained slower.