Haldane and the Emergence of Modern Evolutionary Theory 57This was the first explicit recipe for the construction of models of natural se-
lection. Following this recipe, Haldane constructed thirteen sets of such models
in the first paper and about thirty more in the next nine. The models cover a
bewildering variety of situations. Three themes unify the treatment: (i) Haldane
insists on explicitly quantitative results; (ii) much more than any of his peers, he
attempts to connect each model with experimental data; and (iii) the question of
the speed at which selection can produce change always lurks in the background.
The discussion that follows will emphasize these themes while navigating through
the bewildering variety mentioned above.
What is initially surprising about Haldane’s characterization of evolutionary
theory above is that he did not distinguish between the relative roles of the selection
intensity and the rate of change in the equation connecting (3) and (5) even though,
it is clear that that the observed rate of change was to be explained by the intensity
of selection. This was no oversight. In 1924, there were no direct measurement
of selection intensities. Rates of change, however could be estimated. As Haldane
struggled to connect his models to data, selection intensities became predictions
though, in the 1920s, their measurement did not seem forthcoming.
Part I starts with a haploid model. LetAandB be two phenotypes in a
population in some generation and letpA:1B be their relative proportions. If,
in the next generation, their proportions arepA:(1 -k)B, then Haldane defined
“k” as the coefficient of selection. Ifk=1,noB’s survive, ifk=−∞,noA’s
survive.^15 Let then-th generation haveAandB(which can now be interpreted
as alleles) in the proportionunA:1B. Assuming that generations do not overlap, if
kis the selection coefficient,
un+1=un
1 −k.If|k|<<1, andu 0 = 1 (that is theA’s andB’s are initially present in the same
proportion), the corresponding differential equation yields as a solution:
kn=ln un.All other models, that is, those that allowed sexual reproduction, resulted in slower
rates of change. In the case of non-sexual reproduction, Haldane observed, “speed
must compensate to some extent for the failure to combine advantageous factors”
(p. 22). This eventually became a common argument for the evolution of sex.
In a diploid model, let the two alleles,Aandabe in the proportionsunA:1a
in generationn(that is, the proportion of theAanda gametes produced in
generationn-1isun:1). Assuming random mating, non-overlapping generations
(^15) Haldane’s strategy for solving the models was simple. First, by calculating the change in the
proportion of either type from one generation to the next, he wrote down a difference equation for
the frequency of the alleles. These difference equations were characteristically too complicated for
analytic solution and, in an age when computers were still to be invented, usually too cumbersome
for practical use. However, Haldane found that he could usually convert the difference equation
into a differential equation for two limiting cases, when the selection coefficient was very high
or almost 0. The differential equations, though approximate, could be solved analytically and a
useful result, that could be compared to experimental data, was obtained.