THE GENETiCAl THEoRy of NATuRAl SElECTioN 127
The fundamental theorem of natural selection
and the adaptive landscape
Fisher showed mathematically that evolution by natural selection causes w— to
increase through time. When fitness is normalized so that w— = 1, Fisher found that
the increase in mean fitness per generation is simply equal to the genetic variance
for fitness itself.^3 Fisher named this principle the fundamental theorem of natural
selection. Because a variance can never be negative, the important message from this
result is that natural selection causes populations to evolve so that they become better
adapted to their environment: the average survival and reproduction of individuals
increase through time.
During the spread of a beneficial allele, its frequency increases slowly when the
allele is rare, then accelerates as the allele frequency nears 0.5, and then slows again
as the allele nears fixation (see Figures 5.7 and 5.10). We can understand that rhythm
of change in terms of the fundamental theorem. When the beneficial allele A 2 is rare,
almost all of the population is homozygous for the A 1 allele (see Figure 5.8). There is
little variation among individuals in fitness, so w— increases slowly. The variance for
fitness among individuals is maximized when p = 0.5, and so the change in w— per
generation is greatest. The variance in fitness is again small as A 2 nears fixation, so
the rate of increase in w— is again small.
Fisher’s fundamental theorem leads to the question of just how much genetic vari-
ation for fitness exists in natural populations. The answer depends on the species, the
time, and the place, but it seems that the genetic variance in relative fitness may often
be a few percentages [8, 9]. All else being equal, the fundamental theorem would
lead us to expect that the mean fitness of species should increase by a few percent per
generation. But all else is not equal: what selection gives, other evolutionary forces
take away. The fitness gains made by selection are continuously offset by environ-
ments that change in space and time, deleterious mutations, and other factors.
A complementary perspective on the evolution of fitness was developed by Wright.
He plotted the mean fitness, w—, against the allele frequency, p. This plot, which Wright
called the adaptive landscape, tells us how the population will evolve. His key insight
was that selection causes populations to evolve uphill on the landscape (FIGURE 5.25).
Wright proved mathematically that the allele frequency will change at a rate(5.8)On the right side of this equation, you will recognize p(1 – p) as a measure of
genetic variation: it equals 0 when there is only one allele in the population (that is,
p = 0 or p = 1), and is maximized when the two alleles are equally common [that is,
p = (1 – p) = 0.5]. The last term on the right is a derivative that is equal to the slope of
the adaptive landscape. This measures the direction and strength of selection. When
the slope is positive, selection causes A 2 to spread, and when it is negative, it causes
A 2 to be lost. When the slope of the landscape is 0, selection does not favor either
allele, and the population is at equilibrium.
The adaptive landscape explains the differences in the allele frequency trajecto-
ries that we saw earlier among positive, overdominant, and underdominant selec-
tion (see Figure 5.19). With positive selection, the slope of the adaptive landscape
is always positive, and selection causes the advantageous allele to spread until it
reaches the peak of the landscape at the far right, where p = 1. With overdominance,
the landscape has a peak at an intermediate allele frequency. No matter what the
initial allele frequency is, selection pushes the population to the adaptive peak,
where it reaches the polymorphic equilibrium. Underdominant selection produces(^3) Technically, it is the “additive” genetic variance for fitness that matters. Additive genetic vari-
ance is discussed in Chapter 6.
∆p= p(1 – p) ln(w)
2
1
dp
d –
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