Heterozygous Advantage

Here is a more detailed proof of the final result for heterozygous advantage:

= 0 when = at equilibrium

Clearly, this is true either when p = 0 (a "trivial" equilibrium; if we don’t have any A, there can be no evolution!),

or when

Now, notice that the mean fitness (bottom line)

So now:

at equilibrium

Therefore:-

(why?)

, but, is of course = q!!

Here we have another "trivial equilibrium"; if there are no a, q = 0 and the gene frequency remains fixed for A; alternatively, the thing in brackets (i.e. -sp + tq) = 0, which is the same as saying tq = sp. This latter is the "internal", "non-trivial" equilibrium we want, with both A and a polymorphic (frequency > 0) in the population.

when
t = sp + tp
(s+t)p = t

so: .....