Gaussian adaptation – based on the selection of individuals – is closely connected to the Hardy-Weinberg law for its validity as a model of evolution showing its maximization of mean fitness, which can’t be shown by Fisher’s fundamental theorem, 1930. This blog gives a simple proof of the law for individuals, readable on the high school level. Randomness also seems to be an advantage for the survival of the collective.
The Hardy-Weinberg law for individuals:
If the parents to new offspring in a large population mate at random in proportion to the probability density function, p. d. f., of parents, then the mean of the quantitative characters of the offspring will coincide with the mean of the characters of the parents.
Proof:
Let w(x) be the fitness of the individual having the array of quantitative characters x = (x1, x2, …, xn) and N(m – x) be the p. d. f. of offspring in one generation (non-overlapping) where m is the mean of N. Then the p. d. f. of the parents is
P(x) = w(x) N(m – x) / integral { w(x) N(x – m) dx }.
The mean of parents will be
m* = integral { x P(x) dx }
The mean of a pair of parentfs x1 and x2 with p. d. f. P(x) will be (x1 + x2)/2. If mating takes place at random in proportion to P(x), then, since the mean value of a sum of random variables equals the sum of mean values of the terms, it follows that
m = (m* + m*)/2 = m*
and the law is proved.
A possible consequence is that if parents do not mate at random, then the Hardy-Weinberg law will no longer be satisfied and mean fitness, defined as
W(m) = integral { s(x) N(m – x) dx }
can no longer be maximized using Gaussian adaptation with N as a Gaussian distribution of characters. Thus, randomness seems to be an advantage for the collective survival.
