NATURAL SELECTION (MATHEMATICAL THEORY) Pearson developed a theory of the effects of natural selection on continuously varying characters which was based on the law of ancestral heredity, enunciated by Galton and modified by himself. He produced very strong evidence that natural selection is oc curring in men. The theory is complicated, and somewhat incom plete. Intense selection for a character is effective, being nearly complete in three or four generations, but two points remain un certain, namely (a) whether the population can ever be brought quite to the level selected for, and (b) whether it will regress in definitely towards its original state when selection ceases. It is at least clear that such regression would be slow. The doubt de pends on the uncertainty of the coefficients of correlation between a character in an individual and his remote ancestors.
Haldane and Norton developed a theory of selection in popu lations exhibiting Mendelian inheritance. (See HEREDITY.) A population mating at random and only partly possessing an au tosomal gene A is in equilibrium in the absence of selection when the three genotypes are in the ratio : 2uAa: iaa. If u„ is the value of the ratio in the nth generation, and selection occurs so that the population breeds in the ratios If K and k, which are called coefficients of selection, are both positive, the population reaches an equilibrium with u co = Otherwise either the gene A or a disappears. If K=o, i.e., dominance is complete, we have, if Similarly we may obtain formulae for the effects of differential selection in the two sexes, and of competition limited to members of the same family.
When the two competing types do not mate, owing to self fertilization or incompatibility, or when inheritance is wholly maternal, we find kn = where is the ratio of the types. Fig. I shows the very different rates at which the population would change according to the type of inheritance of the character se lected. In each case k= .00 1, i.e., i,000 of the favoured type sur vive for every 999 of the other. In all cases the reverse series of changes occurs if the sign of k be changed. It is clear that in the
case of an autosomal gene selection is somewhat ineffective so long as the recessive is rare. This is due to the fact that kn is not entirely a sum of logarithmic functions of u„. When, however, dominance is incomplete, recessives, however rare, will increase at a relatively rapid rate.
If there is a moderate degree of self-fertilization, inbreeding or assortative mating, the population assumes different equilib ria in the absence of selection and the rate of change under se lection may be altered. Thus if a proportion l of the population be self-fertilized or mated to sibs we have In the case of sex-linked inheritance, where the proportions of the male sex are u„A :Ia., of the female : 2u„Aa: I a, and selec tion occurs in both sexes with equal intensity, we have These equations can be generalized in various ways. The gen erations may be supposed to overlap, as in man. In this case we must know for each genotype the probability of a member of it producing an offspring between the ages x and x-1-6x (dead as well as living members being included in the calculation). If this probability is K(x)8x for a dominant, [K(x)—k(x)]ox for a re cessive, then when selection is slow the equations for selection are similar to the above, provided we put respectively, so that selection is not very greatly slowed down even when recessives are rare. On the other hand assortative mating does not have this effect.