Z48 



THE QUARTERLY REVIEW OF BIOLOGY 



mixed family where equality would have 

 been expected, the ratio of dominants to 

 recessives will be m-k. For the case 



where k is small, we can let u n = ; ' 



In + r n 



and arrive at the following relationship: 



kUn 



fi n+1 = u n +- -7 ; V which in turn 



Z(l +- U n ) 



may be simplified to give a relationship 

 between u n and k as follows: i/zkn = 

 u n +- \oge-u n — 1. A comparison of this 

 last relationship with the one derived for 

 selection of a simple Mendelian character 



case d: sex-limited characters and uni- 

 sexual SELECTION 



When the characters appear in only one 

 sex, as in the case with milk-yield or 

 other secondary sexual characters, we may 

 let the n-i generation form spermatozoa 

 in the ratio u n A\ 1a, and eggs in the ratio 

 v n A'.ia, in which case the nth generation 

 consists of zygotes in the ratio u n v n AA: 

 (u n -+ *y) Aa'.iaa. In this case, y n = 

 (1 +■ u^- 1 (1 +• O -1 - With complete 

 selection, when k = — <*> , this case reduces 



TABLE i 



Effect of slow selection on an autosomal Mendelian character 



Per cent of favored type 



O.OOOI 



-15.51 



— 1005.0 



O.OOI 



-13.11 



—310.0 



O.OI 



— 10.90 

 — 101 . 60 



0.05 

 -9.194 

 -45-50 



kn when dominants are 

 kn when recessives are J 









O.I 



-8.600 

 -33-04 



0.1 

 -7.905 

 -13.41 



0.5 

 — 6.996 

 -14.71 



1 

 -6.186 

 -10.197 



1 

 -5.580 

 -6.875 



3 

 — 5. 161 

 -4.976 



5 

 -4.614 



-3-717 



10 

 -3.863 

 -1-933 



*5 



-3.190 

 —1. 041 



10 



-1.979 

 — 0.448 



2-5 



-1.711 

 



30 



-1.439 

 +0.366 



35 



-1.180 

 +0.681 



40 



-1.964 

 +0.961 



45 



-1.708 

 +1.110 



50 



-1.467 

 +1.467 



55 



— 1. 110 

 +1.708 



60 



—0.961 

 +1.964 



65 



-0.681 

 +1.180 



70 



—0.366 

 +2-439 



75 





 +1.711 



80 

 +0.448 

 +2-979 



85 

 +1.041 



+3.2-90 



90 



+I-933 

 +3.863 



95 



+3-7I7 

 +4 . 610 



97 



+4-976 

 +5- 161 



98 

 +6.875 



+5.580 



99 

 +10.197 

 +6.186 



99-5 



+14.71 

 +6.996 



99.8 



+ 13.41 

 +7.905 



99-9 

 +33-°4 

 +8.600 



99-95 



+45-50 

 +9-2-94 



99-99 



+101.60 

 +10.90 



99.999 



+310.0 

 +13.11 



99.9999 

 +1005.0 

 +I5-5 1 





shows that the species changes its compo- 

 sition at one-half the rate at which it 

 would change if selection worked on the 

 species as a whole, and not within families 

 only. 



If the family has the mother only in 

 common, but fathers are a random sample 

 of the population, as in the case in cross- 

 pollinating seed plants, we find a relation- 

 ship of the type 3/4^ = u n +- log e «„ — 1 

 for the case where k is small. Thus 

 selection proceeds at three-quarters of the 

 rate which exists in the case of a selection 

 of a simple Mendelian character. 



to y n = 1 + z 1_n (jo 1/2 — *)> where the 

 proportion of dominants is halved in 

 every successive generation. When k = 1 , 

 and all the recessives of one sex die child- 

 less, the proportion of recessives in 

 successive generations, starting from the 

 standard populations, is Z5 per cent, 16.7 

 per cent, iz.5 per cent, 9.56 per cent, 

 7.94 per cent, etc. When the rate of 

 selection is slow, — that is, k is small, — 

 our equation becomes i/±kn = u n + 

 log. e u n — 1, which shows that selection 

 proceeds at one-half the rate for slow 

 selection of a simple Mendelian character. 



