Z46 



THE QUARTERLY REVIEW OF BIOLOGY 



different cases that arise when selection of 

 small and constant intensity operates on 

 Mendelian populations where generations 

 do not overlap either during random 

 mating or when all zygotes are self- 

 fertilized. The cases considered and their 

 results are as follows: 



case a: selection in the absence of 

 amphimixis 



The simplest form of selection is 

 uncomplicated by amphimixis and is 

 illustrated in the following cases: 



1. Organisms which do not reproduce 

 sexually, or are self-fertilizing. 



-l. Species which do not cross, but 

 compete for the same means of support. 



3 . Organisms in which mating is always 

 between brother and sister. 



4. Organisms which are haploid during 

 part of the life cycle, provided that the 

 selection of the character considered 

 occurs only during the haploid phase. 



5. Heterogamous organisms in which 

 the factor determining the character 

 selected occurs in the gametes of one sex 

 only. 



Setting up the functional relationship 

 between y n and k for this type of selection, 



the author finds that y n = 



1 + (i-£)-» 

 when the initial conditions are such that 

 the number of A's equals the number of 

 B's. For small values of k, that is, when 

 selection is slow, this formula becomes 



approximately y n = ^ the logistic 



reaction. For this condition, we see that 

 the proportion of B's falls slowly for a 

 time, then more rapidly, and then slowly 

 again, the most rapid change in the 

 proportion occurring when the number of 

 B's is equal to the number of A's, that is, 

 whenj/ n = |. For this mode of inherit- 

 ance, selection proceeds more rapidly than 

 for any of the other systems of inheritance 



considered. The speed must compensate 

 to some extent for the failure to combine 

 advantageous factors by amphimixis. 



CASE B : SELECTION OF A SIMPLE MENDELIAN 

 CHARACTER 



This is the mode of inheritance that 

 has been subjected to mathematical analy- 

 sis more often than any other. Pearson 

 (6) and Hardy (7) have shown that in 

 the case of random mating, the number of 

 heterozygotes is equal to four times the 

 product of the numbers of the two homo- 

 zygous classes. Thus, if tt n A:xa be the 

 proportion of the two types of gametes 

 produced in the («-i) th generation, then 

 in the nth generation the initial propor- 

 tions of the three classes of zygotes are 

 u n 2 AA:zu n Aa:iaa. The proportion of 

 recessives to the whole population is 

 y n = (1 + Un)- 2 . Now, only i-k of the 

 recessives survive to breed, so that the 

 survivors are in the proportion u n 2 AA: 

 uu n Aa\(i.-TC)aa. This leads to an expres- 

 sion for u n+ i in terms of u n and k as 



U„ (i + Un) 



follows: u n+ i = ; -■ It can be 



1 + u n — k 



easily shown that this result follows from 



random mating. If, now, we know the 



original proportion of recessives, and we 



start with a population u 2 AA:2.UoAa:iaa, 



where u = yo~ 1/2 — 1, we can at once 



obtain tii by use of the equation u\ = 



up (1 + «o) . 



— — —7' and by use of this recurring 



formula, we can determine u n and there- 

 fore y n . 



If the selection is complete, — that is, 

 if all of the dominants are killed off or 

 prevented from breeding,' — the case be- 

 comes very simple, and y n is equal to 

 unity. 



If, on the other hand, all the recessives 

 die, k becomes equal to 1 and the above 

 general equation reduces to y n = yo 

 (1 + nyo 1/2 )~ 2 . This equation shows that 



