14 GENETIC STOCKS AND BREEDING METHODS 



It may be seen at once that the probability of incrosses will steadily rise, since incrosses 

 are produced by incrosses, by backcrosses, and by intercrosses of the preceding genera- 

 tion. Incrosses act as an absorbing barrier. Once a locus reaches an incross, it re- 

 mains as an incross so long as the brother-sister mating system continues. 



If brother-sister mating is started after a cross between unlike strains, so that the 

 initial mating is AA x aa or the reciprocal, that is, q = 1, the probabilities of incrosses 

 in G , G l5 G 2 ,. . . are: 



0, 0, 0.125, 0.281, 0.414, 0.525, 0.616, 0.689, 



0.748, 0.796, 0.835, 0.866, 0.892,.... 



After 12 generations, 89.2 per cent of the matings will be the desired type, incrosses of 

 AA x AA or aa x aa. 



The probability h of heterozygotes is one-half the probability of backcrosses plus 

 the probability of intercrosses : 



h n = (l/2)r n + v n . 

 In G l5 G 2 , . . . , this probability h takes the successive values: 



1, 1/2, 2/4, 3/8, 5/16, 8/32, 13/64, 21/128,..., 



which are the terms of the Fibonacci series. The ratios h n + 1 jh n of the probability of 

 heterozygosity in one generation to the probability of heterozygosity in the preceding 

 generation form a series : 



0.5, 1.0, 0.75, 0.8333, 0.8, 0.8125, 0.8077, 0.8095, 



0.8088, 0.8091, 0.8090, 0.8090, 0.8090,..., 



which shows that, after brother-sister inbreeding has been in progress for several genera- 

 tions, the amount of heterozygosity in G n + 1 tends to be 80.9 per cent of the hetero- 

 zygosity in G n . Or, stated otherwise, the heterozygosity is decreasing at the rate of 

 19.1 per cent per generation, as first discovered by Jennings 662 and established 

 analytically by Wright. 1442 



Successive values of/> n , h n , and h n + 1 /k n up to n — 12 are shown in figure 2. 



It may be assumed that, after brother-sister inbreeding has been in progress for 

 several generations, the rate of depletion of the crosses, backcrosses, and intercrosses 

 is constant. That is, 



?n + l = Mm 

 r n + l = ™n> 

 Z'n + l = to>„, 



where X is a factor of proportionality. It follows that 



~Mn + (1/8X = 0, 



(1/2 - X)r n + (l/2)o B = 0, 



q n + (l/4)r„ + (1/4 - X> n = 0. 



