SYSTEMS OF MATING 



15 



Solving these three simultaneous linear equations for X yields 



X = (1/4), (1/4)(1 + V5), (1/4)(1 - V5), 



of which the largest numerical solution is 



X x = (1/4) (1 + VI) = 0.8090 or 80.9 per cent. 



X x is known as the characteristic root of the determinant made up of the coefficients in 

 the three simultaneous equations and provides a rapid analytical means of finding the 



Fig. 2. The probability of incrosses for brother-sister inbreeding system. 



X 



h n + |/ h n ^ 



0.6 



PROBABILITY 

 0.4 



2 4 6 8 

 GENERATIONS 



The probability of incrosses p n and of heterozygosity h n for the brother-sister inbreeding 

 system when q = 1 ; and the ratios of successive values of h and h n + x /h n . 



expected decrease in heterozygosity after several generations of inbreeding. The 

 largest numerical root is the important one as n gets larger and larger because the 

 probabilities q n + 1 , r n + 1 , and v n¥1 are each functions of X^, X2", and X 3 n and of the 

 initial probabilities q , r , and v . If X 2 and X 3 are smaller than X 1} X 2 n and X 3 n will be 

 of diminishing importance, relative to X x n , as n increases in determining the values of 

 9n + i5 r n + u anQl v n + i- Thus for large n, only the characteristic root need be used for a 

 sufficient approximation in computing the probabilities. 



The above results have been arrived at by direct multiplication of the generation 

 matrix by the frequency of each mating type in G n to obtain the frequencies of the 

 mating types in G n + 1 , essentially by the repetitive application of formula (1) in the 

 preceding section. The same results can be reached by the more complex algebra 

 of the formulas (3), (5), and so forth. 



The generation matrix, G, is given in table 4. The roots X { can be obtained by 



solving equation (8), 



|G - XJI - 0, 

 or 



= 0. 



