8 GENETIC STOCKS AND BREEDING METHODS 



If they are not so related initially, they become so after one generation of random mating 

 as shown above. 



GENERAL METHOD OF ANALYZING REGULAR MATING SYSTEMS 



In the regular mating systems there is a definable probability that matings of any 

 given type will yield matings of the same or any other type (for example, that backcrosses 

 yield incrosses) in the subsequent generation. The probability that a mating of type 

 j yields a mating of type i in the subsequent generation will be called g tj . With s 

 mating types, these probabilities can be arranged in an s x s matrix so that the leading 

 diagonal of the matrix contains the elements g it , row i contains all of the elements 

 g u ,j = 1, . . ., s, and column j contains all of the elements g tj , i = \, . . ., s. Such a 

 matrix will be referred to as the generation matrix G. 



If, in generation n, the mating types have frequencies/^, ?„,..., v n , the frequencies 

 in generation n + 1 will be 



Pn + 1 = Mil + ?n?2l + • • • + v n q sl 

 <7n + l = Ai<7l2 + ?n?22 + • • • + y n? S 2 



Vn + \ = Mis + ?n?2s + • • • + V n q ss . (1) 



The probabilities p n , q n , . . . , v n can be arranged in an s x 1 vector P n , 



Pn = : /Pn' 

 <ln 



If the matrix G is then postmultiplied by this vector P n , following the row-by-column 

 rule of matrix multiplication, a new vector is obtained whose elements are the sums (1). 

 We therefore conclude that 



Pn + l=GP n . (2) 



If, then, the matrix G and the vector P can be evaluated, when P contains the 

 frequencies of mating types in the initial generation, the vector of mating-type 

 frequencies P n can be obtained for generation n by repetitive application of equation (2). 



However, 



Pn = GP n _ l5 



and 



Pn-l = GP n _ 2 ; 

 therefore 



r n — vj r n _ 2 >, 

 and by applying this principle repeatedly, we can show that 



Pn = G"P . (3) 



