322 GENETICS IN RELATION TO AGRICULTURE 



matical relations are much more complex than those given above. We 

 may consider a simple case, however, and show the relations in that case. 

 If we start out with a population consisting of equal numbers AA and aa 

 forms, and assume that a given percentage of crossing occurs, then an 

 equilibrium will be reached when the number of homozygotes produced 

 by the heterozygotes in the population is equal to the number of hetero- 

 zygotes produced by spontaneous crossing. Thus, if we assume 10 

 per cent, of spontaneous crossing in such a population, in the first gen- 

 eration of the 10 per cent, of AA which cross with other plants, half will 

 be fertilized by other AA plants and half by aa. The latter will give 

 heterozygotes, consequently the proportions of different genotypes 

 produced by the AA plants will be 0.95AA : O.OSAa. Similarly aa plants 

 produce 0.05Aa: 0.95aa, so that in the first generation the ratio is 0.95A-A : 

 Q.WAa : 0.95aa. Now in .the next following generation if we assume 

 that random mating occurs among the 10 per cent, of plants which 

 cross with other plants, then one-third of the plants in each genotype will 

 mate with the same genotype, one-third with one of the other two geno- 

 types, and one-third with the remaining genotype. That is, of the 

 0.95AA one-tenth or 0.095 cross, as follows: Y$AA X A A = Q.32AA, 

 YzAA X aa = 0.032Aa and Y$AA X Aa = 0.016AA : 0.0164a. Simi- 

 larly, of the 0.95aa, 0.095 cross: >aa X aa = 0.032aa, >aa X AA = 

 0.032Aa and %aa X Aa = 0.0164 a : 0.016aa. Also of the O.lOAa, 

 one-tenth or 0.01 cross: }$Aa X AA = 0.001644: 0.00164a, %Aa X 

 aa = 0.00164a : O.OOlGaa and ^Aa X Aa = 0.000844 :0.00164a: 

 O.OOOSaa. Summating like genotypes we have 0.0544 : 0.104 a : O.OSaa. 

 The 90 per cent, of AA and aa plants which are self-fertilized produce 

 0.85544 and 0.855aa respectively, while the 0.094a plants which are 

 self-fertilized produce 0.022544 : 0.0454 a : 0.0225aa. Combining 

 these with the results of cross-fertilization we have the ratio for the 

 second generation, 0.92844 : 0.1464a : 0.928aa. Now the ratio of 

 the proportion of homozygotes to the population in the first generation 

 is of course 0.95 and in the second generation it becomes, 



0.928 + 0.928 



0.928 + 0.146 + 0.928 



= 0.927. 



The composition of the third, fourth and fifth generations and the ratio 

 of the proportion of homozygotes to total population for each are shown 

 in Table XL VII. It is evident that, under the conditions assumed in 

 this case, the rate of change in the ratio of homozygotes to the total 

 population becomes very gradual after the first three generations, so 

 that for practical purposes the population has reached a state of 

 equilibrium in the fourth generation. In this generation the ratio of 

 heterozygous dominants to the sum of the heterozygous and homozygous 



