288 AN INTRODUCTION TO MODERN GENETICS 



bom earlier. Let the smaller number of parents born x years ago be 

 represented as ^~'"^; the contribution of these parents to the present 

 birth is e'^^^Jb^dx^ so that a single birth now is the result of the sum 



of all these contributions and we have 



■"'HJbJx= I. This is suffi- 



cient to define the value of m, which is called the Malthusian parameter 

 of population increase. It is positive for an increasing population and 

 negative for a decreasing one. The relative fitness of two varieties A 

 and B can be measured by comparing their Malthusian parameters w^ 

 and m^. HA is stationary in numbers (w^ = o) and B is falling, the 

 ratio of ^ to 5 in the next generation is i : e^*, where w^, is negative . 

 Haldane represents this as i : i — ^5 so that k is nearly equal to — w^ 

 when both are small. 



4. Selection of Single Genes in Infinite Populations^ 



The simplest case of selection is that of a single gene in an infinite 

 population. The theoretical result depends on how the gene is in- 

 herited and the type of mating which occurs. For instance, if the 

 population is one of self-fertiUzing or apogamous plants, heterozygotes 

 will be eliminated by the system of mating and we shall have only 

 types A A and aa. Let the ratio of these in the nth. generation be 

 w„ A A : I aa. If the coefficient of selection (in Haldane 's sense) in 

 favour of AA is k, then in the next generation there will be w„ of A A 



and I — ^ of aa. Thus u„ + ^ = — ^ which is nearly ^u„ when k is 



I — k 



small. Then clearly w„ = ^"wqj where Uq is the initial proportion of A A. 

 A more important case is that of a population mating at random. 

 Suppose we have dominants, heterozygotes and recessives in the pro- 

 portions pnAA : q^jAa : r^aa. If mating occurs at random, the propor- 

 tions of the different types of matings will be as shown below and the 

 offspring they produce can be calculated. 



Mating Proportion Offspring 



AA X AA /)2 p2^^ 



AA X Aa 2pq pq AA : pq Aa 



A A X aa 2pr 2pr Aa 



Aa X Aa q^ . \q^ AA : \q^Aa : \q^aa 



Aa X aa 2qr qr Aa : qr aa 



aa X aa r^ r^ aa 

 1 Cf. Haldane 19326. 



