EVOLUTIONARY MECHANISMS 289 



Total offspring : 



(/>' + Pq + k^) AA : {pq + 2pr + gr + \q^) Aa 



: (r2 -\- qr -\- Iq^) aa 



P + 2? 

 = u^ AA : 2u Aa : I aa. where u = — 



If we repeat the calculation to obtain the next generation, we have 

 simply to substitute u^ for p, 2u for q, and i for r in the above result, 

 when we obtain 



(w2 + uf AA : 2(m2 + w)(i + u)Aa:(i + uf aa 

 i.e. (m + i)^ [w^ -^^ : 2w ^a : I aa] 



Thus in the next and subsequent generations the proportions of the 



different genotypes remain the same. The expression u^ AA : 2u Aa : i 



aa represents the equilibriimi for a population mating at random, and 



this equilibrium is attained after the first generation of random mating.^ 



It is clear that if a is rare, nearly all the a genes will be foimd in hetero- 



zygotes, and homozygous recessives will be extremely rare. 



T u- [AA] + i[Aa] ^ ^ ^ . 



In this expression u = ,, , , ; — - and thus measures the ratio 



of A to a genes. It is more convenient to work with this as a variable 

 rather than the proportion of dominant or recessive zygotes. 



If the wth generation of a randomly mating population is 

 w„2 AA : 2w„ Aa : 1 aa, the next generation, after selection against the 

 recessives, is u„^ AA : 2w„ Aa : (i — k) aa, so that 



[AA] + i[Aa] u„^ + u„ 



i[Aa] + [aa] u„ + 1 - k 



Aw. 



W„ + I - /c 



This is called a finite difference equation, i.e. an equation which ex- 

 presses the relation between two members of a series of discrete values . 

 But if selection is slow, we shall be concerned with a series of many 

 generations, and can regard the difference between successive genera- 

 tions as infinitesimal. That is, we can treat the equation as an ordinary 

 differential equation, and we can also, if ^ is small, neglect it in com- 

 parison with I, and write 



du kn 



dn I + w 



^ Hardy 1908. 



