EVOLUTIONARY MECHANISMS 29I 



At equilibrium this = and we get nearly w + i = \/ -. If the gene 



is lethal and k nearly = i, the frequency of recessive zygotes at 

 equiUbrium is nearly equal to the mutation frequency. 



5. Fisher's Fundamental Theorem of Natural Selection 



Any natural population contains many different genes, and the fitness 

 or selective advantage of an individual depends on the vi^hole assemblage 

 of genes in its genotype. It is clear, in a general way, that the more 

 varieties there are in a population, the faster will natural selection be 

 able to pull it along the evolutionary path by eliminating the unfit and 

 causing the relative increase of the fit. Fisher^ has made this idea precise 

 in his Fundamental Theorem of Natural Selection^ which states that the 

 rate of increase in fitness in a population is proportional to the genetic 

 variance. 



Suppose, in a population whose variabiUty depends on many genes, 

 one gene is present with the proportions pB : qb. Then if a substitu- 

 tion of B for h in the kind of genotypes met with in the population has 

 an average effect on fitness of a, a change dp in the proportion of B will 

 have an effect adp on the fitness of the population. Now suppose the 

 individuals with the gene B have an average advantage of a in fitness 

 (measured as Malthusian parameters) over the individuals with ^; a 

 need not be the same as a, depending on the system of mating, etc. 



p ^-^°- 



Then the rate of increase of the proportion - would be — ^- = e° 



q ^ 



P P 



and we have - = e""' or log - = at. Differentiating this with respect to 



q q 



dp da 



ty and remembermg that — = — -^ we get 



p qy 



dp = adt. 



Whence dp = pqadt, since p ^ q = 1. 



and the rate of increase in fitness a — = pgaa 



dt ^^ 



Now to find the genetic variance we consider a measure of fitness X 



which measures the increments which must, for any genotype, be added 



^ Fisher 1930. 



