Selection
Haploid Model
In a deterministic (selection only, no drift or mutation) model an allele frequency is raised or lowered by multiplying it by the average fitness of the corresponding phenotype. In this simple case there is less of a distinction between allele, genotype, and phenotype, than in diploid models.
Say the fitness of an "A" allele is wA and the fitness of the alternative allele "a" is set to a value of one, wa = 1.
The frequency of the allele pA in the next generation is equal to the frequency times fitness divided by the total frequencies times fitnesses in the population (to maintain this as a proportion out of the total).
[math]p_{t+1} = \frac{w_A p}{w_A p + w_a (1-p)} = \frac{w_A p}{w_A p + 1 - p}[/math]
If p = 0.3 and wA = 1.1, corresponding to a 10% fitness advantage, s = 0.1 and w = 1 + s, then we predict p will be slightly above 32% in the next generation. When the allele is rare a proportion a bit smaller than sp is added to the allele frequency p each generation. This diminishes as the allele becomes more common and there are less alternative alleles to replace (when the allele competes more with itself it looses the relative advantage to the population as a whole).
[math]w_A p + w_a (1-p) = \bar{w}[/math]
is the average fitness in the population, the fitnesses weighted by the frequency of the corresponding alleles.
[math]p_{t+1} = p \frac{w_A}{\bar{w}}[/math]
If the allele fitness is greater than the average fitness in the popualtion then it will increase in frequency (p is multiplied by [math]\frac{w_A}{\bar{w}}[/math] if [math]w_A \gt \bar{w}[/math] then [math]\frac{w_A}{\bar{w}}\gt1[/math]), and vice versa, if the allele fitness is lower than the average fitness it will decrease in frequency (multiplied by a number less than one).
