Consider a two allele system (*A*_{1} and *A*_{2}) and let *p* and *q* represent the frequency of *A*_{1} and *A*_{2}, respectively. Let *w*_{ij} represent the fitness of genotype *A*_{i}A_{j} (assume *w*_{ij} = *w*_{ji} for *i* ≠ *j*). Therefore, for this system *w*_{11} is the fitness of allele A_{1}A_{1}, *w*_{12} is the fitness of allele A_{1}A_{2}, and *w*_{22} is the fitness of allele A_{2}A_{2}. We can express the new frequency of *A*_{1} after one generation of selection using the rational function,
where is the mean fitness of the population. We are interested in finding equilibrium values of *p*, in other words values of *p* such that *p'* = *p*, indicating no change in allele frequency in the next generation. Setting *p*' = *p* gives,
Assuming *p* ≠ 0 we can cancel* p* on both sides of the above equation as,
Bringing all terms to the right-hand side of the equation and substituting *q* = 1 - *p* gives the polynomial,
Therefore, we have deduced that solutions to the above equations are equilibrium values of *p* (we are only concerned with biologically reasonable equilibria). Using fitness values:
where *c* > 0 is a constant, answer the following questions. |