Consider a two allele system (A1 and A2) and let p and q represent the frequency of A1 and A2, respectively. Let wij represent the fitness of genotype AiAj (assume wij = wji for i ≠ j). Therefore, for this system w11 is the fitness of allele A1A1, w12 is the fitness of allele A1A2, and w22 is the fitness of allele A2A2. We can express the new frequency of A1 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.