Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 52 / 5 / a / Solution

Let $u$ be the allele frequency of $A$, with random mating and sufficiently large population size to neglect genetic drift. The Hardy-Weinberg proportions before selection are $u^2,2u(1-u),(1-u{)}^2$. Write the three genotype fitnesses as $w_{AA},w_{Aa},w_{aa}$, with $w_{Aa}$ smaller than either homozygote value. After one viability-selection generation,ConsequentlyPut $a=w_{AA}-w_{Aa}>0$, $b=w_{aa}-w_{Aa}>0$. The underdominant allele-frequency dynamics has stable fixation states zero and one, separated by the unstable thresholdBelow it $A$ decreases to allele fixation of $a$; above it $A$ increases to fixation. Exactly at the threshold the deterministic frequency remains balanced, but any perturbation selects a side. This is underdominance, or heterozygote disadvantage, rather than balancing selection from heterozygote advantage.

In a continuous-time weak-selection or selection-rate convention, the corresponding equation is $\dot{u}=u(1-u)[(a+b)u-b]$. Equal homozygote fitness gives $\dot{u}=su(1-u)(2u-1)$ for a positive selection strength $s$. The next part uses this explicitly stated continuous-time convention; $s$ and migration then have the same time units.