Fitness Model for Gene Frequency in Populations

You will find that in some cases the A allele will truly become dominant and the other allele will go extinct (e.g., see Snapshot 1). Even with the same fitness coefficients, if the initial allele frequency of A is sufficiently small, it will go extinct (Snapshot 2). And in the case of heterozygous advantage (when the Aa fitness coefficient is 100% and the others are less), a stable limiting value for the frequency of A will manifest itself (Snapshot 3). This limiting value is indicated by a green line. The most famous example of heterozygous advantage is that of sickle cell anemia in certain tropical populations.

In some other cases a red horizontal line appears. This is a repelling (rather than attracting) stable value for the allele frequencies. That is, if is less than this value, , while if is greater than this value, . Whether red or green, the limiting value is given by , where , , and are the fitness coefficients for aa, Aa, and AA, respectively.

The mathematics behind this Demonstration is quite simple and is widely documented. See, for example, chapter 6.4 of E. S. Allman and J. A. Rhodes, Mathematical Models in Biology: An Introduction, New York: Cambridge University Press, 2004.