Population genetics: In the study of population genetics, an important measure of inbreeding is the proportion of homozygous genotypes—that is, instances in which the two alleles carried at a particular site on an individual’s chromosomes are both the same. For populations in which bloodrelated individuals mate, there is a higher than expected frequency of homozygous individuals. Examples of such populations include endangered or rare species, selectively bred breeds, and isolated populations. In general, the frequency of homozygous children from matings of blood-related parents is greater than that for children from unrelated parents.
Measured over a large number of generations, the proportion of heterozygous genotypes—that is, nonhomozygous genotypes—changes by a constant factor λ1 from generation to generation. The factor λ1 is a number between 0 and 1.If λ1 = 0.75, for example, then the proportion of heterozygous individuals in the population decreases by 25% in each generation. In this case, after 10 generations the proportion of heterozygous individuals in the population decreases by 94.37%, since 0.7510 = 0.0563, or 5.63%. In other words, 94.37% of the population is homozygous. For specific types of matings, the proportion of heterozygous genotypes can be related to that of previous generations and is found from an equation. For matings between siblings, λ1 can be determined as the largest value of λ for which
rm) in terms of those of the previous two generations (represented by λ for the parents’ generation and by the constant term for the grandparents’ generation).
a. Find both solutions to the quadratic equation above and identify which is λ1. (Use a horizontal span from −1 to 1 in this exercise and the following two exercises.) b. After 5 generations what proportion of the population will be homozygous?
c. After 20 generations what proportion of the population will be homozygous?