Solved: In Example 1 suppose that the initial plants are fertilized with genotype AA

Show that if M = PDP 1, then Mn = PDn P 1 for n = 1, 2,....

In Example 1 suppose that the plants are always fertilized with a plant of genotype Aa rather than one of genotype AA. Derive formulas for the fractions of the plants of genotypes AA, Aa, and aa in the nth generation. Also, find the limiting genotype distribution as n tends to infinity.

In Example 1 suppose that the initial plants are fertilized with genotype AA, the first generation is fertilized with genotype Aa, the second generation is fertilized with genotype AA, and this alternating pattern of fertilization is kept up. Find formulas for the fractions of the plants of genotypes AA, Aa, and aa in the nth generation.

In the section on autosomal recessive diseases, find the eigenvalues and eigenvectors of the matrix M and verify Equation (7).

. Suppose that a breeder has an animal population in which 25% of the population are carriers of an autosomal recessive disease. If the breeder allows the animals to mate irrespective of their genotype, use Equation (9) to calculate the number of generations required for the percentage of carriers to fall from 25% to 10%. If the breeder instead implements the controlled-mating program determined by Equation (8), what will the percentage of carriers be after the same number of generations?

In the section on X-linked inheritance, suppose that the initial parents are equally likely to be of any of the six possible genotype parents; that is, x(0) = 1 6 1 6 1 6 1 6 1 6 1 6 Using Equation (12), calculate x(n) and also calculate the limit of x(n) as n tends to infinity.

From (13) show that under X-linked inheritance with inbreeding, the probability that the limiting sibling pairs will be of type (A, AA) is the same as the proportion of A genes in the initial population.

In X-linked inheritance suppose that none of the females of genotype Aa survive to maturity. Under inbreeding the possible sibling pairs are then (A, AA), (A, aa), (a, AA), and (a, aa) Find the transition matrix that describes how the genotype distribution changes in one generation.

Derive the matrix M in Equation (10) from Table 2.

(a) Use a computer to verify that the eigenvalues and eigenvectors of M = 1 1 4 00 00 0 1 4 0 1 1 4 0 0 000 1 4 0 0 1 4 00 00 0 1 4 1 0 1 4 0 0 000 1 4 1 as given in the text are correct. (b) Starting with x(n) = Mx(n1) and the assumption that lim n x(n) = x exists, we must have This suggests that x can be solved directly using the equation (M I )x = 0. Use a computer to solve the equation x = Mx, where x = a b c d e f and a + b + c + d + e + f = 1; compare your results to Equation (13). Explain why the solution to (M I )x = 0 along with a + b + c + d + e + f = 1 is not specific enough to determine limn x(n).

(a) Given P = 1 0 1 1 1 4 (3 5 ) 1 4 (3 + 5 ) 002 61 1 0 0 1 3 1 4 (1 + 5 ) 1 4 (1 5 ) 0013 1 4 (1 + 5 ) 1 4 (1 5 ) 0 0 26 1 1 011 1 1 4 (3 5 ) 1 4 (3 + 5 ) from Equation (12) and lim n Dn = 100000 010000 000000 000000 000000 000000 use a computer to show that lim n Mn = 1 2 3 1 3 2 3 1 3 0 000000 000000 000000 000000 0 1 3 2 3 1 3 2 3 1 (b) Use a computer to calculate Mn for n = 10, 20, 30, 40, 50, 60, 70, and then compare your results to the limit in part (a). 1