MATLAB Graphics - A Guide to MATLAB for Beginners and Experienced Users - page 90

these numbers for large n and how they depend on the initial population.
Clearly

x n + y n + z n = 1 , n ≥ 0 .

Now we can use the table to express a relationship between the n thand
( n + 1)stgenerations.Becauseofourpresumptiononmating,onlythefirst,
fourth, and sixth columns are relevant. Indeed a moment's reflection re-
veals that we have

1
4 y n

x n + 1 = x n +

1
2 y n

y n + 1 = ( * )

1
4 y n .
(a) Write the equations (*) as a single matrix equation X n + 1 = MX n ,
n ≥ 0. Explain carefully what the entries of the column matrix X n
are and what the coefficients of the square matrix M are.
(b) Apply the matrix equation recursively to express X n in terms of X 0
and powers of M .
(c) Next use MATLAB to compute the eigenvalues and eigenvectors of
M .
(d) From Problem 12 you know that MU = UR , where R is the diag-
onal matrix of eigenvalues of M . Solve that equation for M . Now
it should be evident to you what R ∞ = lim n →∞ R n is. Use that and
your expression of M in terms of R to compute M ∞ = lim n →∞ M n .
(e) Describe the eventual population distribution by computing M ∞ X 0 .
(f) Check your answer by directly computing M n for large specific val-
ues of M .( Hint : MATLAB can compute the powers of a matrix M by
entering Mˆ10 , for example.)
(g) You can alter the fundamental presumption in this problem by as-
suming, alternatively, that all members of the n thgeneration must
mate only witha parent whose genotype is purely dominant. Com-
pute the eventual population distribution of that model. Chapters
12-14 in Rorres and Anton have other interesting models.

z n + 1 = z n +