Chapter 11, Problem 6

In Example 11.9, assume that we start with a hybrid bred to a hybrid. Find u(1), u(2), and u(3). What would u(n) be?

Assume that you start with a hybrid and each time, the hybrid mates with a hybrid

gene_matrix = matrix(c(.5,.25,0,.5,.5,.5,0,.25,.5), nrow=3,ncol=3)
gene_matrix
##      [,1] [,2] [,3]
## [1,] 0.50  0.5 0.00
## [2,] 0.25  0.5 0.25
## [3,] 0.00  0.5 0.50

Since you are starting with a hybrid, we can write the following u vector:

u = c(0,1,0)
u
## [1] 0 1 0

u\(^{(n)}\) = u\(P^n\)
u\(^{(1)}\) = u\(P^1\)

P1 = gene_matrix
u1 = u%*%P1
u1
##      [,1] [,2] [,3]
## [1,] 0.25  0.5 0.25

u\(^{(2)}\) = u\(P^2\)

P2 = gene_matrix%*%gene_matrix
P2
##       [,1] [,2]  [,3]
## [1,] 0.375  0.5 0.125
## [2,] 0.250  0.5 0.250
## [3,] 0.125  0.5 0.375
u2 = u%*%P2
u2
##      [,1] [,2] [,3]
## [1,] 0.25  0.5 0.25

u\(^{(3)}\) = u\(P^3\)

P3 = gene_matrix%*%P2
P3
##        [,1] [,2]   [,3]
## [1,] 0.3125  0.5 0.1875
## [2,] 0.2500  0.5 0.2500
## [3,] 0.1875  0.5 0.3125
u3 = u%*%P3
u3
##      [,1] [,2] [,3]
## [1,] 0.25  0.5 0.25

u\(^{(n)}\) = (.25,.5,.25)