Chapter 1, Problem 8E

The following table describes a seven-stage model for the life cycle of the loggerhead sea turtle.

Table I Seven-Stage Model for Loggerhead Sea Turtle Demographics

The corresponding Leslie matrix is
$L=\left[\begin{array}{ccccccc}0& 0& 0& 0& 127& 4& 800\\ 0.6747& 0.7370& 0& 0& 0& 0& 0\\ 0& 0.0486& 0.6610& 0.6907& 0& 0& 0\\ 0& 0& 0.0147& 0.0518& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0.8091& 0& 0.8089\\ 0& 0& 0& 0& 0& 0.8091& 0.8091\end{array}\right]$
Suppose that the number of tunics in each stage of the initial turtle population is described by the vector
${\text{x}}_0={\left(\begin{array}{ccccccc}200,000& 130,000& 100,000& 70,000& 500& 400& 1100\end{array}\right)}^T$
(a) Enter L into MATLAB and then set
${\text{x}}_0={\left(\begin{array}{ccccccc}200000& 130000& 100000& 70000& 500& 400& 1100\end{array}\right)}^{\prime }$
Use the command
${\text{x}}_{\text{5}0}=\text{round}\left(L^{\wedge }50\ast \text{x}0\right)$
to compute ${\text{x}}_{\text{5}0}$
${\text{x}}_{100},{\text{x}}_{150},{\text{x}}_{200},{\text{x}}_{250}$ , and ${\text{x}}_{300}$
(b) Loggerhead sea tunics by their eggs on land. Suppose that conservationists take special measures to protect these eggs and, as a result, the survival rate for eggs and hatchlings increases to 77 percent. To incorporate this change into our model, we need only change the (2, 1) entry of L to 0.77. Make this modification to the matrix L and repeat pan (a). Has the survival potential of the loggerhead sea turtle improved significantly?
(c) Suppose that, instead of improving the survival rate for eggs and hatchlings, we could devise a means of protecting the small juveniles so that their survival rate increases to 88 percent. Use equations (1) and (2) from Application 2 of Section 1.4 to determine the proportion of small juveniles that survive and remain in the same stage and the proportion that survive and grow to the next stage. Modify your original matrix L accordingly and repeat part (a), using the new matrix. Has the survival potential of the loggerhead sea turtle improved significantly?

AX and $A^{\text{2}}X$ for the given matrices A and X. Use format long and enter these matrices in MATLAB Compute $A^k$ and $A^{k}X$ for $k=5,10,15,20$ . What is happening to $A^k$ as k gets large? What is the long-run distribution of married and single women in the town?