Solve each problem.
Several years ago, mathematical ecologists created a model to analyze
population dynamics of the endangered northern spotted owl in the Pacific
Northwest. The ecologists divided the female owl population into three
categories: juvenile (up to \(1 \text { yr old }),\) subadult \((1\) to 2 yr old )
and adult (over 2 yr old). They concluded that the change in the makeup of the
northern spotted owl population in successive years could be described by the
following matrix equation.
$$
\left[\begin{array}{c}
j_{n+1} \\
s_{n+1} \\
a_{n+1}
\end{array}\right]=\left[\begin{array}{rrr}
0 & 0 & 0.33 \\
0.18 & 0 & 0 \\
0 & 0.71 & 0.94
\end{array}\right]\left[\begin{array}{c}
j_{n} \\
s_{n} \\
a_{n}
\end{array}\right]
$$
The numbers in the column matrices give the numbers of females in the three
age groups after \(n\) years and \(n+1\) years. Multiplying the matrices yields
the following.
\(j_{n+1}=0.33 a_{n}\)
Each year 33 juvenile females are born for each 100 adult females.
\(s_{n+1}=0.18 j_{n}\)
Each year 18\% of the juvenile females survive to become subadults.
\(a_{n+1}=0.71 s_{n}+0.94 a_{n} \quad\) Each year \(71 \%\) of the subadults
survive to become adults, and \(94 \%\) of the adults survive.
(a) Suppose there are currently 3000 female northern spotted owls made up of
690 juveniles, 210 subadults, and 2100 adults. Use the matrix equation on the
preceding page to determine the total number of female owls for each of the
next 5 yr.
(b) Using advanced techniques from linear algebra, we can show that in the
long run,
$$
\left[\begin{array}{c}
j_{n+1} \\
s_{n+1} \\
a_{n+1}
\end{array}\right] \approx 0.98359\left[\begin{array}{c}
j_{n} \\
s_{n} \\
a_{n}
\end{array}\right]
$$
What can we conclude about the long-term fate of the northern spotted owl?
(c) In the model, the main impediment to the survival of the northern spotted
owl is the number 0.18 in the second row of the 3 \(\times 3\) matrix. This
number is low for two reasons.
The first year of life is precarious for most animals living in the wild.
In addition, juvenile owls must eventually leave the nest and establish their
own territory. If much of the forest near their original home has been
cleared, then they are vulnerable to predators while searching for a new home.
Suppose that, thanks to better forest management, the number 0.18 can be
increased
to \(0.3 .\) Rework part (a) under this new assumption.
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