Chapter 5.7, Problem 87E

To determine

To find:

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 yr old), sub adult (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.

$\left[\begin{array}{c}{j}_{n+1}\\ s_{n+1}\\ a_{n+1}\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0.33\\ 0.18& 0& 0\\ 0& 0.71& 0.94\end{array}\right]\left[\begin{array}{c}{j}_n\\ s_n\\ a_n\end{array}\right]$

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.

$j_{n+1}=0.33a_n$	Each year 33 juvenile females are born for each 100 adult females
$s_{n+1}=0.18j_n$	Each year 18% of the juvenile females survive to become sub adults.
$a_{n+1}=0.71s_n+0.94a_n$	Each year 71% of the sub adults survive to become adults, and 94% of the adults survive.

(Source: Lumberton, R. H., R. McKinley, B. R. Noon, and C. Voss, “A Dynamic Analysis of Northern Spotted Owl Viability in a Fragmented Forest Landscape,” Conservation Biology, Vol. 6, No. 4.)

(a) Suppose there are currently 3000 female northern spotted owls made up of 690 juveniles, 210 sub adults, and 2100 adults. Use the matrix equation to determine the total number of female owls for each of the next 5 yr.

To determine

(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?

To determine

(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.

• 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.