Chapter 2, Problem 2P

Population Dynamics

In 1991, the U.S. Fish and Wildlife Service proposed logging restrictions on nearly 12  million acres of Pacific Northwest forest to help save the endangered northern spotted owl. This decision caused considerable controversy between the logging industry and environmentalists.

Mathematical ecologists created a mathematical model to analyze the population dynamics of the spotted owl. They divided the female owl population into three categories: juvenile (up to 1 year old), subadult (1 to 2 years old), and adult (over 2 years old). Suppose that in a certain region there are currently 2950 female spotted owls made up of 650 juveniles, 200 subadults, and 2100 adults. The ecologists used matrices to project the changes in the population from year to year. The original numbers can be displayed in the column matrix

$X_0=\left[\begin{array}{c}650\\ 200\\ 2100\end{array}\right]$.

The populations after one year are given by the column matrix

$X_1=\left[\begin{array}{c}693\\ 117\\ 2116\end{array}\right]$.

The subscript 1 tells us that the matrix gives the population after one year. The names of the matrices for subsequent years will have subscripts 2, 3, 4, etc.

Did the total population of females increase or decrease during the year?

Let A denote the matrix

$\left[\begin{array}{ccc}0& 0& .33\\ .18& 0& 0\\ 0& .71& .94\end{array}\right]$.

According to the mathematical model, subsequent population distributions are generated by multiplication on the left by A. That is,

$A\cdot X_0=X_1,A\cdot X_1=X_2,A\cdot X_2=X_3,\dots$

If the population distribution at any time is given by $\left[\begin{array}{c}j\\ s\\ a\end{array}\right]$, then the distribution one year later is $A\cdot \left[\begin{array}{c}j\\ s\\ a\end{array}\right]$.