Chapter 3.2, Problem 47E

Population dynamics Suppose that for a certain metropolitan area and the surrounding counties, during each 5-year period, an average of $20\%$ of the metropolitan population $M$ moves to the surrounding counties and the rest remains. Similarly, suppose that in the same period, an average of $30\%$ of the surrounding counties’ population S moves to the metropolitan area and the rest remains. This population dynamic can be represented as the matrix

$\begin{array}{l}\begin{array}{cc}M& S\end{array}\\ D=\left[\begin{array}{cc}80\%& 20\%\\ 30\%& 70\%\end{array}\right]\begin{array}{c}M\\ S\end{array}=\left[\begin{array}{cc}0.8& 0.2\\ 0.3& 0.7\end{array}\right]\end{array}$

Where row $1$ shows how the metropolitan population $M$ changed ( $80\%$ remained in $M$ and $20\%$ moved to $S$) and row $2$ shows how the population of the surrounding counties changed. Currently, the population is evenly divided between the two areas, which can be represented by the row matrix

$P=\left[\%in\text{}M\%inS\right]=\left[50\%50\%\right]=\left[0.50.5\right]$

(a) Form the product $PD$ and interpret its entries.

(b) Predictions of this population distribution after $10$ years and after $15$ years could be found from what matrix products?

(c) Suppose that at some future time, the populations reach the point where $60\%$ live in the metropolitan area and $40\%$ live in the surrounding counties. Find the predicted distribution $5$ years later.