The annual migration patterns between urban, suburban, and rural areas within a region are shown below. In the year 2021, the urban population, u, was 800,000, the suburban population, so, was 500,000, and the rural population, ro, was 40,000. Write the transition matrix, T, that describes how each member in the region moves between populations and the column matrix, P, to describe the initial sizes of each population

Answer 1

The transition matrix, T, is:

\[ \begin{pmatrix} 0.90 & 0.05 & 0.03 \\ 0.07 & 0.85 & 0.08 \\ 0.03 & 0.10 & 0.89 \end{pmatrix} \]

where the rows represent the initial populations in urban, suburban, and rural areas respectively, and the columns represent the probabilities of moving to urban, suburban, and rural areas respectively.

The column matrix, P, to describe the initial sizes of each population is:

\[ \begin{pmatrix} 800,000 \\ 500,000 \\ 40,000 \end{pmatrix} \]