In a certain region, about 6% of city’s population moves to the surrounding suburbs each year, and about 4% of the suburban population moves into the city. In 2015, there were, 10,000,000 residents in the city and 800,000 in the suburbs. Set up a difference equation that describes this situation, where\({{\bf{x}}_{\bf{0}}}\)is the initial population in 2015. Then estimate the population in the city and in the suburbs two years later, in 2017.

As the population is moving from two origins to two destinations, the order of the migration matrix is \(2 \times 2\).

For the first column of the migration matrix, 6% of the city’s population migrates to the surrounding of suburbs, i.e., 94% is in the city every year.

The first column of the migration matrix is

\(\left[ {\begin{array}{*{20}{c}}{0.94}\\{0.06}\end{array}} \right]\).

For the second column of the migration matrix, 4% of the suburb population moves to the city, i.e., 96% population remains in the suburbs.

The second column of the migration matrix is

\(\left[ {\begin{array}{*{20}{c}}{0.04}\\{0.96}\end{array}} \right]\).

So, the migration matrix is \(M = \left[ {\begin{array}{*{20}{c}}{0.94}&{0.04}\\{0.06}&{0.96}\end{array}} \right]\).

The initial population matrix is \({{\bf{x}}_0} = \left[ {\begin{array}{*{20}{c}}{10,000,000}\\{800,000}\end{array}} \right]\).

The difference equation is \({{\bf{x}}_{k + 1}} = M{{\bf{x}}_k}\).

For \(k = 0\), by the equation \({{\bf{x}}_{k + 1}} = M{{\bf{x}}_k}\),

\(\begin{array}{c}{{\bf{x}}_1} = M{{\bf{x}}_0}\\ = \left[ {\begin{array}{*{20}{c}}{0.94}&{0.04}\\{0.06}&{0.96}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{10,000,000}\\{500,000}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{9,420,000}\\{1,080,000}\end{array}} \right].\end{array}\)

\(\begin{array}{c}{{\bf{x}}_1} = M{{\bf{x}}_0}\\ = \left[ {\begin{array}{*{20}{c}}{0.94}&{0.04}\\{0.06}&{0.96}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{10,000,000}\\{500,000}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{9,420,000}\\{1,080,000}\end{array}} \right].\end{array}\)

For \(k = 1\), by the equation \({{\bf{x}}_{k + 1}} = M{{\bf{x}}_k}\),

\(\begin{array}{c}{{\bf{x}}_2} = M{{\bf{x}}_1}\\ = \left[ {\begin{array}{*{20}{c}}{0.94}&{0.04}\\{0.06}&{0.96}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{9,420,000}\\{1,080,000}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{8,898,000}\\{1,602,000}\end{array}} \right].\end{array}\)

So, the population of the city after two years is 8,898,000 and of suburbs is 1,602,000.