Question

Exercise 7.3.5 The following is a Markov (migration) matrix for three locations        1 5 1 5 2 5 2 5 2 5 1 5 2 5 2 5 2 5        7.3. Applications of Spectral Theory 397 (a) Initially, there are 130 individuals in location 1, 300 in location 2, and 70 in location 3. How many are in each location after two time periods? (b) The total number of individuals in the migration process is 500. After a long time, how many are in each location?

Answer 1

Both get the same results that is,

`\left[\begin{array}{ccc}140\\160\\200\end{array}\right]`

Explanation:

Given :

`\bf M=\left[\begin{array}{ccc}(1)/(5)&(1)/(5)&(2)/(5)\\(2)/(5)&(2)/(5)&(1)/(5)\\(2)/(5)&(2)/(5)&(2)/(5)\end{array}\right]`

and initial population,

`\bf P=\left[\begin{array}{ccc}130\\300\\70\end{array}\right]`

a) - After two times, we will find in each position.

`P_2=[P].[M]^2=[P].[M].[M]`

`M^2=\left[\begin{array}{ccc}(1)/(5)&(1)/(5)&(2)/(5)\\(2)/(5)&(2)/(5)&(1)/(5)\\(2)/(5)&(2)/(5)&(2)/(5)\end{array}\right]* \left[\begin{array}{ccc}(1)/(5)&(1)/(5)&(2)/(5)\\(2)/(5)&(2)/(5)&(1)/(5)\\(2)/(5)&(2)/(5)&(2)/(5)\end{array}\right]`

`=(1)/(25) \left[\begin{array}{ccc}7&7&7\\8&8&8\\10&10&10\end{array}\right]`

`\therefore\;\;\;\;\;\;\;\;\;\;\;P_2=\left[\begin{array}{ccc}7&7&7\\8&8&8\\10&10&10\end{array}\right] *\left[\begin{array}{ccc}130\\300\\70\end{array}\right] = \left[\begin{array}{ccc}140\\160\\200\end{array}\right]`

b) - With in migration process, 500 people are numbered. There will be after a long time,

`After\;inifinite\;period=[M]^n.[P]`

`Then,\;we\;get\;the\;same\;result\;if\;we\;measure [M]^n=(1)/(25) \left[\begin{array}{ccc}7&7&7\\8&8&8\\10&10&10\end{array}\right]`

`=\left[\begin{array}{ccc}140\\160\\200\end{array}\right]`