Question

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]
(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

(a)
`\mathbf{P_2 = \left[\begin{array}{c}140 \\ 160 \\ 200 \end{array}\right]}`

(b) After an infinite period of time; we will get back to a result similar to after the two time period which will be
`= \left[\begin{array}{c}140 \\ 160 \\ 200 \end{array}\right]}`

Explanation:

The Markov Matrix can be interpret as :

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

From (a) ; we see that the initial population are as follows: 130 individuals in location 1, 300 in location 2, and 70 in location 3.

Le P represent the Population; So ;
`P = \left[\begin{array}{c}130 \\ 300 \\ 70 \end{array}\right]`

The objective is to find How many are in each location after two time periods;

So, after two time period ; we have the population
`P_2 = [M]^2 [P]`

where;

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

`[M]^ 2 = (1)/(25) \left[\begin{array}{ccc} 1+2+4 & 1+2+4 &1+2+4 \\ \\ 2+2+4&2+2+4&2+2+4\\ \\ 2+4+4&2+4+4& 2+4+4 \end{array}\right]`

`[M]^ 2 = (1)/(25) \left[\begin{array}{ccc}7&7&7 \\ \\ 8 &8&8\\ \\10&10& 10 \end{array}\right]`

Now; Over to after two time period ; when the population
`P_2 = [M]^2 [P]`

`P_2 = (1)/(25) \left[\begin{array}{ccc}7&7&7 \\ \\ 8 &8&8\\ \\10&10& 10 \end{array}\right] \left[\begin{array}{c}130 \\ 300 \\ 70 \end{array}\right]`

`\mathbf{P_2 = \left[\begin{array}{c}140 \\ 160 \\ 200 \end{array}\right]}`

(b) The total number of individuals in the migration process is 500. After a long time, how many are in each location?

After a long time; that is referring to an infinite time (n)

So;
`P_n = [M]^n [P]`

where ;

`[M]^n \ can \ be \ [M]^2 , [M]^3 , [M]^4 .... \infty`

; if we determine the respective values of
`[M]^2 , [M]^3 , [M]^4 .... \infty` we will always result to the value for
`[M]^n`; Now if
`[M]^n` is said to be a positive integer; then :

After an infinite period of time; we will get back to a result similar to after the two time period which will be
`= \left[\begin{array}{c}140 \\ 160 \\ 200 \end{array}\right]}`