Question

The following is a Markov (migration) matrix for three locations

Answer 1

Answer:

(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 $= \left[\begin{array}{c}140 \\ 160 \\ 200 \end{array}\right]}$

Step-by-step explanation:

The Markov Matrix can be interpret as :

$M = \left[\begin{array}{ccc} \dfrac{1}{5} & \dfrac{2}{5} &\dfrac{1}{5} \\ \\ \dfrac{2}{5}&\dfrac{2}{5}&\dfrac{1}{5}\\ \\ \dfrac{2}{5}& \dfrac{2}{5}& \dfrac{2}{5} \end{array}\right]$

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;

$[M]^ 2 = \left[\begin{array}{ccc} \dfrac{1}{5} & \dfrac{2}{5} &\dfrac{1}{5} \\ \\ \dfrac{2}{5}&\dfrac{2}{5}&\dfrac{1}{5}\\ \\ \dfrac{2}{5}& \dfrac{2}{5}& \dfrac{2}{5} \end{array}\right] \left[\begin{array}{ccc} \dfrac{1}{5} & \dfrac{2}{5} &\dfrac{1}{5} \\ \\ \dfrac{2}{5}&\dfrac{2}{5}&\dfrac{1}{5}\\ \\ \dfrac{2}{5}& \dfrac{2}{5}& \dfrac{2}{5} \end{array}\right]$

$[M]^ 2 = \dfrac{1}{25} \left[\begin{array}{ccc} 1+2+4 & 1+2+4 &1+2+4 \\ \\ 2+2+4&2+2+4&2+2+4\\ \\ 2+4+4&2+4+4& 2+4+4 \end{array}\right]$

$[M]^ 2 = \dfrac{1}{25} \left[\begin{array}{ccc}7&7&7 \\ \\ 8 &8&8\\ \\10&10& 10 \end{array}\right]$

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

$P_2 = \dfrac{1}{25} \left[\begin{array}{ccc}7&7&7 \\ \\ 8 &8&8\\ \\10&10& 10 \end{array}\right] \left[\begin{array}{c}130 \\ 300 \\ 70 \end{array}\right]$

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

(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 $= \left[\begin{array}{c}140 \\ 160 \\ 200 \end{array}\right]}$