Answer:
(a) ![\mathbf{P_2 = \left[\begin{array}{c}140 \\ 160 \\ 200 \end{array}\right]}](https://tex.z-dn.net/?f=%5Cmathbf%7BP_2%20%3D%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D140%20%5C%5C%20160%20%5C%5C%20200%20%5Cend%7Barray%7D%5Cright%5D%7D)
(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]}](https://tex.z-dn.net/?f=%3D%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D140%20%5C%5C%20160%20%5C%5C%20200%20%5Cend%7Barray%7D%5Cright%5D%7D)
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]](https://tex.z-dn.net/?f=M%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%20%5Cdfrac%7B1%7D%7B5%7D%20%26%20%5Cdfrac%7B2%7D%7B5%7D%20%26%5Cdfrac%7B1%7D%7B5%7D%20%5C%5C%20%5C%5C%20%5Cdfrac%7B2%7D%7B5%7D%26%5Cdfrac%7B2%7D%7B5%7D%26%5Cdfrac%7B1%7D%7B5%7D%5C%5C%20%5C%5C%20%5Cdfrac%7B2%7D%7B5%7D%26%20%5Cdfrac%7B2%7D%7B5%7D%26%20%5Cdfrac%7B2%7D%7B5%7D%20%5Cend%7Barray%7D%5Cright%5D)
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]](https://tex.z-dn.net/?f=P%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D130%20%5C%5C%20300%20%5C%5C%2070%20%5Cend%7Barray%7D%5Cright%5D)
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]](https://tex.z-dn.net/?f=P_2%20%3D%20%5BM%5D%5E2%20%5BP%5D)
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]](https://tex.z-dn.net/?f=%5BM%5D%5E%202%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%20%5Cdfrac%7B1%7D%7B5%7D%20%26%20%5Cdfrac%7B2%7D%7B5%7D%20%26%5Cdfrac%7B1%7D%7B5%7D%20%5C%5C%20%5C%5C%20%5Cdfrac%7B2%7D%7B5%7D%26%5Cdfrac%7B2%7D%7B5%7D%26%5Cdfrac%7B1%7D%7B5%7D%5C%5C%20%5C%5C%20%5Cdfrac%7B2%7D%7B5%7D%26%20%5Cdfrac%7B2%7D%7B5%7D%26%20%5Cdfrac%7B2%7D%7B5%7D%20%5Cend%7Barray%7D%5Cright%5D%20%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%20%5Cdfrac%7B1%7D%7B5%7D%20%26%20%5Cdfrac%7B2%7D%7B5%7D%20%26%5Cdfrac%7B1%7D%7B5%7D%20%5C%5C%20%5C%5C%20%5Cdfrac%7B2%7D%7B5%7D%26%5Cdfrac%7B2%7D%7B5%7D%26%5Cdfrac%7B1%7D%7B5%7D%5C%5C%20%5C%5C%20%5Cdfrac%7B2%7D%7B5%7D%26%20%5Cdfrac%7B2%7D%7B5%7D%26%20%5Cdfrac%7B2%7D%7B5%7D%20%5Cend%7Barray%7D%5Cright%5D)
![[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]](https://tex.z-dn.net/?f=%5BM%5D%5E%202%20%3D%20%5Cdfrac%7B1%7D%7B25%7D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%201%2B2%2B4%20%26%201%2B2%2B4%20%261%2B2%2B4%20%5C%5C%20%5C%5C%202%2B2%2B4%262%2B2%2B4%262%2B2%2B4%5C%5C%20%5C%5C%202%2B4%2B4%262%2B4%2B4%26%202%2B4%2B4%20%5Cend%7Barray%7D%5Cright%5D)
![[M]^ 2 = \dfrac{1}{25} \left[\begin{array}{ccc}7&7&7 \\ \\ 8 &8&8\\ \\10&10& 10 \end{array}\right]](https://tex.z-dn.net/?f=%5BM%5D%5E%202%20%3D%20%5Cdfrac%7B1%7D%7B25%7D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D7%267%267%20%5C%5C%20%5C%5C%208%20%268%268%5C%5C%20%5C%5C10%2610%26%2010%20%5Cend%7Barray%7D%5Cright%5D)
Now; Over to after two time period ; when the population
![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]](https://tex.z-dn.net/?f=P_2%20%3D%20%5Cdfrac%7B1%7D%7B25%7D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D7%267%267%20%5C%5C%20%5C%5C%208%20%268%268%5C%5C%20%5C%5C10%2610%26%2010%20%5Cend%7Barray%7D%5Cright%5D%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D130%20%5C%5C%20300%20%5C%5C%2070%20%5Cend%7Barray%7D%5Cright%5D)
![\mathbf{P_2 = \left[\begin{array}{c}140 \\ 160 \\ 200 \end{array}\right]}](https://tex.z-dn.net/?f=%5Cmathbf%7BP_2%20%3D%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D140%20%5C%5C%20160%20%5C%5C%20200%20%5Cend%7Barray%7D%5Cright%5D%7D)
(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]](https://tex.z-dn.net/?f=P_n%20%3D%20%5BM%5D%5En%20%5BP%5D)
where ;
![[M]^n \ can \ be \ [M]^2 , [M]^3 , [M]^4 .... \infty](https://tex.z-dn.net/?f=%5BM%5D%5En%20%5C%20%20can%20%5C%20be%20%5C%20%5BM%5D%5E2%20%2C%20%5BM%5D%5E3%20%2C%20%5BM%5D%5E4%20....%20%5Cinfty)
; if we determine the respective values of
we will always result to the value for
; Now if
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]}](https://tex.z-dn.net/?f=%3D%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D140%20%5C%5C%20160%20%5C%5C%20200%20%5Cend%7Barray%7D%5Cright%5D%7D)