A country is divided into three demographic regions It is fo

A country is divided into three demographic regions. It is found that each year 5% of the residents of region 1 move to region 2, and 5% move to region 3. Of the residents of region 2,15% move to region 1 and 10% move to region 3. And of the residents of region 3, 10% move to region 1 and 5% move to region 2. What percentage of the population resides in each of the three regions after a long period of time?

Solution

First of all find the transition matrix which is given by

The entry of transition matrix is positive, hence it will be a regular stochastic matrix

in order to find the steady state population

lambda*I - P = I - P, will give the matrix as

(3.25 + 1.75 + 1)t =1

t = 1/6

and we get the final values as

Output = (0.5417, 0.166. 0.2916)

Percentage of population in country 1 = 54.17%

Percentage of population in country 2 = 16.66%

Percentage of population in country 3 = 29.16%

0.90	0.15	0.10
0.05	0.75	0.05
0.05	0.10	0.85