Definition 12.2.1Let n locations be denoted by the numbers 1,2,
⋅⋅⋅
,n. Also suppose it is the casethat each year aijdenotes the proportion of residents in location j which move to location i. Alsosuppose no one escapes or emigrates from without these n locations. This last assumption requires∑iaij = 1. Thus
(aij)
is a Markov matrix referred to as amigration matrix.
If v =
(x ,⋅⋅⋅,x )
1 n
T where xi is the population of location i at a given instant, you obtain
the population of location i one year later by computing ∑jaijxj =
(Av)
i. Therefore, the
population of location i after k years is
(Akv)
i. Furthermore, Corollary 12.1.7 can be used to
predict in the case where A is regular what the long time population will be for the given
locations.
As an example of the above, consider the case where n = 3 and the migration matrix is of the
form
i will converge to the ith component of a steady
state. It follows the steady state can be obtained from solving the system
.6x +.1z = x
.2x + .8y = y
.2x+ .2y+ .9z = z
along with the stipulation that the sum of x,y, and z must equal the constant value present at the
beginning of the process. The solution to this system is
{y = x,z = 4x,x = x}.
If the total population at the beginning is 150,000, then you solve the following system
y = x, z = 4x, x+ y +z = 150000
whose solution is easily seen to be
{x = 25000,z = 100000,y = 25000}
. Thus, after a long time
there would be about four times as many people in the third location as in either of the other
two.