Linear Algebra and Analysis

12.2 Migration Matrices

If v =

^T where x_i is the population of location i at a given instant, you obtain the population of location i one year later by computing ∑ _ja_ijx_j = _i. Therefore, the population of location i after k years is _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

( ) .6 0 .1 |( .2 .8 0 |) . .2 .2 .9

Now

( ) 2 ( ) | .6 0 .1 | | .38 .02 .15 | ( .2 .8 0 ) = ( .28 .64 .02 ) .2 .2 .9 .34 .34 .83

and so the Markov matrix is regular. Therefore,

_i will converge to the i^th 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

. 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.