Suppose that a species of bird only lives in Canada the Unit

Suppose that a species of bird only lives in Canada, the United States, or in Mexico. Every year, 4% of the Canadian birds travel to the US, and 1% of them travel to Mexico. Every year, 6% of the US birds travel to Canada, and 4% go to Mexico. From Mexico, every year 10% travel to the US, and 0% go to Canada. Give the transition matrix. Is there a way for the three countries to have constant populations?

Solution

(a)The transition matrix P is as under:

Canada

USA

Mexico

Canada

0.95

0.06

0.00

USA

0.04

0.90

0.10

Mexico

0.01

0.04

0.90

(b) The steady state vector X is given by the equation P.X = X or, (P-I₃) .X = 0

      The matrix A = P-I₃ is

-0.05

0.06

0.00

0.04

-0.10

0.10

0.01

0.04

-0.10

We will reduce A to its RREF as under:

Multiply the 1st row by -20

Add -1/25 times the 1st row to the 2nd row

Add -1/100 times the 1st row to the 3rd row

Multiply the 2nd row by -250/13

Add -13/250 times the 2nd row to the 3rd row

Add 6/5 times the 2nd row to the 1st row

Then the RREF of A = P-I₃ is

1

0

-30/13

0

1

-25/13

0

0

0

Thus, if X = (x,y,z)^T , then x –(30/13)z = 0 and y –(25/13)z. Let z =13t. Then x = 30t and y = 25t.Therefore, X = t( 30,25,13)^T. The steady state vector, thus, is ( 30,25,13)^T

	Canada	USA	Mexico
Canada	0.95	0.06	0.00
USA	0.04	0.90	0.10
Mexico	0.01	0.04	0.90