Page 294 - A Course in Linear Algebra with Applications

278 Chapter Eight: Eigenvalues and Eigenvectors
1
1
n
It is now easy to find X n; for A n = (SOS' )™ = SD S~ .
Therefore
= n = n 1
X n A X 0 SD S- X 0
(I 2 \ (2 n M (-1 2 /100
)
-{i iJ(,o 3 n J \ 1 -lj \ 10

lich leads to
n n
= f 180 • 3 - 80 • 2
n n n
~ \ 90 • 3 - 80 • 2
The solution to the problem can now be read off:

n n
r n = 180 • 3" - 80 • 2 and w n = 90 • 3 - 80 • T.
Let us consider for a moment the implications of these equa-
tions. Notice that r n and w n both increase without limit as
n —> oo since 3 n is the dominant term; however

lim ( ^ ) = 2.
n—>oo t o n
The conclusion is that, while both populations explode, in the
long run there will be twice as many rabbits as weasels.

Having seen that eigenvalues provide a satisfactory so-
lution to the rabbit-weasel problem, we proceed to consider
systems of linear recurrences in general.

Systems of first order linear recurrence relations
A system of first order (homogeneous) linear recurrence
relations in functions y„ ,..., j/n of an integral variable n is
a set of equations of the form
t (i) (i) , , M
Vn+l = a HVn + ••• + O-lmVn
(2) (1) , , {rn)
Vn+l = a 2l2M + • • • + 0,2mVn
m
(m) _ (!) i i ( )
a
V n-\-l — miyn -r ' • • -r 0, mrny n