Page 292 - A Course in Linear Algebra with Applications

276 Chapter Eight: Eigenvalues and Eigenvectors

8.2 Applications to Systems of Linear Recurrences
A recurrence relation is an equation involving a function y
of a non-negative integral variable n, the value of y at n being
written y n. The equation relates the values of the function at
certain consecutive integers, typically y n+i,y n,..., y n- r. In
addition there may be some initial conditions to be satisfied,
which specify certain values of j/j. If the equation is linear in y,
the recurrence relation is said to be linear. The problem is to
solve the recurrence, that is, to find the most general function
which satisfies the equation and the initial conditions. Linear
recurrence relations, and more generally systems of linear re-
currence relations, occur in many real-life problems. We shall
see that the theory of eigenvalues provides an effective means
for solving such problems.
To understand how systems of linear relations can arise
we consider a predator-prey problem.

Example 8.2.1
In a population of rabbits and weasels it is observed that each
year the number of rabbits is equal to four times the number
of rabbits less twice the number of weasels in the previous
year. The number of weasels in any year equals the sum of
the numbers of rabbits and weasels in the previous year. If
the initial numbers of rabbits and weasels were 100 and 10
respectively, find the numbers of each species after n years.

Let r n and w n denote the respective numbers of rabbits
and weasels after n years. The information given in the state-
ment of the problem translates into the equations

r n + i = 4r n - 2w n
w n+1 = r n + w n

together with the initial conditions ro = 100, w 0 = 10. Thus
we have to solve a system of two linear recurrence relations
for r n and w n, subject to two initial conditions.