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 £© Section
 ¢¡£¡¥¤§¦¨¤§©  £
 £© Nonlinear Difference Equations
In Section we discussed the difference equation
xn+1 = αxn, ()
n = 0, 1, 2, . . ., as a model for either growth or decay and we saw that its solution is given
by
n
xn = α x0,
n = 0, 1, 2, . . .. Now
0, 0 < α< 1,
lim αn = ( 1, α= 1, ()
→∞
n ∞, α> 1,
from which it follows that if {xn} is a solution of () with x0 > 0, then
0, 0 < α< 1,
lim x = x lim αn = ( x , α= 1, ()
→∞ n 0 →∞ 0
n n ∞, α> 1.
These limiting values are consistent with our radioactive decay example since, in that case,
0 < α< 1 and we would expect the amount of a radioactive element to decline toward
0 over time. The case 0 < α< 1 also may make sense for a population model if the
population is declining and heading toward extinction. However, the unbounded growth
indefinitely into the future implied by the case α> 1 is very unlikely for a population
model: eventually ecological or even sociological e to the forefront, such as
when the population begins to overreach the resources available to it, and the rate of
growth of the population changes. Even for bacteria growing in a Petri dish, diminishing
food and space eventually cause a change in the rate of growth. Hence the equation
xn+1 = αxn, ()
for n = 0, 1, 2, . . . and α