Chapter 2: Discrete-Time Models, Sequences, and
Difference Equations
Section 2.1: Exponential Growth and Decay
Suppose that a bacterial colony starts with 100 bacteria and the bacteria divide every 20
minutes. How will the population size change over time?
Time (Minutes)
Population
0
20
40
60
80
100
120
Assume that one unit of time is 20 minutes. Then the table can be written as
Time (20 Minutes)
Population
Use the table to find an expression for the population size, Nt , at time t = 0, 1, 2, . . . and
find an equation for Nt+1 in terms of Nt .
7000
6000
Population Size
5000
4000
3000
2000
1000
0
0
1
2
3
Time
1
4
5
6
Definition: Suppose that a bacterial colony starts with N0 bacteria and each bacterium
produces R bacteria per unit time, then the population size Nt satisfies the recursion or
recursive equation
Nt+1 = RNt .
The solution of this recursive equation is
Nt = N0 · Rt .
If R > 1, then R is called the growth constant and if 0 < R < 1, then R is called the
decay constant.
Example: Suppose that a population begins with 62 individuals and quadruples every day.
(a) Find the recursion for this population.
(b) Find the solution of this recursion.
(c) How long will it take for the population to reach 15,872 individuals?
2
Example: Suppose that a bacterial colony starts with 4 bacteria and triples every 30 minutes.
(a) Find a formula for Nt if the unit of time is 30 minutes.
(b) Find a formula for Nt if the unit of time is one hour.
(c) Find a formula for Nt if the unit of time is 40 minutes.
Example: Suppose that Nt = 50 · 2t , t = 0, 1, 2, . . ., and one unit of time corresponds to 2
hours. How long does it take the population to triple in size?
3
Example: A strain of bacteria doubles every 50 minutes. If initially there are 10 bacteria,
how long will it take until there are 640 bacteria?
4