Exponential Growth
Galen Grimes (gag5@psu.edu)
7 May 2000
From "The Trouble With Tribbles"
Capt. Kirks opens a grain storage compartment aboard space station K-7only to be pummeled with a virtual tidal wave of Tribbles cascading out of the storage compartment. As he slowly emerges out of the near avalanche of furry creatures he hears Mr. Spock utter his worst fears.
Mr. Spock: | They appear to be gorged! |
. . . | |
Nils Barris: | There must be thousands of them. |
Capt. Kirk: | ...hundreds of thousands. |
Spock: |
One million, seven hundred seventy-one thousand, five hundred sixty-one. That's assuming one Tribble, multiplying with an average litter of ten, producing a new generation every 12 hours, over a period of three days. |
Kirk: | That's assuming they got here three days ago. |
Spock: |
And allowing for the amount of grain consumed and the volume of the storage compartment. |
Considering the extent to which Star Trek, The Original Series has been shown in syndication over the last 30, it is highly likely that most of us have been amused at least once by this episode, unaware that Mr. Spock was presenting us with a brief lesson in exponential growth as he calculated the likely number of Tribbles occupying the storage compartment.
The formula Mr. Spock used for calculating exponential growth is:
M = M_{o} e^{kt}
Where:
M is the population after the period of time
expressed in "t".
M_{o} is the initial number of the
population
k is the growth constant (it is a positive
number if you are calculating exponential growth; a negative number if you are
calculating exponential decay)
e is the base of natural logarithms
(approximately 2.71828).
Here is a fairly simple way to understand the Star Trek 'Tribbles' example of exponential growth by producing a simple chart.
Mr. Spock states that one Tribble produces an average litter of ten every 12 hours. So one Tribble enters the storage area and starts eating. Twelve hours later that one Tribble produces its first litter of 10. So at the end of 12 hours there are eleven Tribbles in the storage compartment -- the original Tribble and its litter of 10 baby Tribbles. Now these eleven Tribbles start eating (or more precisely, the first one continues to eat and the 10 babies begin eating) and 12 hours later each of these eleven Tribbles has a litter, averaging 10 baby Tribbles each. So at the end of 24 hours there are now 121 Tribbles in the storage compartment. Every 12 hours you multiply the numbers of Tribbles that were in the storage compartment 12 hours before by 10 and add to that, the previous number of Tribbles in the storage compartment, to get your new total. Your chart for a three day period should look something like this:
Hours | Tribbles | |
0 | 1 | |
12 | (1 * 10) + 1 | 11 |
24 | (11 * 10) + 11 | 121 |
36 | (121 * 10) + 121 | 1,331 |
48 | (1,331 * 10) + 1,331 | 14,641 |
60 | (14,641 * 10) + 14,641 | 161,051 |
72 | (161,051 * 10) + 161,051 | 1,771,561 |
If you need more information on exponential growth check your text or research the subject on the Internet.
Homework:
1. How fast would the Tribble population inside the storage compartment have reached 1,771,561 (or approximately this number) if two Tribbles had initially crawled into the storage compartment instead of just one? If there had initially been two Tribbles, how many Tribbles would there be at the end of three days?
2. How many Tribbles would have been in the storage compartment if they had not been detected for four days (assuming enough space and grain in the storage compartment)?
3. What if Tribbles produced only eight babies in an average liter instead of ten. How many Tribbles would Kirk and Spock have found after 3 days?