EXPONENTIAL POPULATION GROWTH IN
TRIBBLES:
Background:
In an episode of the
original Star Trek series (The Trouble With Tribbles; 1967), a furry little
animal called a Tribble is brought on board the U.S.S. Enterprise. Three days
later, the ship is overrun with these prolific critters, and Captain Kirk is
literally up to his ears in Tribbles. Kirk claims that there must be thousands
-- or hundreds of thousands -- of Tribbles on board, at which point Mr. Spock
calculates aloud that there are actually 1,771,561 Tribbles .
The Question:
Was Spock right?
(or, more realistically, were the writers of the script accurately calculating
population growth?)
The
Conditions:
Spock provided the following parameters for calculating
Tribble growth:
1. Initial population = 1
2. 3 days on
Enterprise
3. No deaths while on board
4. Unlimited food (grain)
5. 10
offspring per breeding event
6. 12 hours between generations
7. All
individuals can have babies (they are born pregnant)
The
Calculations:
In a very simple model, population growth rate (G) is
calculated as the product of reproductive rate (r) and the number of breeding
individuals in the population (N), or:
G = r * N
So, given
the parameters provided by Spock,
Time (h)
|
r
|
N
|
G
|
Total Population Size
|
12
|
10
|
1
|
10
|
(1+10 = ) 11
|
24
|
10
|
11
|
110
|
(110+11 = ) 121
|
36
|
10
|
121
|
1210
|
1331
|
48
|
10
|
1331
|
13310
|
14,641
|
60
|
10
|
14641
|
146410
|
161,051
|
72 [3 days]
|
10
|
161051
|
1610510
|
1,771,561
|
Conclusions:
We
should have never doubted the logic of a Vulcan. Spock was right!
(And, for the record, while Tribbles did indeed prosper, they did not live
long...but you will have to watch the episode to find out why!)
Why
should we care about the number of Tribbles on the Enterprise? For one thing,
this is an example of exponential growth:
This type of population growth, indicating a population that
has few or no checks on its growth, is also characteristic of human population
growth. There are many well-established problems associated with human
overpopulation, including resource depletion, increased levels of pollution, and
increased incidence of disease outbreaks. Will humans breed themselves
to self-destruction? This is exactly what happened in an introduced
reindeer population on St. Matthew
Island, off the coast of Alaska. After an initial population explosion, the
reindeer population crashed as food resources were depleted (an example of
density-dependent factors).
Text and figures copyright 2003 by
Greg Pryor.