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.