A New Doomsday Clock

Probability Level 2

If rabbits suddenly became immortal, invincible, and if they matured and reproduced independently of their environment, about how long would it take until rabbits literally covered the earth?

Helpful (but also somewhat disturbing) facts:
- The earth has about $5 \times 10^{14}$ square meters of surface area.
- Approximately 20 rabbits can fit in one square meter.

Rabbit litters range from 1-14 babies. 6 is a conservative average.
Rabbits can begin reproducing when they're about 6 months old.
Rabbit gestation lasts about one month.
Rabbits are "induced ovulators" and can be impregnated again within minutes of giving birth.
There are probably around $10^{9}$ rabbits currently alive.

about 5 years about 5000 years about 50 years about 500 years

1 solution

Zandra Vinegar Staff
Aug 16, 2015

Here is one approximation method, but many other methods are just as valid!

$5 \times 10^{14}$ square meters, and 20 rabbits/square meter means we need $100 \times 10^{14} = 10^{16}$ rabbits.

There are currently about $10^9$ or 1 billion rabbits. So let’s make that the unit: billions of rabbits.

Similarly, let’s make the time unit 6 months and approximate that rabbits are old enough to mate if they were born not in the past 6-month block, but any time before that.

Below, the parts of each sum highlighted in yellow are those rabbits mature enough to breed. The number highlighted in blue is the number of new baby rabbits.

Every half-year we multiply the yellow highlighted sum by 18 to get the number of new babies (since 2 rabbits breed and produce about 6 x 6 = 36 offspring every 6 months.)

Since the number of mature rabbits is simply the total population 6 months ago, this progression can be setup as a recurrence relation:
$R_n = R_{n-1} + 18(R_{n-2})$ ; $R_0 = 1$ ; $R_1 = 19$

The important observation is the exponential nature of this growth. Here’s a plot of the first 15 years:

And here are the values of the relation for the first 10 6-month periods:

the number of rabbits we need / our rabbit unit = $\frac{10^{16}}{10^9} = 10^7$

So it will take around 5 years for the rabbit population to explode to $10^{16}$ rabbits, enough to completely cover the world.

But man that's not right the rabbits that born in first month only who will breed after 6 months other won't breed other who born in second ,third,fourth,fifth,sixth month won't breed in sixth month You should have mentioned that in assumptions if you want problem go so

Mahmod Abass - 5 years, 9 months ago

A New Doomsday Clock

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