Race to Extinction

Tasks

Starting with five dice, how long can you survive?

On average, will the population increase or decrease?

How much will the population increase or decrease?
Maths

In nature, we might want to know whether a particular species will go extinct. These dice show how a population can change over several generations. Extinction occurs when the population shrinks to zero.

Extinction is guaranteed when the population doesn’t grow. With our dice, we expect the population to decrease by one-sixth every time we roll.

This might not happen every time, but eventually the trend toward extinction will win. Starting with five dice, we expect extinction to happen around the fifth roll.

When the population is growing, extinction is not guaranteed, but neither is survival. Mathematicians can use population growth to estimate the probability of extinction.

Sometimes a population can show no growth and no decay but fluctuate around a stable figure. However, even in this case, extinction is guaranteed, as eventually we will expect there to be a few unlucky generations that shrink the population to zero, even if this takes a long time.

The maths of predicting extinctions is called the Galton-Watson process.
History

In the 1800s, there was concern that aristocratic surnames would become extinct.

In 1874, English mathematicians Francis Galton and Henry Watson published a method to calculate the probability that a surname would go extinct. This assumed surnames were simply passed from fathers to sons and didn’t reflect many of the other reasons family names change or die out over time.
People

Francis Galton 1822-1911
Darwin’s theory of evolution sparked Galton’s interest in measuring variations in human populations, such as height and fingerprints. To do this, Galton invented new methods of collecting and analysing data. He also wrote about the best way to cut a round cake

Henry Watson 1827 – 1903
Henry Watson was an English mathematician and ordained priest. In 1873, Francis Galton posed a question asking how likely was it that certain surnames would go extinct. Watson replied with a solution, and together they wrote a paper called “On the probability of the extinction of families”.
Applications

The original application of this process was to determine the probability of a family surname dying out (particularly surnames of the aristocracy). Although this method ignores many of the real-world reason why surnames might change or die out over time.

However this method is model for Y-chromosomes that are passed on from father to son, and may explain why there are so few distinct Y-chromosomes in the current population.

Family trees are a kind of branching process. A similar branching processes is nuclear fission, where the nucleus of atoms split into smaller nuclei.

Maths at Home

Try this game at home. You will need some counters (for example you could use tiddlywinks, or lego, or just pieces of paper) and a dice.

Roll the dice to determine how many “babies” each counter will have. Roll 1 for 1 baby and 2 for 2 babies (and keep the original counter). If you roll 3, 4, 5, 6 then remove counter.

In this game, we expect the population to decrease by one-sixth for every roll. Starting with five counters, we expect the game to last about five rolls. What is your record?

You could also experiment with changing the rules.
What about if you roll 1, 2 or 3 to make 1, 2, or 3 babies (plus the original counter). With 4, 5, 6 removing the counter as before. In this case your population should grow, and extinction becomes very unlikely (about 11% if you start with five counters).

What about if you roll 1 or 3 to make 1 or 3 babies (plus the original counter). Roll 2, 4, 5, 6 to remove the counter. Then the population should fluctuate around the starting value. But eventually we do expect extinction – even if it takes a long time. We expect this to take about 20 rolls, but sometimes it will be much longer!