## Tuesday, 14 August 2012

### Population and Exponential growth ... (part 2)

Exponential Growth produces some surprising results.
The next example has been discussed in different forms - I prefer my version.

A clump of alien life (A.L) falls on Earth in the middle of an abandoned wall garden somewhere in Italy.
It can grow by using energy from the Sun and gases from the atmosphere - doubling in size every week.  It is bright orange in colour so easy to see, and spreads in a single layer to maximise energy capture. And,
It can be killed by spraying coke.

Let us assume that by the end of 40th weeks, A.L had covered 1/16th  or 6.25% of the surface area in the walled garden
By the end of 41st week, the area covered has increased to 1/8 - 12.5%
By the end of 42nd week, the area covered has increased to 1/4 - 25%
By the end of 43rd week, the area covered has increased to 1/2 - 50%
By the end of 44th week, A.L fills the whole area in the walled garden!

Notice that it took A.L 40 weeks to occupy the first 6.25% of the garden (and during the first 40 weeks of observation, the garden would have looked relatively free from alien life) but in the next 4 weeks A.L occupied the remaining 93.75% leaving no free space.

In 4 weeks, the unoccupied area reduced from 93.75% to 0%
In the last week, A.L added as much area as it had done since the start.

Getting worried by the spread of alien life, coke was sprayed killing 99% of A.L
1% of A.L, that survived, grew as before doubling every week and in just over 6 weeks it had covered the whole area again.

Cloudy weather would prevent Sun reaching A.L and reduce its growth rate to say half - A.L doubling every two weeks. Then A.L will take 88 weeks to cover the garden.  The growth rate could vary from one week to the next and an average rate will prevail.  Different parts of the garden could grow at different rates but again an average rate will determine how long it takes A.L to cover the garden.

Extinctions, variable rates, geographically different rates etc. only affect the time it takes to cover the garden - end result is the same.
What matters is the number of doubling periods.

Remember 1 grows to 1000 (actually 1024) after 10 doubling periods
1 grows to 1,000,000,000 after 30 doubling periods
1 grows to 1,000,000,000,000,000 after 50 doubling periods; that is 1000 trillion.
Rule of thumb is that numbers increase by 1000 after 10 doubling periods.