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Rate of change

Filed in Notes on September 2nd 05 .

  • The BBC reports that the rate of deforestation in the Amazon has halved
  • Last year the rainforest lost 9,000 Km2, compared with roughly 18,000 Km2 in previous years

Halving the rate of change does not stop the forest shrinking, but the shrinking will slow down. Many environmental organisations point out that these figures can’t show a permanent trend yet. The Brazillian government hopes that the fall in destruction of the rainforest is in response to a tightening of the regulations.

Suppose we say the annual rate continues to fall by a factor of two each year. Starting at 18000, we get a series like 9000 and 4500, 2250, 1125, 562.5, 281.25 and so on. Each successive year is adding in a smaller contribution to the total loss. After 10 years, you are looking at a loss of 18 Km2 or roughly one thousandth of the loss in 2003 (210 = 1024). This series is called a geometric progression that happens to have a fractional ‘growth rate’ – like compound interest in reverse.

Plotting the series against year in Excel, you get something like

Geometric progression with growth factor 0.5

If we do a running subtotal of the loss assuming that the loss is halved each year – rather like a cumulative frequency curve in statistics, the result looks something like this…

Cumulative loss of Amazon rainforest over 10 years assuming a growth rate of 0.5

As you can see, the total loss will approach around 36 000 Km2 and each subsequent year will add a smaller and smaller amount, so that the total will never quite reach twice the starting figure. The sequence tends to a limit of twice the starting rate.

Challenges

  • can you devise a spreadsheet that will calculate the deforestation for each year and the cumulative total?
  • If the deforestation actually ‘grows’ by a factor of 0.75 instead of 0.5 each year, what will the new limit be? Use your spreadsheet to find out. You may need to change the formulas
  • Can you use the formula from the Wikipedia page for the sum of an infinite geometric progression to work out what the maximum loss to the rainforest will be based on a growth rate of 0.5 per year? hint – write the sequence like 18000 × ( 1 + 0.5 + 0.25 + 0.125…) and identify X in the wikipedia formula…

We have built a simple mathematical model of deforestation of the Amazon rainforest. Alas I suspect the assumption of halving each year is going to be too optimistic. We can hope.

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