Toward
a stewardship of the Global Commons:
engaging “my neighbor” in the issue of sustainability
By
members of the Critical Issues Committee, Geological Society of America
Part III
Doubling time:
it works for ANY rate of change
A. R. Palmer,
Institute of Cambrian Studies, Boulder, CO
A parable: “When was the pond half full?”I
lived by a large pond with a thriving community of fish, so fishing
was good. One day not too long ago some algae began to
grow in the pond. Their population
was doubling every minute. Yesterday
morning I went fishing and everything was fine. Yesterday noon when I looked out at the pond, it was suddenly filled
with green algal scum and the fish were dying from lack of oxygen.
Why didn’t I see the disaster coming and do something?
When was the pond half full [11:59]?
Onequarter full [11:58]? One eighth full [11:57]? Suppose, instead of my pond, we were considering an island, or a
continent, or Spaceship Earth?
At the heart of the concept of doubling
(or halving) is the exponential function familiar to many from mathematics,
science and engineering. Geologists are perhaps most familiar with this
in its backwardrunning version, i.e. the description of the rates of
decay of radioactive isotopes. Most
of us, however, learned about exponential growth as compound interest
in the context of a personal savings account.
If we put our money in a bank and let the interest accumulate,
our annual income grows as the capital increases. Even though the interest
rate remains constant  our capital grows at an exponential rate. Recognition of doubling (or halving) time for
ANY rate of change was not always emphasized.
Because we are discussing a “doubling”
time, i.e. the time for a quantity in question to double (or halve),
the exponential function has a userfriendly aspect that is very helpful.
It turns out that any finite rate of change expressed as a percent
(e.g. 5% per year) can be converted, to a good approximation, to a doubling
time simply by dividing it into 70.
For example, if the rate is 2%, you might expect to have twice
the number, i.e. 100% more stuff, in 100/2=50 years, but because of
the compounding effect the correct answer is 70/2 = 35 years.
The proof of this statement makes a good exercise for a math
class at the appropriate grade level.
Rates of loss (for instance, depletion of resources) have a halving
time that is calculated the same way.
For anything diminishing at 2% per year there will only be half
as much in 35 years.
This simple way
to calculate doubling time (or halving time) should be an essential
part of everyone’s education. When
the mayor is proud because the city has a healthy growth rate of 3%
per year, that means the city will double in about 23 years if that
rate continues, and double again in another 23 years, and double yet
again in another 23 years, thus octupling from its original size in
69 years. One might ask if a
city with 8 times the present population is viable. Garbage also has a rate of growth, as does
traffic, pollution, schools and housing.
A city is not a closed container, thus
its limits to growth will be determined by cultural factors, but Earth
IS a closed container for all practical purposes (more on this in
Part IV
). Growth in a closed container will ultimately fill it up. Thus, we should look carefully at anything
in our culture with a growth rate, calculate its doubling (quadrupling
and octupling) time and make a judgment about whether we think this
is healthy for our future. We
should do similar calculations and make similar judgments about those
aspects of our culture where, as a result of our consumption habits,
a resource is diminishing at a measurable rate.
Thus, calculations of doubling (or halving) time are critical
components of the issue of sustainability – by which we mean the indefinite
continuation of the entire human enterprise within some steadystate
limits imposed by space and resource availability (see
Part X
in this
series).
Global population growth rates may be
diminishing, but they are still positive. Thus population is still growing and it has
a doubling time. Currently the
rate is about 1% per year (thus doubling in 70 years).
Consumption of resources (Part IV)
increases with population size, even if individual rates of consumption
do not increase.
If we could slow
the present population growth rate to 0.1% per year, it would still
be double its present size of about six billion in 700 years, quadruple
in 1,400 years, and octuple in 2,100 years – equivalent only to the
time represented by the Christian era.
Fortyeight billion people may make things a bit crowded.
Such a population, with its attendant FOOTPRINTS (the areas of
productive land necessary to support each one of us, see the upcoming
June essay), may not be sustainable.
It is not even clear that we can handle one more doubling with
a reasonable quality of life for all.
Such doublingtime
scenarios should make us wonder if there is such a thing as the politically
popular “smart growth” – perhaps it’s a euphemism for “predictable and
voluntary disaster”. Every time
you see a headline or magazine article mentioning rates of change (either
increase or decrease), do the quick mental math to calculate the doubling
(or halving) time. It is a very
revealing exercise.
DEMONSTRATION 1 
Challenge the computer skills of the students by having them
write a program that can check the statement that any rate of growth
or loss, divided into 70 will give you its doubling or halving time.
DEMONSTRATION 2 
Have the class check the newspapers and news magazines for mentions
of rates of change (these could include population size, waste generation,
growth of GNP, inflation, depletion of forested land, or demands on
natural waters) and then do the calculations of doubling or halving
time. Discuss the implications of the calculations.
