13. Predicting the world population
time
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
Here we look at the world population data (in billions) over the last half
of the century. The figures come from the U.S. Bureau of the Census:
http://www.census.gov/ftp/pub/ipc/www/world.html
and at the same site they also provide estimates of what the population
might be all the way to 2050. The models they have used to make
these predictions are no doubt complex and must take account of a
large variety of forces which operate differently in different parts of the
world. But back in 2001, I decided to take the previous 50 years of
data and use it to construct a simple model, and then see how the predictions it gave me compared with the wisdom and experience of the
U.S Bureau of the Census.
The data are tabulated and graphed at the right at 5-year intervals
I guess the first thing to ask about any data set is whether it looks like it
wants to be a straight line, and the answer here seems to a clear NO.
Put a ruler along these points and see what you get. The data are reasonably straight, but there’s a definite concave-up character to it.
Well, this is a biological population, so the next thing we ask is about
exponential growth. There are a couple of things we could do to check
that out––plot logz against t and see if we get a straight line, or plot the
5-year multiplier m against z and see if it is constant. [Those are two
different ways of doing the same job––exponential growth has both a
straight-line log plot and a constant multiplier.]
First let’s look at logz. This is plotted against t at the right. Well looks
pretty straight to me, and I have taken the liberty of including the trend
line I got from my spreadsheet (see problem 2).
Pop(×109)
2.556
2.780
3.039
3.345
3.707
4.086
4.454
4.851
5.279
5.688
6.083
z
7
6
5
4
3
2
1
0
1950
t
1960
1970
1980
1990
2000
logz
0.8
0.7
What do you think?
0.6
Well, predicting the future is a rather special enterprise. For this, we
want to be sensitive to slight patterns in the data which might signal a
trend, and there’s a definite one to be seen in the graph––a concave-up
trend over the first 20 years and a concave-down trend in the data over
the last 30 years. And that’s precisely the sort of behaviour we want to
pick up on.
0.5
0.4
1950
t
1960 1970
1980 1990
To go forward into the future, we want to understand that last concavedown trend a bit better. How should we proceed?
I throw the question out to the class, and the first suggestion I get is that
we might fit a parabola to those last 30 years of data. Well that might
work well––who knows? (problem 2(b)) But that’s not what I did. The
point is that for biological populations, exponential growth is the standard and when we don’t have, I like to go right back to the multiplier m
and see just how far away from “constant” it is. And that’s what I did.
13. predicting world population
1
2000
That concave-down trend announces a decreasing percentage growth
rate (recall that a straight line logz graph means a constant percentage
growth rate). What we want to do (if we want to predict) is discover
exactly how it is decreasing. To do that we look at the multipliers:
nextz
m =
z
which give us the ratio of population sizes taken over one time step,
which here is 5 years. These are tabulated at the right and plotted (multiplier against population size) below.
The first graph is my first attempt to plot m. What I got was a set of
points which certainly appeared to lie in a horizontal line. Good––that
fits our observation that m is nearly constant. But to understand the
small variations in m I needed to magnify the vertical scale, and that’s
what’s done in the second graph.
And wow––look what I got––a striking display of the variation in percentage growth rate, increasing at first and then decreasing (just as the
log plot suggested) with a sharp change in behaviour around 1965 when
z is around 3.3 billion.
t
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
z
2.556
2.780
3.039
3.345
3.707
4.086
4.454
4.851
5.279
5.688
6.083
m
1.088
1.093
1.101
1.108
1.102
1.090
1.089
1.088
1.077
1.069
m
2.0
1.5
1.0
0.5
One thing this graphing exercise clearly warns us about is that predictions can be a tricky business. Suppose that in 1965, at the peak of that
m-graph, we had tried to predict the world’s population over the next
15 years based on the past 15 years of data. We’d be way off. Something major clearly happened in 1965, some population growth bubble
clearly burst. Anyone who bases predictions on past trends has to ask
whether the circumstances of the next few years is going to be similar
to that of the recent past.
In particular, this question of understanding the world’s population
growth is enormously complicated, with a considerable spread of economic and cultural factors from region to region. But here, we will
treat this as a simple mathematical problem of predicting future growth
from past data, and just for fun we’ll see how our predictions compare
with more sophisticated efforts.
We’ll use the decreasing m data of the past 30 years, assuming that,
whatever it was that happened in the mid 60’s to change the face of
population growth, its effects are still holding sway and will continue to
do so. The nice thing about the m-z data of this 30-year period is that it
lies in a pretty good decreasing straight line data, and that’s easy to
describe mathematically.
z
0.0
0
1
2
3
4
5
6
7
m
1.12
1.11
1.10
1.09
1.08
1.07
z
1.06
0
1
2
3
4
5
6
7
Different students have different
ideas as to which points to ignore for
the purposes of prediction. So a variety of models emerges, and some
good discussion. See problem 3.
m
So that’s our strategy. Fit a straight line to the m-data since 1965, and
then project it 25 years into the future.
A trend line fit to the last 7 points gives us:
m = 1.1581 – 0.01534z.
To predict the future we take this as the formula which tells us how the
5-year multiplier m depends on population size z.
1.12
1.11
1.10
1.09
1.08
1.07
z
1.06
0
13. predicting world population
1
2
3
4
5
2
6
7
If we put this formula for m into our growth equation, we get
nextz
= m = 1.1581 – 0.1534z
z
nextz = mz = (1.1581 – 0.1534z)z.
It’s not possible to solve this equation analytically (that is, to
get an expression for z at any time), but the equation can easily
be iterated numerically over successive 5-year periods to move
into the future.
In the table at the right, the actual z-data is tabulated up to the
year 2000, and the data below that, from 2005 to 2025 (in the
box) is calculated from the above “nextz” formula. The righthand column allows us to compare these prediction with those
of the U.S. Bureau of the Census. We find that our values are a
tiny bit higher at the beginning, and a small amount lower at
the end. It’s of course a very complicated business, but it will
be fun seeing who’s got the better predictions. [Put a copy of
this table away in your drawer of treasured things; take it out
every 5 years and gloat.]
t
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
2015
2020
2025
z
3.345
3.707
4.086
4.454
4.851
5.279
5.688
6.083
6.476
6.856
7.219
7.560
7.877
US bureau
6.468
6.849
7.227
7.585
7.923
Problems
1. What is the asymptotic world population size predicted by our
model?
logz
0.8
2.(a) The trendline for logz against t for the period 1950-2000, plotted
at the right, has equation
0.7
logz = –14.65366 + 0.007725t
0.6
This model gives us an exponential equation for z in terms of t. What
prediction would it give us for the population in 2025?
0.5
(b) Suppose instead we fitted a parabola to the data of the last 35
years (1965 to 2000 inclusive). What 2005 prediction would that give
us?
0.4
1950
t
1960 1970
1980 1990
2000
m
1.12
3. In our analysis of the world’s population growth, we might well
decide that for the purposes of prediction, the trend of the last 10 years
should have far more effect than the trend of the last 30 years, particularly since the last three data points in the m-z graph do seem to form
a line of their own. Obtain a trendline through the last three points,
and use it to predict the world’s population at 5-year intervals to 2025.
Compare with the above predictions, ours and the census bureau.
1.11
1.10
1.09
1.08
1.07
z
1.06
0
13. predicting world population
1
2
3
4
5
3
6
7
4. At the right is tabulated the population of Canada in millions at 5year intervals from 1951 to 1996. Your task is to construct a model
for this data which might permit you to predict the future population.
Use technology to assist you as appropriate.
(a) Begin with a plot of the population against time over the entire
period of the data.
(b) Tabulate the 5-year multiplier m and plot it against z over the period of the data.
(c) Either with a spreadsheet or by ruler and eye, fit this data to a
straight line. Find the equation of the line.
(d) Starting with the population size in 1996, use this line to predict the
population size in 2001, 2006 and 2011.
5. In 1910 a couple of scientists tabulated the population of the U.S.A
from 1790 to 1910, and fitted a model similar to ours to estimate the
growth over the coming century. Use a spread sheet to
(a) Plot z against t.
(b) Calculate the 10-year multipliers m and plot m against z.
(c) Calculate and plot the trendline for m against z.
[Answer: The line is m = 1.3641-0.00221z.]
(d) Calculate the asymptotic population size obtained using this m-z
equation––that is, the population size that would be attained after an
infinite time.
(e) Following the method of this section, use the spread sheet to predict the population size in 2000. [The actual figure was about 272 million. Be prepared for a poor prediction.]
13. predicting world population
Pop of Canada (×106)
t
z
1951
13.648
1956
16.081
1961
18.238
1966
20.015
1971
21.568
1976
23.518
1981
24.900
1986
26.204
1991
28.111
1996
29.959
Pop of USA (×106)
t
z
1790
1800
1810
1820
1830
1840
1850
1860
1870
1880
1890
1900
1910
3.929
5.308
7.240
9.638
12.866
17.069
23.192
31.443
38.558
50.156
62.948
75.995
91.972
4