^{1}
and Dr. Manuel Alfonseca^{2}
Mathematical models of demography
and population trends have been with us since two and a half centuries ago. As
the subject matter is extremely controversial, experts frequently disagree. In
this paper we have selected six of the main milestones in this field, which we
have completed with a short description of our own work in the area.
In this paper we have
selected six of the main milestones in the field of population dynamics, namely:
Edmond Halley, Leonhard Euler, Thomas Robert Malthus, Pierre François Verhulst,
the Club of Rome and Robert McCredie May, following mainly a recent historical
book (Bakaër 2011), which we have completed with a short description of our own
work in the area.
Around 1690, Caspar Neumann
collected data in Breslau (Wroclaw) about the number of births and deaths in
his city and the age of people at death. Neumann sent his data to Henry Justel
(secretary of the Royal Society). Halley got hold of the data, analyzed them, noticed
that the population, the number of births and the number of deaths remained
constant in those five years (1687-1691)
and built a life table showing the population in the city as a function of age.
His formula was simple: given that P
where D
The famous Swiss
mathematician Euler was also one of the pioneers of mathematical demography.
Rather than tackling a stable population, as Halley had done, Euler considered
the situation where a population grows at a constant rate x. As the situation
is exactly the same as that in compound interest, the formula he reached was:
where P In a later work ( ·
A population in
exponential growth:
·
The birth rate
is constant: m = B ·
The death rate
is constant, ergo . From here we
get: . Also, . ·
Let P ·
The maximum age
reached by a member of the population is 100 years. From these assumptions,
Euler derived his demographic equation, namely:
which is the same as
Halley’s equation when r=1. Although Euler was the first
to prove that in a population in exponential growth the shape of the age
pyramid remains constant, this discovery was not taken into account until it
was rediscovered in the twentieth century. Of course, assuming an exponential
growth for extended periods would be an unrealistic assumption.
In 1798, Malthus published
(anonymously) a book entitled
In other words, population
grows exponentially (or in a geometric progression) as Euler had assumed, but
food production grows in an arithmetic progression, i.e. linearly. Therefore,
the geometric increase of the population must be stopped by the effects of
famine. Malthus made two mistakes: 1.
Extending the
exponential growth of population indefinitely. As Gordon Moore expressed it in
a meeting of the IBM Academy in 2003: 2.
Assuming that
food production cannot grow exponentially. However, the agricultural revolution
of the nineteenth and twentieth century produced exactly that result. Charles Darwin and Alfred
Russell Wallace applied (and cited) the ideas of Malthus to develop the theory
of evolution by means of natural selection.
In 1838, as a reaction
against Malthus theory, the Belgian Verhulst proposed (in his
This is the
Verhulst tested the logistic
curve against real data from France, Belgium and the U.S.A. and got a good fit. However, his predictions
for longer times (around 50 years) did not come true. The problem in his case
is that the two parameters in his equation (the growth rate
In 1972, the Club of Rome
published its first report, the best-selling book In 1974, the second report
of the Club of Rome ( The reports of the Club of
Rome have been checked with real data in the forty years passed since their
publication, the reviewers sorting themselves among those who assert that the
predictions of the Club are being confirmed, and those who claim that they have
been proved incorrect.
In 1974, the Australian born
Robert May published in
This equation works for
0<a≤3. Figure 2 shows the solutions for the equation for several
values of parameter a, and for K=1000.
It can be seen (and proved
mathematically) that for a<2, the equation converges to a single value; for
2≤a<2.45, it converges to a period 2 cycle; for 2.45≤a<2.545,
it converges to a period 4 cycle; and for 2.57≤a≤3, the equation
becomes chaotic. Populations, therefore, can be subject to quite unpredictable
dynamics, which makes it possible that the optimistic or pessimistic
predictions by different authors may not be according to reality. Equation (5) can be
simplified to (6) by means of the following variable changes:
Using UN population data for
the interval 1900-2010, we have simulated its time evolution by means of rate
equations inspired on condensed matter physics (Gonzalo et al 2012). The final
differential equation to be solved was:
whose solution is:
where t Equation
(8) describes a step up in population due to an increase in fertility rate, a
decrease in death rate (life expectancy increase) or a combination of both,
resulting in a net growth rate. This equation fits the UN data satisfactorily
and shows clearly that the population increase in 1950–2010 should be
attributed more to the transient decrease in death rate level (related to the
increase in life expectancy) than to a non-existent increase in birth rate,
which was decreasing consistently already even before the 1950s, even before chemical
contraceptives and legalized abortion begun to play any role. An equivalent approximate form of equation (8) is:
Using this equation we
analyzed the three future scenarios estimated from the UN data and came to the
following conclusions (Gonzalo et al 2015): Today the world is not
overpopulated and is unlikely to be so in the foreseeable future. Extrapolating
present trends shows that total world population may reach a maximum of 7.74
billion by 2050, and by the end of the current century it may have decreased.
Assuming that the smooth natural decrease in birth rate by the mid-seventies
had continued all the way down, the estimated population maximum would have
reached 8.4 billion by 2065, rather than 7.73 billion by 2050; therefore the
policies aimed at lowering the birth rate supported by the UN should be
reconsidered. The population rise during this century (1950-2050) is due to the
high and sustained decrease in death rate (and the corresponding high increase
in life expectation) rather than an increase in fertility that actually never
happened.
1.
Nicolas
Bacaër, 2011. 2.
Meadows, D. H.; Meadows, D. L.; Randers, J.; Behrens
III, W. W., 1972. 3.
Mesarovic, M.; Pestel, E., 1974. 4. Gonzalo, J.A., Muñoz, F.F., Santos,
D.J., 2012. 5. Muñoz, F.F., Gonzalo, J.A., 2013.
6.
Gonzalo, J.A., Muñoz, F.F., 2014. - Gonzalo, J.A., Alfonseca, M., 2015.
*Quantitative estimates of the future world population decline*. Journal of Global Issues and Solutions, The Bimonthly Journal of the BWW Society, 6 pp.
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