bstract
Decrease or growth of population comes from the interplay of
death and birth (and locally, migration). We revive the logistic
model, which was tested and found wanting in early-20th-century
studies of aggregate human populations, and apply it instead to
life expectancy (death) and fertility (birth), the key factors
totaling population. For death, once an individual has legally
entered society, the logistic portrays the situation crisply.
Human life expectancy is reaching the culmination of a
two-hundred year-process that forestalls death until about 80
for men and the mid-80's for women. No breakthroughs in
longevity are in sight unless genetic engineering comes to help.
For birth, the logistic covers quantitatively its actual
morphology. However, because we have not been able to model this
essential parameter in a predictive way over long periods, we
cannot say whether the future of human population is runaway
growth or slow implosion. Thus, we revisit the logistic analysis
of aggregate human numbers. From a niche point of view,
resources are the limits to numbers, and access to resources
depends on technologies. The logistic makes clear that for homo
faber, the limits to numbers keep shifting. These moving edges
may most confound forecasting the long-run size of humanity.
Introduction: Runaway Growth or Slow Implosion?
As Charles Darwin said, in the struggle for life number gives
the best insurance to win [1]. The Bible (Genesis 22:17) records
that when God wanted to boost the elected ones, he promised that
they would become more numerous than the grains of sand on the
sea shore (i.e., >>1012).
In fact, world population since the mid 20th century has grown
by about 2% per year, a rate that doubles the population in
roughly 35 years. Actual data fitted over five centuries with
reasonable equations show that the secular rate of growth kept
increasing until around 1970, leading, at least from a
mathematical point of view, to an infinite population in a
finite time (Figure 1). (See [2] for a numerical history and [3]
for an infinite prediction.) Such growth worries
environmentalists and many others and leads to a first question:
Where is the world population moving?
A second worry occupies rich, mostly white populations, in
particular Europeans. The anxiety of their politicians and
demographers (and the Pope) roots in the fact that on average a
European woman now bears only about 1.4 children along her
fertile span. To preserve a population, the rate should be
around 2.1. The gap means that European populations are slowly
imploding. An "oldies" boom preserves numbers for a while but
empties society of the vis vitalis, the vital force carried by
youth. Thus, a second question: Where is the population of the
advanced industrialized nations moving?1
Two mechanisms control the size of a population: life expectancy
at various ages and the fertility rate. (Migration also affects
the size of local populations, but we will not consider it
here.) For both death and birth, demographers still are
searching for working models, that is, numerical models
corresponding to theory whose parameters are set by independent
data and whose results pass the test of conforming to still
other data. (See [5] for a classic introduction to demography;
[6] for a severe critique of progress a decade later; and [7, 8]
for the current state of affairs.) The absence of working models
means that the demographers cannot forecast the evolution of
either life expectancy or fertility. Lee and Tuljapurkar, for
example, state: "Doubtless the most important source of error in
population forecasts is uncertainty about future [fertility]
rates, because these rates are changing over time in ways that
so far have been difficult to predict or even to explain after
the fact"[9]. Thus, population predictions are based on
numerical assumptions, or guesses, or "scenarios."2
So far, demographic predictions tend to diverge from the real
numbers after about 20 years. To give some examples, in 1951 the
Population Division of the United Nations (UN) estimated that
the world population in 1980 [10] would be between 2.976 and
3.636 X 109. The use of four significant figures for a scenario
is certainly worth a note. The number in 1980 was actually 4.45
x 109. In 1986 the UN predicted 6 x 109 for the year 2000. The
1995 world population is 5.7 x 109. The UN mark will probably be
reached in 1997. The UN 1992 prediction for 2150 is a la carte
[11]. One can choose seven different world population levels
placed between 4.299 x 109 and 694.213 x 109. However, the
preference is for 11.543 x 109. Unabated is the love for
significant figures.
Predictions are always made with ifs. Because everybody seems
scared by increasing human populations, fertility values are
tamed in such a way-as to produce a maximum psychologically
acceptable number of humans, usually between 1010 and 2 x 1010
by 2100. The reckoning date is well beyond the life expectancy
of present politicians and demographers. These soothing
predictions are obviously based on the if that current total
fertility rates will fall everywhere to the conservation value
of 2.1 (see Figure 6, later). As in weather forecasting,
building the analysis from the bottom up becomes more and more
complicated when one details to regions (and social status), and
forecast results are not better.
Suffice it to say that the problem of the future size of
humanity is unsolved. Whether the answer is unknown or
unknowable, the problem partly lies in the methods, and no sign
of breakthrough has appeared in the literature. Our response is
to go back to the numbers and have a fresh look. We seek
quantitative regularities to see whether it is possible to
forecast with some internal logic where and when the growth will
stop.
For both mortality and fertility all the mechanisms involved are
regulatory and require social and cultural intervention. Because
changes in culture and social behavior can be described by
diffusive processes, basically captured in logistic equations
(or their derivatives or sums), our fresh look will refuter the
numbers along these lines. The fact that we can model with good
precision over long periods several parameters usually looked at
in charts in a qualitative way will show the strength of our
method.
The use of the logistic model is widely established in many
fields of modeling and forecasting [12, 13]. It has a
controversial history in population ecology, a point to which we
return near the end of this article. One of a family of
density-dependent functions, the logistic law of growth assumes
simply that systems grow exponentially under the constraints of
an upper limit producing a typical S-shaped curve [14]. The
three parameters of the logistic curve, which recur in our
figures, are characteristic duration Dt, the limit K, and the
midpoint tm. The characteristic duration Dtis the time needed
for the curve to grow from 10% to 90% of the limit K. Appendix 1
offers a mathematical description of the logistic model.
There is obvious need for demographic statistics of reasonable
quality and consistent definitions. See [2, 11, 15-28] for data
sources. We also offer precise definitions of terms in the
Glossary, Appendix 2. Shortcomings arise in several ways. For
example, although local demographic registrations are of ancient
origin and reliable, their patching up into national statistics
may not be. African states may have made written records only
recently. Changing cultural values affect what is recorded.
Years for which detailed survey data are available are few and
do not include all countries. In Appendix 3 we give some
quantitative examples of the uncertainty associated, even in the
present day, with fertility rates. Nevertheless, we believe that
the long-run and comparative nature of our approach makes the
analyses robust.
Our plan is to look first from the bottom up, using the logistic
to model life expectancy and then, in much greater detail,
fertility. When cases are intractable analytically, as modeling
human population has been, the alternative is to look from the
top down with phenomenological insights. The master case is that
of thermodynamics, where a couple of well-centered axioms
permitted almost two centuries ago the construction of a branch
of physics unchallenged to date. Its analytical counterpart,
statistical mechanics, took a full century to develop. In the
case of demography, analysis of the aggregate behavior or niche
started for animal populations in the mid-1800s. Before
concluding, we briefly reapply the logistic model to the
analysis of aggregate human populations with the help of some
extra hindsight.
Modeling Life Expectancy
Life expectancy is an important parameter in defining the size
of a population because for a given birth rate the number of
people is proportional to it.
Life expectancy in the developed world started changing in about
1800, improving slowly. The maximum gains have been in reducing
infant mortality, but octogenarians also gained a few years.
Demographers and medical doctors still struggle to define the
future of the process. A simple solution can be found by
assuming that each of us is endowed with longevity by DNA.
Dangers along the way impede reaching the final age. However, by
removing the dangers through nutrition, hygiene, medicine, and
various coatings and protections, finally one can reach an age
corresponding to longevity.
Because the removal of the dangers is a process of social
learning, the equation most apt to describe it is a logistic
[29]. For a time, knowledge and experience enable people to gain
years of life with increasing speed. Then the process slows as
we near the limits of efficacy of our various strategies. In
fact, evolution of life expectancy during the last two centuries
can be precisely mapped using logistics. In Norway the gain in
life expectancy at birth forms a neat logistic taking off at the
1% level in 1810 and eventually adding 39 years to the life of
the new-born Norwegian child (Figure 2). The process is logistic
at each age, with 20-year-olds eventually gaining 20 years,
50-year-olds 11, and 80-year-olds 3 (Figure 3). In fact, one can
also map with a logistic the final gain versus age, as we have
done for the Dutch population (Figure 4).
All such analyses show that in developed countries we are near a
limit [30]. Barring genetically engineered defense against
senescence, life expectancy for women will stay in the mid-80s
and for men about 5 years less.3 Consequently, the effect of
increasing life expectancy on population, which for a while has
masked the decrease in fertility in some rich countries, will
disappear.
For developing countries we have not attempted to analyze
comparable trends because the series of credible population
statistics are not long enough. We would expect similar results.
The basic processes of social development are the same, though
perhaps operating more rapidly than they did for the countries
that industrialized early. In developing countries, increase in
life expectancy will sum up quickly, boosting the size of their
populations on top of the effect of fertility.
In the long term, life expectancy acts as a fixed multiplier on
population and is thus much less important than fertility, which
acts exponentially.
Modeling fertility
Reproduction is at the center of life. As Manfred Eigen showed
in his seminal papers 30 years ago, survival is the axiom from
which the mathematics of life can be deduced [31]. But survival
in an abrasive context means starting again and again, i.e.,
reproduction. Being central to survival, reproduction naturally
also occupies a central place in the thinking and action of
human societies. However, species are normally endowed with
excessive reproductive capacity to take care of critical
transients and occasional opportunities. Where survival rates
are high, as in birds of prey who have few natural enemies,
total fertility tends to be low. A chick every year or two for
each female can suffice. Animals with lots of hungry enemies and
poor defenses, such as snails, tend to astronomical prolificity.
These generic observations may have trickled down into the
concept that in humans the transition to lower mortality will
lead to a fertility transition to replacement levels. Visual
observation of mortality and nasality curves for many nations
shows in fact that both fall starting in the latter part of the
19th century (as shown for Finland in Figure 5). The reported
decrease in mortality typically precedes that of fertility.
The post hoc propter hoc is necessary, but rarely sufficient, to
determine a causal relationship. The two phenomena can descend
from the same cause. To give a whimsical but not impossible
mechanism, eating peanut butter could increase health and
inhibit fertility. The phenomena could also be completely
unrelated. A famous chart shows the decreasing number of storks
flying to Germany during the last 30 years, closely matching the
number of children born, by a constant multiplicative factor.
Certainly decreasing the mortality of infants and the young
before reproductive age reduces the need to produce many
offspring. But the human species is endowed with excess
fertility that had to be, and was, pruned even before the
decrease in mortality that came after 1800, to fix a round
reference date.
A free-wheeling human female can produce a dozen or more
children during her fertile period. Although this number was
fairly frequently reached in agricultural families up to 1900,
most families stopped at much lower levels, say 4 children born.
This essential fact means that fertility was always under
control, helped in case of necessity by infanticide, a practice
widely used up to our days [32]. In Western countries
infanticide is mostly substituted by abortion, which is the same
act at a different time. Because fertility has always been under
control, we must ask then why people stop at one number instead
of another and, whether the choice, probably made without
explicit reasoning, can nevertheless be rationalized.
Some years ago a striking attempt at rationalization was made
for India. It was observed that India had a mortality
transition, but did not seem to have a marked fertility
transition. The mean number of children per family hovers around
four. The rationale was that females in India usually do not
hold stable paid jobs. The male provides cash for the family.
Simple calculation shows that to have at any time at least one
male in earning age, the family must shoot for two sons, that
is, four children in the mean.
In a system where mechanisms external to the family do not
provide old-age benefits, children are the only insurance for
old age. Clearly, and the fact usually is not stressed enough,
mechanisms for social security can be internal or external to
the family but require children in both cases. With external
mechanisms, as in the welfare state, the children in the system
become a "common." As extensively analyzed by socioecologists,
this commons can suffer a tragedy if everyone takes away and
nobody restores the resources. In fact, Western countries
currently do not have enough children for ensuring the pension
system, a point to which we return.
In short, the fertility situation is very confused, much more
than transpires from the short considerations just outlined, and
up to now fertility has escaped all model descriptions [33, 34].
Demographic books continue to be littered with puzzling charts
and lists of numbers [35]. The UN projections of fertility
express hopes for the future but no consistent view of
mechanisms or continuity with the past (Figure 6). However,
encouraged by our success in describing the evolution of life
expectancy during the last two centuries, we believe that our
logistic model deserves a chance.
Concerning the strategy of attack, the guidelines are simple:
Concentrate on age-specific fertility rates (number of children
produced by 1,000 women of a given age cohort) to avoid the
complexities of age structure in a given population and on total
fertility rates (the average number of children per woman per
lifetime).4
Spread the fertility rate analysis over countries of different
cultural background and economic status.
Also look at the male side of the problem. Fertility rates are
usually seen from the side of the female, but sometimes
statistics also are available on birth rates according to age of
the father.
Look at the hierarchical position of children according to birth
order in different contexts in time and culture. Most women in
Europe (and Japan) have only one child. The concerns about
population stability move in opposite directions depending on
the probability of multiple births.
Model the fertility transition per se, that is,
phenomenologically, without theoretical (and emotional)
constraints. Theory should adapt to facts.
Keep an eye on social moods. Because making children is deep, it
will inevitably be moody. (The annual birth peak in late winter,
reflecting increased conception in the spring and early summer,
shows mood [36].)
As a reference case, we fit first the fertility data for
Finland, a country with unusually good long records. Figure 7
reports the results for the time course of the total fertility
(average number of children per female per lifetime). That
course appears well approximated by a standard three-parameter
logistic inserted between an early, rugged high plateau and a
current low one. The year 1926 marks the midpoint of the
logistic transition. Finland moved from a fertility rate of 4.95
in 1890 (99% of the upper limit) to a rate of 1.55 in 1962 (1%
above the lower limit). The characteristic duration is 36 years.
On this secular evolution of total fertility rate, we find
superposed a short pulse of extra fertility, as appears in
Figure 7. Colloquially, it looks like the outcrop of the "baby
boom." This transient we analyze separately, integrating it in
time. It also reduces to a logistic. The pulse centers in 1953,
as seen in Figure 8. With the "baby boom" perturbation, we have
described completely 200 years of total fertility in Finland.
The model seems to work.
Now let us start our digging into the inner logic of fertility
with a relation taken as extremely important as a causative
agent in the present reduction of fertility in Western
countries: fertility versus mortality. Many convoluted
discussions exist on how and why one lags the other in the
"demographic transition" from high to low death and birth rates.
Comparing the evolution of fertility with that of mortality
requires some reflections. Mortality is usually fairly
selective, hitting mostly children and old people. If we take
total mortality, it mixes the two phenomena, which differ in
psychological impact. We do not think reasonable a feedback
process where female fertility is inhibited because
octogenarians overcrowd the area. Certainly fertility in terms
of population is diminished because of the dilution by infertile
people. But, in terms of total fertility rates this dilution
will not appear because the calculations refer to age cohorts of
fertile females only.
In fact, our Finnish case also provides a comment on the
possible mortality-fertility relation. As the "baby boom" fit
appears good, we dare to backcast where the signals (good data)
disappear in the noise of the recession and war years. In Figure
8 we then see that 1% of the phenomenon existed in 1929 (1953
minus 24 years). The situation for the world then was more bust
than boom. It would be curious to explain the pulse as a
compensation for the mortality of the war, if it began a decade
before the war.
A narrower argument in the current literature is that when
females see their children do not die, they make fewer. In 1800
in Europe one child out of four would die in the first years of
life. Taking the reasoning ad litteram for Finland, compensation
would occur in relation to a fertility in terms of survivors of
4.95 x 0.75 = 3.71. Total fertility in Finland in 1993 was 1.55,
only 42% of the 3.71 we might now expect if this explanation
sufficed. Certainly perceived values (e.g., of child survival)
may differ from statistical ones, but other demographic examples
show that people tend to perceive precisely, at least in the
means.
Perhaps we can learn by looking at the secular evolution of
infant mortality across cultures. Norway, analyzed logistically
in Figure 9, shows that the model works again and that the
reduction has a midpoint in 1920 and a characteristic duration
of 115 years. France and Italy (not shown) have the same
midpoint and characteristic durations of 88 and 110 years,
respectively.
Returning to the Finnish case (Figure 10), when we compare the
fitted logistics on fertility end infant mortality, we see
similar midpoints, 1926 versus 1916, but very different root
points (defined as 1% of the process), 1890 versus 1799. This
temporal gap does not disprove a cause-and-effect mechanism, but
it certainly weakens the argument. One must explain a delay of a
century in setting up the process of reducing births.
Part of the explanation could be biased reporting. Clearly,
fertility rate statistics can be falsified. Even in modern
Europe, infanticide at birth was widely practiced, with no
registration of the newborn if the decision had been taken to
kill [37].
In our opinion, anthropologists Marvin Harris and Eric Ross [38]
offer the key to the problem (see also [39] for an economist's
formalization). Looking at reproductive control in historical
perspective, Harris and Ross show that people always had the
tools. In other words, family planning always existed, as the
decision to have or keep a child was taken inside the family. In
the analysis of Harris and Ross, this planning tends to have an
economic arrière pensée: are children a burden or an asset?
Both, naturally. But the burden tends to fall on the female, and
the asset accrues to the family as a whole.
In agricultural societies children are a clear plus. They become
useful already about age 4. They run errands in place of adults,
bringing food and messages to people working in the fields. They
care for little stables, growing, for example, rabbits for the
family. All at very little extra cost for the parents. The
family systematically exploits youngsters until they marry and
later if they remain in the patriarchal house. Old people run
the system. They have, perhaps, the experience and, certainly,
the authority to do so. The elders find in their command both
social position and material support.
The patriarchal family was a welfare organization, perhaps more
efficient and effective than the welfare state. A counterproof
of this point of view is the presence of large families in
proto-industrial English cities. In the words of Harris and Ross
[38, 100], "A more plausible explanation for the early decline
in mortality, which was most dramatic among infants, is that it
was produced by a relaxation of infant mortality controls and by
more careful nurturance in response to a new balance of
child-rearing costs and benefits brought on by the shift to wage
labor and industrial employment."
With the expansion of factories, shops, mines, mills, transport,
and other industrial capitalist enterprises, wage labor
opportunities for children increased. Children became relatively
more valuable, and infanticide, direct and indirect,
yielded--though never entirely--to more positive nurturance.
This insight may resolve a paradox at the heart of demographic
transition theory: that Europe experienced an explosive rise in
population precisely at a time when, demographers have argued,
Europeans were beginning consciously to control their fertility.
What was new were the emergent material conditions, in
particular, the magnitude of the incentive that industrial
capitalism in the late 18th and early 19th centuries presented
for the alteration of behaviors that had heightened the risk of
infant and child mortality.
This is not to say that the overall living standards of the
multi-child family necessarily improved, but simply that
wage-earner parents who reared more children were better off
than those who reared fewer children under the existing
conditions. Indeed, for the children themselves life was likely
more "mean and brutish" than ever. Children were commonly fed at
near starvation level until such time as it was necessary to
fatten them up to go out and seek work. Descriptions in
Victorian fiction abound. Charlotte Bronté's Jane Eyre and
Charles Dickens' Nicholas Nickleby contain well-known examples.
The relaxation of infant mortality controls could manifest
itself in the statistics. A diminution of direct and indirect
infanticide is likely to show in demographic tables not only as
a decrease in mortality but also as an increase in fertility.
Live births previously regarded as spontaneous abortions or
stillborn and never registered would, under a more nurturant
behavioral regime, be registered as live births and distort the
rate of fertility change in an upward direction. We have now
reported Indian and European cases for pegging the "wanted
children" number to a well-defined value. Harris and Ross [38]
give examples since antiquity.
The "pill," although scientifically made and certainly more
reliable, did not introduce an essential discontinuity in birth
control. Many types of contraceptives always existed in the form
of vegetables and seeds that contained hormone mimics [40]. The
Greeks and Romans, for example, extensively used a plant similar
to fennel as an anticonceptional. The plant grew wild in Libya.
To stress its commercial value, the Romans minted its image on
coins, so that the plant can be exactly identified.
Significantly, it was harvested to extinction.
Assuming that economic considerations prevail, let us now look
at the present situation in developed countries. All of them are
in a phase of low fertility. The current wisdom assumes that
wealth is an opportunity for selfishness where personal pleasure
is put before the toils of rearing children.
Historically, even in periods of high fertility, the wealthy
have had few children. This argument often pops up, helped by
the argument of feminist power and female careers. Clearly, in a
well-off family children are not assets. They are costly to grow
and educate at the appropriate standard. They bring no income
when they are young. They are unnecessary for the support of
their aging but still wealthy parents. If static property such
as land forms the wealth, many children would inevitably split
it. These trivial reasons neatly explain the premodern
demographic practices of the rich. Fertility control has never
been a real problem, although infanticide would have been more
complicated in a wealthy environment.
Nowadays wealth at large is linked more to financial assets than
to static property, but child costs still can be described the
same way. Nurturing the oldies is left to third parties financed
by the income or assets of the old people. In this situation of
no economic incentive, only one basic reproductive instinct
remains, that of continuity. Adults beyond reproductive age who
realize that there is nothing after them rage and despair. Their
genes will disappear. Metaphorically, the rocket went into space
without a payload.
Assuming that the basic instinct for continuity is finally
stronger than bare economic considerations, then every couple
may long for a child. With the very low level of child mortality
at present (around 1%), one child should be enough. But here
another argument, or instinct, comes in. The child should be
male. If we put biological mechanisms in control, this request
makes sense, as otherwise the Y gene would be lost.
It is difficult to demonstrate that the cultural biases leading
to the same conclusion are an externalization of the basic
instinct under folkloric disguise. However, suppose couples
reproduce starting with the idea of the boy. 50 percent of them,
or a few more, get one. The other half get a girl and a dilemma:
what to do next. We may assume that they decide on a second try.
The last, if unsuccessful.
With this strategy in mind, and taking into account that about
15% of the females never give birth for various reasons, we find
a reproduction rate of about 1.3 per female, almost exactly the
present reproduction rate in European countries.5 If our
reasoning is correct, their situation is unlikely to change,
because of a lack of driving forces in the short term.
On the other hand, the potential exists for further decline,
because modern techniques permit the determination of the sex of
the fetus at an early age. Such potential is realized already,
for example in India, where, as described, two males per family
are in request. Ninety-nine percent of the abortions following
sex determination in India are females. (One study of 8,000
cases of abortion in India showed 7997 female fetuses [41].)
This excess female "mortality" has been a historically
omnipresent phenomenon. In China l9th-century surveys report
male-female childhood sex ratios up to 4 to 1 as late as the
1870s [42]. Prostitution obviously flourished, and concubines'
flagged status.
In Western countries, the economic detachment of parents from
children may bring another scenario and perhaps set restoring
forces into action. A population with a total reproductive rate
of 1.3 per female is unstable and converges to very small
numbers in a few generations. Children are decoupled from the
family, but they are still coupled to society because,
collectively, they must earn the pensions paid to old-timers.
As suggested, shrinking the total number of children will wreck
the pension system, bringing a feedback signal already strong
after two generations, well ahead of the shrinking to zero of
the total population [43]. History shows such reasoning to be
correct. We report one case in which the same social context,
land, and population under different laws produce different, and
predictable, results.
Before the French Revolution, primogeniture determined the
inheritance of land. The eldest child inherited all. The 1789
revolution brought the abolition of the rights of the eldest
child in 1790-1791. Equal partition followed under the
Napoleonic code. Consequently, fertility shrank to the point of
producing only one son per family. The loss of inheritance by
splitting would mean a loss of liberty and downward social
mobility. The only lever left to the family than was to reduce
the number of offspring, and they used it. Pierre Le Play, the
19th-century developer of the social survey, observed that the
Ancien Regime produced the eldest son, whereas the new produced
the only son [44].
When opportunity knocks, the rich may also procreate profusely.
Royal families provide a typical case. They use children to
consolidate power by putting them into the administration and
the army and to penetrate external territories via marriages.
Empress Maria Theresia, a career woman with heavy duties had,
nevertheless, 17 children. Her Habsburg family actively
procreated for 1,000 years, perhaps one of the roots of its
continuing power. They still appear now as a big bunch, and they
might come back, as adumbrated by the recent proposal to Otto to
become king of Hungary.
Conversely, civilizations have simply melted away because of
poor reproductive rates of the dominant class. We should not
forget that the white man's supremacy started with a
reproductive stir in Europe during the last part of the first
millennium and continued with ups and downs until the end of the
19th century. The question may be whether underneath the
personal decision to procreate lies a subliminal social mood
influencing the process, as endorphins do. The fact that crude
birth rates in Austria jumped up by 60% in 1939, the year after
the German annexation, may not be pure chance. The subliminal
mood of the Europeans could now be for a blackout after 1,000
years on stage.
Sweden counters the trend to the lowering (to 1.3) of the total
fertility in Europe. Swedes, after a decrease from a value of
2.5 in 1964 to 1.6 in 1977, started a rise in 1983, bringing the
value back to 2.0 in 1992 (Figure 11). Examining the
phenomenology of this change by fine analysis of fertility
pattern would be useful.
Logistic equations have successfully described the growth of an
individual and the evolution of the vis vitalis in integral
form, counting, for example, the publications of a scientist or
the works of an artist [45]. Thus, it is natural, heuristically,
to try them on fertility versus age. Children are a form of DNA
publication, after all. Fertility statistics include reports of
female fertility as a function of age, usually in blocks of 5
years. Fathers are not neglected; births according to the age of
the father are sometimes available.
Figure 12 shows the result for Finland at intervals from 1776 to
1976. When we think of all the whimsical forces presiding over
reproduction, the fitting appears excellent. The first datum is
usually low with respect to the fitted curve, a general
characteristic of the fitting of vis vitalis interpretable to
mean that the young have the drive but not yet the tools (for
the artist). In the case of young girls, society constrains.
But, in short order, the lost activity is made up, and the
second point is perfectly in line.
Our frugal condensation of the characteristics of fertility into
a few numbers makes quantitative comparisons convenient. We can,
for example, compare the fertility rates over time. Comparison
shows that the reproductive activity concentrates in younger and
younger years, the midpoints moving from 30 to 31 years old in
1776-1926 and to 26 in 1976. The characteristic duration of the
process shortens from 20 years to 14. The time structure
remains, as the profile of the histograms in Figure 12 basically
stays the same. In accounting terms, clearly children of higher
rank (n + 1) are falling off. The perfect self-consistency shows
that the planning, if subliminal, precedes the accident (e.g.,
an abortion).
This analysis also makes comparison sharp for different
cultures, and certainly the biological dictates are
cross-cultural. In Egypt (Figure 13) a woman now produces the
number of children more or less that a Finnish woman did 100
years ago. We find an analogous distribution in time, as the
Egyptian midpoint is 30 years (as for Finland to the 1920s), and
the characteristic duration is 18 years (vs. 19). One might
think that the preset number of children determines the time
pattern of pregnancies.
We can zoom into further detail, such as the distribution in
time of deliveries according to the sequence of children (first,
second, etc.), for example, for Canadian females (Figure 14).
The time structure in the production of the first child is
basically identical to that of the following ones. The distance
between the child waves is about 2.5 years (the numbers for the
midpoints are rounded). Such Prussian order was certainly
unexpected in such amateurish activity. The probability for a
third Canadian child falls off rapidly.
Very ordered, if not in a Prussian row, are also Malawi women
(Figure 15), who seem to shoot in faster and faster flashes.
Probabilities decrease very slowly for successive children, and
so Malawi grows rapidly. Only the first four children are
analyzed in Figure 15, but the total nears seven (Figure 16).
The pattern may resemble that of Finland around 1700. The
physical fertility span seems busy all over. The characteristic
duration is longer for Malawi women.
Speaking of time constants, at the other extreme we find Japan,
where the characteristic duration of the age-specific fertility
rate is only 9 years, meaning that all the children are produced
near the midpoint, 27 years (Figure 17). The number is only 1.6
per female.
These few examples, extracted from a large portfolio, do not
show any special feature that characterizes female fertility in
a way that permits the forecast of the fertility of a certain
group. Obviously the fertile period expands or contracts to
accommodate more or fewer children. The central point does not
move much away from 28 years. The consistency of the process
impresses, as if the decision to produce n children were taken
before starting. Recall that the statistics are
longitudinalized, that is, taken at a certain date for women of
various age brackets. The integration assumes that data refer to
a certain cohort (by date of birth) observed along its life
span.
The total fertile period for women is about 35 years. For the
few places where statistics are available, it is interesting to
integrate for a given actual cohort across years of large
fertility change to see whether the presumed family plan is kept
across the change. We perform the experiment for Finnish women,
for whom we know 1926 was the peak year of change in fertility
rates. (Long time-series data on cohort fertility are also
available for France and several other European countries [15].)
We take cohorts born in the periods 1881-1885 and 1921-1925 with
midpoints around 1914 and 1952. We need to keep in mind the
reproductive doldrums circa 1930.
Figure 18 compares the cohort and period fertility rates in
Finland and shows that the actual cohorts do experience the
lower fertility into which they will grow. We might interpret
this to indicate that the women in some way anticipate the
future trend or that the trend of the time when they were around
the center point had a dominant effect on their behavior, and
the tails were somehow adjusted.
Let us check the symmetric case of male fertility. For animals,
including humans, the female obviously makes the biological
decision, but the male behavior might mirror the decisions in a
revealing way. Figure 19 reports male fertility (age of father
when a legitimate child is born) for Egypt. Here we see a
curious phenomenon involving two logistics, or "bilogistic
growth" [46]. A first wave of procreation, similar to that of a
female, generates most children. Then a second wave follows, at
a midpoint distance of about 15 years, as the aging father has a
second pulse of procreation (midpoint 47 years) with about
one-fourth of the children of the first. One might think that in
predominantly Moslem Egypt, men, having attained with age a
certain economic success, refresh their harems. But the same
phenomenon also appears in Canada, a country of very mixed
religion (Figure 20). The midpoint distance between the spurts
is less than in Egypt, 11 years, and the size of the second
spurt is only about one-tenth of the first. Divorce, common in
countries such as Canada, permits longitudinal polygamy, and
consequently the final result may be the same. Canadian men
reach 90 percent of their second wave of fathering by 45,
whereas Egyptian men take until 55.
In a nutshell, high fertility appears to be the effect of
protracted fertility, both for male and female. Simple
observation makes this conclusion fairly obvious, but here we
report it in crisp mathematics that may help the next step in
conceptualization. More generally, we have seen that many
individual demographic processes are well modeled by the
logistic. Although interesting per se, the analysis of fertility
does not give clues for the future. For example, our historical
analysis of Finnish women (Figure 7) shows the transient but
carries no logic to foresee or negate a new transient either up
or down. Notwithstanding the considerable success in the
morphology of fertility, the assembly of the mechanisms that
enable population forecasts from the bottom up has yet to come.
Modeling the Niche
Alternatively, forecasts of population may be made by methods
that look at the aggregate numbers and neglect the mechanisms.
After all, animal societies growing in a given niche have
numerical dynamics neatly fitted by logistic equations with
constant limit K. The idea originated in Europe in the middle of
last century and reached its maximum splendor in the United
States in the 1920s with the work of Pearl, Reed, and Lotka
(reproduced in [47]; see also [48]). Putnam resumed this work
after World War II [49]. His brilliant recapitulation remains
worth attentive reading.
What these investigators found is that logistics usually fit
well the growth of a human population for a while, but then
often problems come. The mathematicians and statisticians who
looked into the problems tried to solve them with the tool they
were most familiar with: mathematics. They invented
"generalized" logistics of various descriptions and increasing
complication, until it was no longer worthwhile to do with these
logistics what could be done with polynomials [12]. In any case,
the capacity to predict eluded the analysts.
The reason why the logistics work well with most animal
populations is that the niches that encase the populations are
of constant size. When the animals can invent new technologies,
such as when bacteria produce a new enzyme to dismantle a sleepy
component of their broth, then we face a problem. New logistics
suddenly pop up, either growing from the limit of the prior one
or, if the invention came early, in the course of the first
logistic.
This expansion of the niche happens with humans. In fact, homo
faber keeps inventing all the time, so that logistics have
fleeting limits. To give an example, if we take the "industrial
revolution" as one very large innovation (embracing the changes
discussed earlier in mortality and fertility), we can reconceive
the population history of England as a sum of two logistics, or
bilogistic, with the first limit set by medieval technology at 5
million and the second limit rising to accommodate 48 million
more in the modern era [46] (Figure 21). Japan, which was
largely impervious to Western technology under the Tokugawa
Shogunate and then absorbed it eagerly under the Meiji beginning
in the 1860s, provides an even cleaner example with an addition
of 103 million to a base of 33 (Figure 22).
Abandoning restraint, we have also mapped the growth of the U.S.
population with a sum of overlapping logistics keyed to long
cycles of economic expansion (Figure 23). The fit is not bad,
but equally good alternative fits exist, and interpretation is
fuzzy. As with forecasting fertility, the method does not say
what will come next. One trick we can try, which has worked
post-mortem in other analyses, is to envelop the sum of
logistics in a "superlogistic" constructed by taking as base
points the centerpoints of the single logistics, loaded with the
values of the respective limits. We bring the masses into the
gravity centers, so to speak. The result for the United States,
shown in the inset of Figure 23, is that the population grows to
a limit of around 390 million roughly in 2100. The midpoint is
around 1940 when the United States became world power #1. All
the figures appear plausible, but the exercise is acrobatic.
On niche approaches, we might summarize by saying that so far no
one has built a solid structure capable of propelling itself
into the future. As the case of the growth of the population of
the United States shows, just fitting equations provides no
roots to keep the results standing. However, we cannot help but
be impressed by the factors of increase between apparent
population limits, 10 in England and 4 in Japan.
Conclusions
The revival of the logistic brings substantial progress in the
modeling of the evolution of life expectancy and of fertility
versus age. Life expectancy grows at all ages, with different
rates, according to a logistic path. In looking at the limits of
these logistics, when no more gain should be expected, we find
convergent paths. Thus, mortality all along the life span will
be reduced to small numbers and mortality concentrated toward
the end. Death comes through senility, and the mean age of
death, when the limit of our logistic life span occurs, can be
defined as longevity. For Europeans (and probably everyone
eventually), it is about 80 years for men and 85 for women.
The longevity "pill" could come with genetic engineering, but in
any case we doubt it will be demographically important until
after, say, 2050. Its diffusion will be limited by
experimentation and investigation of its effects. Although life
expectancy above 50 years of age greatly affects social
organization, its change has only transitory effect on total
population numbers.
Fertility dynamics also follow logistic paths. The retrospective
analysis of fertility (and mortality) afforded by the logistic
undermines several popular arguments in demographic theory. It
supports the view that fertility has always been under cultural
control and that family plans, beyond one son, are essentially
economic. In the absence of fears about the future of social
security or other conceptional stimuli (or massive immigration),
the populations of the advanced industrialized nations will
slowly implode. Admitting such "ifs," we have not solved the
problem of modeling total fertility in the future, on which
prediction of the population as a whole depends.
However, the logistic can offer a consistent approach for
predicting values for variables in demographic models, including
those that take into account the changing age structure of a
population, provided the logistic character of the process is
reasonably established and no "new logistic" arises. These
values are logically superior to the guesses now incorporated
into many models.
With respect to niche approaches, logistic analysis at least
formally and quantitatively defines the problem of limits. The
growth of human populations demonstrates the elasticity of the
human niche, determined largely by technology. For the homo
faber, the limits to numbers keep shifting, in the English case
by a factor of 10 in less than two centuries. In the long run,
these moving edges probably most confound forecasting the size
of humanity.
We are grateful to Thomas Buettner, Joel Cohen, and Paul
Waggoner for assistance.
Appendix 1: The Logistic Model
This appendix defines the logistic model and the terms we use
for demographic analysis. The logistic model assumes that the
early growth of a population (or other variable) N(t) increases
exponentially with a growth rate constant a. As the population
N(t) approaches a limit k, the growth rate [dN(t)]/dt slows,
producing the characteristic S-shaped curve. The mechanisms that
cause this slowing depend on characteristics of the population
or system being modeled, but empirical studies have shown that
this slowing is present in many growth and diffusion processes.
Thus, the logistic is a useful generic model for both systems
where the mechanisms are understood and where the mechanisms
might be hidden.
The continuous nonlinear ordinary differential equation that
describes this growth process is:
For the analytic form we need a third parameter, tm, which
specifies the midpoint of the sigmoidal curve and is related to
the initial population No(t) by
We also replace a with Dt, the characteristic duration, that is,
the time it takes the population N(t) to grow from 10% to 90% of
the limit k. Dt is related to a by Dt = In81/a. The analytic
form of this differential equation, with our parameterization,
is:
It also is possible to define a change of variables that allows
a normalized logistic to be plotted as a straight line (often
known as the Fisher-Pry Transform):
If FP is plotted on a logarithmic scale, the S-shaped logistic
is rendered linear and the period between 10-2 and 102 equals
At. The FP transform also allows more than one logistic to be
shown on the same graph with the same scale, as each curve is
normalized to the limit k.
Literally thousands of examples of the dynamics of populations
and other growth processes have been well modeled by the simple
logistic. Classic examples include the cumulative growth of a
child's vocabulary and the adoption of hybrid corn by lowa
farmers. The excellent fits obtained are a major reason for our
preference for the logistic. Another advantage of our
formulation of the model is that its parameters have clear,
physical interpretations. In addition, recent studies have shown
that the simple logistic often outperforms more complicated
parameterizations, which have the disadvantage of losing clear
physical interpretations for their parameters [50].
Appendix 2: Glossary of Terms
Fertility: The childbearing performance of individuals, couples,
groups, or populations. General fertility rate (GFR): The ratio
of the number of live births in a specified period (often I
year) to the average number of women of childbearing age
(usually taken as age 15-49) in the population during the
period. For example, the U.S. GFR for 1990 was 31.8 births per
1,000 women age 15-49 per year [24].
Age-specific fertility rate (ASFR): The number of live births
occurring to women of a particular age or age group per year,
normally expressed per 1,000 women. For example, the ASFR for
U.S. women age 20-24 was 122.1 births per 1,000 women in 1990
[24].
Total fertility rate (TFR): The TFR can be interpreted as the
number of children a woman would have during her lifetime if she
were to experience the fertility rates of the period at each
age. The TFR is obtained by summing the age-specific fertility
rates (ASFR) over the whole range of reproductive ages for a
particular period (usually 1 year). Although one of the most
frequently quoted measures of fertility, the TFR sometimes
requires a certain caution in interpretation. It is a
hypothetical measure, not necessarily applicable to any true
cohort, and may be of dubious value when the level or timing of
fertility are changing. A TFR of 2.1 is a widely cited benchmark
for a stable population. The U.S. TFR for 1990 was 1.92 births
per woman, the world average in 1990 was 3.45. Africa averaged
the highest TFR in 1990 at 6.24 [24].
Cumulative age-specific fertility rate: This rate is equivalent
to the total fertility rate. As stated in the definition for
TFR, the TFR is the sum of the Age-specific fertility rates
(ASFR) over the whole range of reproductive ages. An example
will make this clearer. Egypt in 1982 had an ASFR distribution
as shown in Appendix Table 1.
Because there are 5 years per age group, the sum of the ASFR
values in the table (1,055.2) multiplied by 5 gives the TFR in
births per 1,000 women (5,276 births per 1,000 woman per
lifetime). It is customary to give the TFR in births per woman,
in this case approximately 5.3 births per woman per lifetime.
Thus, the Cumulative ASFR divided by 200 equals the TFR.
Notice that the distribution of the ASFR, when plotted as a
histogram, approximates a bell-shaped curve. The cumulative sum
of a bell-shaped curve is an S-shaped curve. Thus, we use the
well known S-shaped logistic growth curve to characterize the
cumulative ASFR of different countries. The three parameters of
the logistic curve are characteristic duration Dt, limit - k,
and midpoint - tm. The characteristic duration Dt is the length
of time needed for the curve to grow from 10% to 90% of the
limit, which in this case roughly translates to the length of
the childbearing process for a given ASFR distribution. The
limit k is equivalent to the TFR, and the midpoint tm, the
center of the curve.
Age-specific fertility rate by birth order (ASFR): Very rarely,
the ASFR of a population is broken down by birth order. For
example, in Egypt in 1982, women age 20-24 had 173.9 births per
1,000 women. Of those births, 56.7/1,000 were from women having
their first child, 68.0/1,000 were from women having their
second child, and so on [23].
Crude birth rate: The ratio of live births in a specified period
(usually 1 year) to the average population (normally mid-year
population) in that period, usually in births per 1,000 persons.
Varies from 10 per 1,000 in developed countries to 60 per 1,000
in the developing countries. The U.S. crude birth rate for 1990
was 16 per 1,000 persons [24].
Crude death rate: The ratio of deaths in a year to the
population, usually given in deaths per 1,000 persons. The crude
death rate (also called simply the death rate) is strongly
influenced by the age-sex structure of a population. The lowest
death rates are to be expected in rapidly growing or youthful
populations with a high life expectancy. For example, Singapore
in 1980 had a death rate of 5 per 1,000, whereas the U.S. death
rate for 1980 was 8.6 per 1,000 [24]. In historical times, the
crude death rate might have been as high as 30-40 per 1,000,
with crises years reaching rates twice as high.
Mortality: The process whereby death occur in a population.
Infant mortality rate: The ratio of the number of deaths during
a specific period (usually 1 year) of live-born infants who have
not reached their first birthday to the number of live births in
the period. It is usually given in deaths per 1,000 live births.
The infant mortality rate for the United States in 1990 was 10
deaths per 1,000 live births [24].
Life expectancy: The average number of additional years a person
would live if the mortality conditions used in the calculation
remain valid. Usually given as life expectancy at birth, which
can vary from 80 + years for females in the developed countries
to 40 to 50 years in the developing countries. Sometimes given
as life expectancy at age X, which gives the average additional
number of years a person at age X can be expected to live.
Appendix 3: Note on the Problems with Fertility Data
A major obstacle to accurate demographic modeling is uncertainty
in the available data, especially for the developing countries.
In countries where births and deaths are not recorded, it is
hard to construct accurate population estimates and even harder
to reconstruct age-specific fertility rate (ASFR) data needed
for accurate population modeling. For these countries, a few
widely quoted publications contain data derived from different
sources.
The United Nations Demographic Yearbooks contain data collected
from the national statistics offices of the member countries.
For countries with good national statistics agencies, the data
are accurate, but for the others the data can be unreliable. For
example, some countries' statistics agencies might, for
political or social reasons, underreport the data on various
subpopulations. For this reason, the United Nations World
Population Prospects projections do not rely exclusively on the
data provided by the national statistics agencies, but
supplement them with data from independent surveys conducted by
academics, other national or international nongovernmental
agencies, or the World Bank. Another source of data is the
Demographic and Health Surveys (DHS) project run by the
Institute for Resource Development, Inc., Columbia, Maryland.
The main objective of this 9-year project is to "advance survey
methodologies [in the developing countries] and to aid in the
development of the skills necessary to conduct demographic and
health surveys." The data for the participating countries are
considered reliable.
To illustrate the uncertainty in fertility data, Appendix Figure
1 and Appendix Figure 2 show comparisons between the ASFR data
on Thailand and Tunisia from the United Nations Demographic
Yearbook (national statistics) and the DHS surveys. In the case
of Thailand, the estimated total fertility rate is either 2.34
births per woman or 1.85, the former being above and the latter
below replacement level. The data for Tunisia also differ by one
child per woman. Uncertainty of this magnitude could
significantly bias population projections. Clearly, better data
are necessary for accurate population modeling.
ENDNOTES
1 Such fear dates back to the 1930s [4].
2 For example, Lee and Tuljapurkar [9] assume that the U. S.
fertility rate will converge to 2.1, the replacement level:
"Conventional time series models for fertility can lead to
unrealistic forecasts (including negative fertilities), so we
examined two alternative models that incorporate prior
information.... In both alternatives we constrain the models so
that they yield a prescribed ultimate average value of the total
fertility rate. We used a mean of 2.1, chosen in part because it
is close to the ultimate level of the 1992 Census projection and
in part because many demographers view such an assumption as
appropriate."
3 See also the following article in this issue by Marchetti for
additional examples and details.
4 Measuring fertility inside cohorts rather than against the
whole population avoids the problem of the age structure of the
population. Integrating fertility over age, as we do, gives a
conceptual structure at a given time. However, fertility of
cohorts can change in time, so that this number cannot be
applied to a given woman (or 1,000 of them) to forecast their
longitudinal fertility. The same model as for fertility rates
can be applied to the age of mother or father at the birth of
children, but the results are not directly comparable, because
in the case of the rates all women in a certain age cohort are
counted. In the case of age at birth (father or mother), only
the ones having children are counted. Depending on country and
period, about 10% to 20% of women have no children.
Consequently, the integration of fertility rates leads to lower
values than integrating cumulative births (father or mother).
5 If, 15% of females have no children, 43% of females have a
boy, and 42% have two kids, the total fertility rate is equal to
1.27.
FIGURES
You can download the figures as a Microsoft Powerpoint file for
easier printing and viewing.
Fig. 1. World population growth, 10,000 B.C. to present. Sources
of data: McEvedy and Jones [2] and United Nations [25].
Fig. 2. Increase in life expectancy at birth, Norway, 1840-1990.
A logistic curve fits the increase in life expectancy at birth.
The logistic curve is displaced by 45 years, the life expectancy
in 1845. As in most figures in this article, the logistic curve
is plotted in both the traditional S-shaped form and in the
Fisher-Pry transform. See Appendixes I and 2 for a description
of the three parameters of the logistic model. Sources of data:
United Nations [24] and Flora [16].
Fig. 3. Increase in female life expectancy, by age, Norway,
1800-2050. Logistic curves fit the increase in life expectancy
at various ages. For example, an 80-year-old person could be
expected to live 6 more years in 1815 and 9 more years in 1975,
an increase of life expectancy of 3 years. A logistic is fit to
the points between 1815 and 1975. Source of data: Flora [16].
Fig. 4. Increase in life expectancy versus age, The Netherlands,
1825-1975. The figure shows that the increase In life expectancy
is greater for the young than for the old, which implies an age
limit. For example, a 10-year-old in 1975 had a life expectancy
24 years greater then in 1815. The actual life expectancy at
each age is not shown on this graph. Source of data: Flora [16].
Fig. 5. Crude birth and death rates, Finland, 1722-1993. Birth
and death rates have fallen by a factor of 4 since the 18th
century. The fluctuations from the mean also have decreased
drastically. Sources of data: Lutz [17] and United Nations [24].
Fig. 6. United Nations total fertility rate data and
projections, 1950-2025. The United Nations provides data and
projections for 187 countries from 1950 to 2025. Source of data:
United Nations [24].
Fig. 7. Logistic decline of total fertility rate, Finland,
1776-1983. This figure fits a logistic curve to the decline in
total fertility from a stable value in 1776 of 4.95 births per
woman to the current value of 1.55 births per woman. The pulse
of fertility in the 1940s and 1950s (squares) is modeled in
Figure 8. Source of data: Lutz [17].
Fig. 8. Logistic "pulse" of fertility during a logistic decline,
Finland, 1930-1983. This figure integrates the "pulse" of
fertility evident in Figure 7 (the portion of the data plotted
with squares instead of circles). The tbeoretical declining
logistic curve was subtracted from this "bell-shaped" portion of
the data, and the integral (cumulative sum) was then plotted and
fit to a logistic curve to show the shape of the "babyboom"
process. Source of data: Lutz [17].
Fig. 9. Logistic decline of infant mortality, Norway, 1850-1990.
The decline from 121 deaths per 1,000 infants born to 10 infants
is dramatic and remarkably regular. The logistic was fit
assuming a final goal of zero deaths, where the theoretical
limit might be around 3 to 4, depending on advances in medical
technology and screening procedures. Sources of data: Flora [46]
and United Nations [24].
Fig. 10. Logistic declines of infant mortality and total
fertility rate, Finland, 1800-1983. The Fisher-Pry transforms of
the fitted logistics are plotted together for comparison. The
data are not shown so as to ease comparison. Source of data:
Lutz [17].
Fig. 11. Total fertility rate, Sweden, 1900-1993. Sources of
data: Conseil de l'Europe [15] and United Nations [24].
Fig. 12. Logistic analysis of age-specific fertility rates,
Finland, 1776-1976. (A) Histograms of the age-specific fertility
rates in 25-year intervals. (B) Integral (cumulative sum) of the
bell-shaped histogram data, resulting in an S-shaped curve that
is fitted with the three-parameter logistic. As explained in
Appendix 3, the cumulative sum of the ASFR divided by 200 is
equal to the total fertility rate (TFR). (C) Fisher-Pry
transforms of the corresponding logistics (rendering them
linear). Source of data: Lutz [17].
Fig. 13. Logistic analysis of age-specific fertility rates,
Egypt, 1982. See Figure 12 and Appendix 3 for a description of
the method of analysis used here. Source of data: United Nations
[23].
Fig. 14. Logistic analysis of age-specific fertility rates by
birth order, Canada, 1977. The method of analysis used in this
figure is similar to that used in Figure 12 and explained in
Appendix 3, but here the fertility data are broken down further
by the birth order, that is, first child, second child, and so
on. The percentages refer to the number of women who go on to
have more children (53% of Canadian childbearing women had a
second child, but only 16% had a third child.) Source of data:
United Nations [23].
Fig. 15. Logistic analysis of age-specific fertility rates by
birth order, Malawi, 1977. See Figure 14. Source of data: United
Nations [23].
Fig. 16. Logistic analysis of age-specific fertility rates,
Malawi, 1977. See Figure 12 and Appendix 3 for a description of
the method of analysis used. Source of data: United Nations
[23].
Fig. 17. Logistic analysis of age-specific fertility rates,
Japan, 1990-95. See Figure 12 and Appendix 3 for a description
of the method of analysis used. Source of data: United Nations
[24].
Fig. 18. Logistic analysis of age-specific fertility and actual
cohort age-specific fertility, Finland, 1891 and 1921. Top:
Histograms of both the ASFR and the Period ASFR, which follows
the fertility rates of a 5-year cohort of women throughout their
actual reproductive careers. Bottom: Corresponding logistics.
Period ASFR differ from the cohort ASFR when the fertility rates
are rapidly changing, as shown by the 5-year age cohort from
1921 to 1961. Source of data: Lutz [17].
Fig. 19. Logistic analysis of age-specific birth rates by age of
father, Egypt, 1982. This figure is similar to Figures 12-18 in
that the cumulative ASFR is analyzed with logistics. However,
this figure shows the cumulative ASFR by the age of the father
as the sum of two logistic pulses. Source of data: United
Nations [23].
Fig. 20. Logistic analysis of age-specific birth rates by age of
father, Canada, 1977. This figure shows the cumulative ASFR by
the age of the father as the sum of two logistic pulses. Source
of data: United Nations [23].
Fig. 21. The population of England fit with a bilogistic growth
curve, 1541-1975. The sum of two logistics is used to analyze
the population history of England. The first logistic curve has
a 132-year characteristic growth time and a limit of 5 million
and is centered in 1540. The second has a characteristic growth
time of 166 years and a limit of 48 million and is centered in
1892. See Meyer [46] for a description of the bilogistic model.
Sources of data: Wrigley [28] and Flora [46].
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