The phenomenological theory of world population growth
Sergei P Kapitza
Originally from Physics-Uspekhi 39(1) 57-71 (1996). Copyright S.P.Kapitza
Modified by Mark Hopkins, 2004.

Transcriber's Notes:
Most of the figures, graphs and tables are from the original article intact, with improvements in the visual format. Many, however, are superseded by more recent developments. In particular, the conclusions following from the model posed, itself, about the future of the world population, even with the modifications made here to the model to better incorporate recent data, have been strongly refuted by the population data that has emerged since the paper was written. The maximum figure of 14 billion (or 10 billion with the modification made) will likely not be approached, but only 8 billion instead.

Abstract
Of all global problems world population growth is the most significant. Demographic data describe this process in a concise and quantitative way in its past and present. Analyzing this development it is possible by applying the concepts of systems analysis and synergetics, to work out a mathematical motel for a phenomenological description of the global demographic process and to project its trends into the future. Assuming self-similarity as the dynamic principle of development, growth can be described practically over the whole of human history, assuming the growth rate to be proportional to the square of the number of people. The large parameter of the theory and the effective size of a coherent population group is of the order of 10⁵ and the microscopic parameter of the phenomenology is the human lifespan. The demographic transition - a transition to a stabilised world population of some 14 billion in the foreseeable future - is a systemic singularity and is determined by the inherent pattern of growth of an open system, rather than by the lack of resources. The development of a quantitative non-linear theory of the world population is of interest for interdisciplinary research in anthropology and demography, history and sociology, for population genetics and epidemiology, for studies in evolution of humankind and the origin of man. The model also provides insight into the stability of growth and the present predicament of humankind, and provides a setting for discussing the main global problems.

1. Introduction
The growth of world population is the single most important global problem and is certainly the most complex of all these issues. An understanding of the process of growth can be sought by extending the scope of analysis and sensibly simplifying our approach. Growth is usually expressed by the data of demography as presented for countries or regions¹.

These statistics contain the clue to working out the quantitative laws that could in principle describe the dynamics of world population growth. Starting with demographic data, one can develop an approach to discern the laws that could describe development over many generations and encompassing the whole world. By applying the methods of systems analysis and synergetics for the phenomenological description of the human population system a mathematical model is suggested.

An alternative approach is pursued by global modellers, mainly known by the first reports of the Club of Rome. In these studies an attempt is made at developing a complex computer based model using comprehensive inputs from extensive data bases. Following a reductionist methodology most of the factors which influence human development are purportedly taken into account. At present the results of this detailed analysis have been extensively criticised and such an approach has not led to any recent advances.

On the other hand, one can apply the methods developed initially in physics to describe systems with many particles and degrees of freedom. These methods, based on first principles, were later successfully applied to complex physical and chemical systems, then to biology and the environment, and finally to economics and social phenomena. Extending the methods so successfully used in physics to the population of the world is not self-evident, if a meaningful result is expected. In the course of study this point will be reviewed in the light of the results of modelling, since growth and development of humankind over an immense timescale is to be described. In pursuing interdisciplinary research with methods and data coming from very different fields of knowledge it is not easy to establish a common frame of reference. One of the main complications is that what may seem to be obvious, if not trivial, to one party is far from the usual concepts of others. In these cases some facts and ideas will necessarily demand a more detailed and in-depth presentation for the sake of interdisciplinary understanding. Bringing together facts and figures of anthropology and history, and of demography -- a well established science with an old and thorough mathematical tradition, has proved that a dialogue, a constructive intercourse is not only possible, but most necessary in dealing with global problems.

Table 1. UN Data (1992)

Region	Population, billions			% total, mid 1990
	1990	2000	2025	0-4	15-24	65-up
World	5.3	6.3	8.5	12	19	6
Developed	1.2	1.3	1.4	7	15	12
Developing	4.1	5	7.1	13	20	4
Europe	0.50	0.51	0.51	6	15	13

Finally, the whole issue of world population growth as a complex global problem deals with data of a very different nature and involves issues well beyond science. Ideology and conceptions of some societies and people inevitably touch on the convictions and values of others.

At present, our common horizon is dominated by the population explosion. The annual addition to the population of the world is approaching 100 million with 250,000 people arriving daily. This ever increasing growth sets the pace for demands of food and energy. The growth rate is alarming and has, when simplistically extrapolated, led to apocalyptic forecasts and doomsday projections² (see Table 1. By using up more and more resources of the Earth humankind is exerting a growing pressure on our environment and biosphere. Ultimately, this pressure will have an effect on our growth. In this case it is all the more important to develop lines of rationale in assessing factors which determine the growth rate and what limits there are to our numbers. Only if we obtain a broader and in-depth vision of human growth and evolution can we hope to further our understanding of these universal issues.

In spite of all the drama of the population explosion and the high emotions it generates, what really matters is that at present we are passing through a population transition. This phenomenon is experienced by all populations and involves a rapid rise in the growth rate followed by a similarly fast decrease in growth, finally leading to the stabilisation of the population (Fig.l). This transition has been experienced by all developed countries, and now the same process is happening worldwide. The demographic transition is accompanied by a general increase in economic growth and the corresponding resettlement of people from rural areas to towns. At the closing stages of the population transition a marked change in the age profile is observed, with a predominance of the older, rather than younger generation.

In the modern interconnected and interdependent world this transition will practically come to an end within the next 50 years. It is happening much faster than in Europe, where a similar process began at the end of XVIII century. The demographic transition is certainly the most significant worldwide transformation and is intensively studied by modern demography³.

Following the approach and methods of demography the changes in a population can be worked out for the next one or two generations.

For these projections the world is divided into a number of regions, where certain growth scenarios are assumed. Patterns of changes in fertility and mortality, considered to be more elementary processes, lead to a description of the demographic transition. Unfortunately, it is difficult, if at all possible, to provide more than an adequate description of the transition following this reductionist approach passing from a more fundamental level of concepts to the next level of complexity. But in no case can fertility or mortality be considered to be elementary processes. These concepts are also phenomenological, generalising in these indicators many factors. In a system of such complexity as humankind, most, if not all, connections and interactions are essentially nonlinear and cannot be summed up, assuming linear cause-and-effect relations, a direct connection betweer part and the whole.

The alternative is to pursue a systemic approach, whereby the entire population of the world is seen as an evolving and self-organising system. This is the main feature of the mathematical model for the development of all humankind. The feasibility of this approach is far from obvious. What should be considered in the first place is the extent to which the concept of a system may be meaningfully applied to the total population of the world and the question of whether the process of growth is statistically regular and historically predictable.

The acceptance of a synthetic rather than analytic, reductionist treatment is the key to the model, much as in modern cosmology the dynamic treatment of the Universe is only possible when the whole of the world is considered to be the object of study.

2. The world population as a system
For many years the mere possibility of considering the population of the world as a single system was not taken seriously. Most demographers saw in world population data a mere sum of the populations of all countries, a number that had no objective dynamic meaning⁴. At present this extreme attitude has been abandoned, and in modern demography a systemic and historic approach is becoming prevalent¹.

Data on the population for the main regions of the world show both an overall growth trend and at the same time a diversity in the individual patterns of development. The sum growth of the world population follows a more regular pattern, culminating in a rapid escalation of the population explosion (see Figures 1 and 2).

One prerequisite for a system as applied to the world population is the interaction of all its regions. Interactions are instrumental for a system and the interconnections by transport and trade, migration and information bring all interdependent parts of the world together. Now these connections unequivocally permit one to treat the population of the modern world as a system. In the course of this study on a number of occasions we shall see proof of the systemic behaviour of the world population and phenomena that support this attitude.

To what extent is this true for the past? In the development of the model the criteria for systemic behaviour will be obtained. Even in the distant past, when the number of people was much less than today and the world was divided, the populations of different regions slowly but surely did interact. The time of this interaction may be estimated and it can be shown that in most cases the systemic approach to the world population is valid.

In the demographics of regions and countries the change of population by migration is a noticeable contribution to the balance of people. But on a global scale, migration should not be taken into account, as there is yet no possibility to leave the Earth¹⁶. On a planetary scale, migration is simply one of the internal interactions of the system. Because of migration and wars, on a historically significant time scale we get a mixing of people and cultures that is part of the systems dynamics. When these interactions and exchanges are for a long time cut off, this isolates a part of the population and leads to the subsequent stagnation of a subsystem.

It should also be noted that biologically all people belong to the same species Homo Sapiens. We all have the same number of chromosomes-46, different from other primates. All races can intermix and socially interact⁵ and, biologically, humankind is a comparatively uniform species⁶. But in terms of numbers we are by five oders of magnitude more numerous than any mammal of comparable size and position in the food chain. Only domestic animals, living with and around humans, are not naturally limited in their numbers, as other creatures in the wild, occupying their own area and ecological niche. Humans practically inhabit the whole world, all parts that are fit to live in. By industry humans have created an environment, a habitat much of our own making. There are good reasons for assuming that during the last million years mankind has hardly evolved biologically and human development and self-organisation is primarily social.

These processes of social and technological development are to be described by a quantitative phenomenological model. The data of demography and concepts of anthropology and history will be interpreted in terms of systems analysis and synergetics^7,8. This may help to introduce some new ideas and concepts into traditionally humanistic studies. The model is an attempt to develop a quantitative statistical approach to anthropology, a theory that may lead to new insights into a problem of concern for us all.

3. Modelling world population growth
Constructing a model entails the application of methods developed for the study of dynamic systems to data provided poorly by demography and anthropology. These facts and figures are poorly known to physicists and one of the purposes of this paper is of introduce them to a new set of problems. In a number of cases it will be possible to identify concepts and recognise well-known ideas in a new setting. On the other hand it should be kept in mind that both the data and the model itself are but crude images of the real world. In interdisciplinary studies, one of the main difficulties that of taking ideas and methods from one field and transferring them into another. This process is perhaps best developed in mathematics when models are suggested, and then in theoretical physics, although it is difficult to say when a model can reach the status of a theory, serve not as a description of events, but lead to greater insight into the nature of the phenomena treated.

For a system as complicated as the population of the world it is this complexity that provides an opportunity. When many factors are relevant and different interactions simultaneously occur, it can be expected that a statistical approach is feasible. In this case, most of the spatial and temporal variations will average out with the consequence that of all the different processes taking place, those finally determining the systemic behaviour are left. Then we can expect to explore the resulting pattern of changes that have an objective nature and can be expressed numerically. The following treatment of the world population is based on the development of such a model^9-12.

The population of the world y at the time t years is to be determined by the function y(t). Changes in y will be considered over a large number of generations. In this case it can be assumed that the time span of a single generation, of the human life-time will not explicitly enter the formula, just as the distribution of people by age and sex. In systems, where a multifactored process is considered with many degrees of freedom and development can be treated as statistically uniform, one may expect that growth will follow a self similar pattern and scale in time. This main assumption of scaling can be expressed as the invariance of the relative rate of change in the system

(1)

where t₀ and y₀ are the points of reference for time and population. In most cases y₀ = 0.

In self-similar processes of growth and development, the ratio of relative changes in population and time is constant and as a necessary consequence of scaling (1) growth is described by a power law y = C(t₀-t)^a, where C and a are constants. The simplest case is linear growth, where a = 1. Exponential growth or growth following a logistic law^13,14 are not self-similar, as they have an internal time scale - the time of doubling, so these growth laws are not scaleable.

Forster¹⁵ was an early advocate of the following law:

(2)

as an empirical formula to describe world population growth, where the following values for the constants were obtained by the least square fit of a large collection of population data from A.D. to 1960. The accuracy implied for a seems somewhat excessive and in the following treatment a = -1 is assumed. Later Horner¹⁶ suggested a similar expression

(3)

where a = -1 and C = 200 billion people-years. These formulae describe with good accuracy world population growth over thousands of years and more, now culminating in the population explosion. The best-fitting model, fitting the logarithms linearly to the historical data comprising the Facts On File reference up to 1900, the UN 1999 reference up to 1950 and the USA data up to 2004 (all at the US Census Bureau⁵¹) gives y = 914 billion (2140 - t)^-1.52.

Expressions for growth (2) and (3) should be seen as a physically meaningful law for self-similar growth, following a hyperbolic growth curve. These self-accelerated blow-up processes in great detail have been studied in recent research of non-linear phenomena^17,18. In other words, the population explosion is seen as a global instability, well known in the behaviour of essentially non-linear systems.

On the other hand (2) and (3) are limited in their validity both in the past and present for, as we approach year 2025 (or 2140) the population of the world will go off to infinity. This has led some to believe that the world is approaching an infinite Population Singularity, indicating an impending crisis. In a very similar way these expressions are not valid going back to the past in cosmic time. For 20 billion years into the past, it would yield about 10 witnesses to the Big Bang, itself. In other words, the self-similar expression (2) is limited in the past and present due to the divergences of hyperbolic growth. This is exactly what should be expected, for scaling power laws are only asymptotic during an intermediate phase¹⁹.

Consider for a moment the 10 mythical witnesses of the Big Bang. Had they actually existed, they would have had to live a billion years to bide for the time before the population incremented to 11, which this is just as meaningless as their very existence. Likewise, the Singularity approaching t₀ means that just one year before, everything would double in less than a year, which is equally nonsensical (unless the engineering of reproduction, itself, suddenly went haywire in a runaway process). What has not been taken into account is the duration of the human life, an interval of time that characterises our reproductive capacity and lifespan. Both of these considerations place limits on the applicability of the scaling law.

It is well established that all countries pass through a maximum growth rate of their demographic transition, followed by a rapid decline. This has been observed for all developed countries and is now seen in countries of the developing world representing regions of Africa, Asia and South America (Fig. 3). In fact, the transition has taken a decided turn downward in recent years. Whereas, as late as 1999, there were several nations over 4% and many over 3%, by 2002 a few years later, none exceeded 4% and only a few remained over 3%. Most of the high-population nations (e.g., India, China, Egypt, the United States) were already under 2% by 2002. The demographic transition for the world population is illustrated in Fig. 8. It is significant that the population growth rate passes through a pronounced maximum, and does not stabilise at its highest point.

At the very outset of human development at t_- it can be assumed that the growth rate is also limited. It cannot in general be less than P = 1 person, presumably a hominid, per generation, a requirement for the continuity of growth. As a very approximate description of this early stage, it provides the necessary asymptotic cut-off for the divergence in the past

(4)

and the cut-off of growth rate as the demographic transition at t₀ is approached.

By introducing a cut-off time constant t the divergences both in the past and present can be eliminated, regularising the growth rate

	(5a)
	(5b)

and for the last stage of growth and the demographic transition

(5c)

A characteristic time t is introduced into the model as a phenomenological microscopic parameter, allowing the expressions for the growth to be extended into something more appropriate into the past, and indefinitely future, beyond what had been the singularity at t₀. The singularities are removed, while retaining the required asymptotic growth in the intermediate period, so that the result should more accurately describe the human population curve over the entire history.

It turns out that we're fortunate enough to be at a special time where the inflection point of the curve described by equation (5c) has been passed. This can be seen clearly in the mid-year population estimates provided by the US Census Bureau⁵¹ for the past 30 years. There is a near exact symmetry about 1989 and the totals of the populations on either side of this time add up to a constant, plus or minus 5 million, which is within the margins of error of the estimates (figures provided in the millions)

y_1989-X + y_1989+X = constant

1989-X	1974	1975	1976	1977	1978	1979	1980	1981	1982	1983	1984	1985	1986	1987	1988	1989
y1989-X	4014	4087	4159	4231	4304	4379	4454	4530	4610	4691	4771	4853	4937	5023	5110	5197
y1989-X+y1989+X	10392	10392	10392	10391	10389	10389	10387	10385	10386	10387	10386	10388	10390	10392	10395	10394
y1989+X	6378	6305	6233	6160	6085	6010	5933	5855	5776	5696	5615	5535	5453	5369	5285	5197
1989+X	2004	2003	2002	2001	2000	1999	1998	1997	1996	1995	1994	1993	1992	1991	1990	1989

The numerical values for the constants appearing in the equation (5c) and the constant of integration that arises after integrating the equation to yield:

(6)

can then be determined by fitting the data against this result.

The following values for the relevant constants are provided by the original author:

The author's data comprises Model III the best fit (at the time) to demographic data (see Table 2); the new data comprise Model X. Note that population estimates, even when quoted down to the final digits, as is tradition, are typically only accurate to around 3% to 5%²¹.

Table 2

Model	ymax billions	C bn.-yr.	t years	t0 year	dy/dtmax mn./yr.	d ln y/dtmax %/year	tmax year	y1990 millions	t0.9ymax year	K 1000s	-t- mn. yr.	ytot billions
X	10	125	38	1989	87.5	1.92	1973	5108	2105	57.5	3.4	65
I	10	180	55	1998	60	1.31	1964	5260	2157	57.2	4.9	99
II	13	185	45	2005	92	1.60	1986	5135	2143	64.1	4.5	102
III	14	186	42	2007	105	1.73	1989	5253	2138	66.6	4.4	103
IV	15	190	40	2010	119	1.81	1993	5259	2133	68.9	4.3	106
V	18	195	33	2017	180	2.18	2003	5230	2119	76.9	4.0	110
VI	25	200	25	2022	320	2.88	2011	5306	2099	89.4	3.5	114
VII	¥	200	(20)	2025	—	—	—	5713	—	(100)	(3.1)	115

The results of modelling show that t_- and C are not sensitive to the value of t, while on the other hand a clear value for t only emerges from recent data that follows the occurrence of the demographic transition. For this purpose it is useful to compare demographic data for the growth rate (5c) and the relative growth rate

(7)

that passes through its maximum value before (5) and reaches its maximum value of 1.7% at t_max = t₀ - 0.43t, which for Model III comes out to around 1989. (The number l = 0.43 is the unique solution greater than 1/2p to the equation tan(1/2l) = 1/l.) The growth rate (5c) will pass through its high point in t₀ = 2007. With the introduction of a finite t = 42 years renormalisation shifts t₀ from the year 2027 to the year 2007 (Fig. 4).

The maximum rate actually occurred between 1962 and 1972, going over 2.0% during this interval and peaking in 1963-1964 and 1968-1969 respectively at 2.2% and 2.1%.

Two values of t can be introduced - for the past and present, but the model shows that a plausible estimate for t is obtained with the same t as for the present. This value for t = 42 years is a reasonable time constant to describe the human life from our every-day experience, although the value is obtained from the global demographic transition, as an average for many countries at very different stages of development.

Rescale to dimensionless variables for time

(8)

and population

(9)

with time is reckoned from t₀ in units of t and as the unit of population the constant K = 67000 appears. K is really the single large parameter of the theory that enters into all formulae and describes all relevant proportions in the phenomenology. The dimensional constant PK then represents a number of people, so that PK serves as a unit of population (9). K, itself, serves as a real dimensionless number. The distinction originates ultimately from the two scales in (4), where the unit P originally appears. The constant PK can be interpreted as the natural size of a coherent population unit and numbers of this order turn up in human genetics, city planning etc.

Introducing t and y into (5) and on integrating the rate the growth the following expressions are obtained

	(10a)
	(10b)
	(10c)

In these expressions time and population appear in a symmetric way indicating the reciprocal connection of the variables. The singularities arise as y ® 0 and t ® 0. In the beginning of growth at t_- the independent variable is t, as it should be. But as we approach t_- the significant variable is y that really determines the change during the transition at t_-, although time certainly is physically independent. This is the consequence of the essentially nonlinear nature of the population system for which three distinct epochs can be identified. The first - epoch A is dominated by linear growth, epoch B is described by a hyperbolic growth curve and epoch C is the transition to a stabilised world population.

The asymptotic transition of one solution into another is best seen in the series expansion

(11)

and

(12)

These functions intersect near t_* = -K^1/2 at an angle 2/(3K), which for large K is effectively a smooth transition. The beginning of growth at epoch A is determined by

(13)

or 4.4 million years ago for Model III. This expression shows that t_- is rather insensitive to t.

Clearly it is possible to join the two solutions together to form a single solution that starts at t_- = -pK/2 and, with minor changes to (10) best seen by writing dy/dt in terms of y, to get a single equation for y ranging over the all times

(14)

By integrating (14) a global solution for the growth curve can be obtained

(15)

where n_- = 0. These rather simple calculations provide a general description of the dynamics of the global demographic system. Developing this theory further will require applying more advanced methods of nonlinear mechanics to the demographic problem, a problem this model has helped to bring into the realm of physics.

To gain insight into the growth mechanism it is best to consider the primary expression for the rate of growth during epoch B where K is the main parameter determining the rate of growth for a collective binary interaction in a generation. This growth rate is well known in chemical kinetics and has been extensively studied in systems analysis. In the case of the human system this growth rate should be seen as the net result of all partial processes, that contribute to growth. Thus the growth rate is the outcome of all inputs of an economic, social and biological origins. The human capacity to multiply, that taken alone could lead to exponential growth is only a part of all mechanisms contributing and limiting growth in this essentially nonlinear expression. To break up growth into the separate channels that sum up to (15) is beyond the agenda of phenomenology, expressed by the quadratic growth rate. In fact, this should be seen as the effective mechanism of growth described in phenomenological terms. The parameters t and K are indeed constants, as no evolution is implied to adjust their values to changing circumstances (Fig 5).

In a formal sense the y² above can be considered as an effective field and first in an expansion in powers of y to describe higher order interactions. Since K is large and since both spatial distributions and temporal fluctuations have to be taken into account, it would, at any rate at the present stage, be premature to extend the expansion in terms of powers of y.

The term corresponding to the first-order interaction, dy/dt = y²/K², is then that of a cooperative, global societal interaction ranging over the entire span of human history. The microscopic time constant t then provides a natural cutoff for the phenomonology. Starting in the remote past, where initial growth is postulated, the trend becomes a self-accelerating quadratic growth rate, which reaches its limit, culminating in the break down of scaling at the demographic transition. The hyperbolic singularity diverges in a finite time - exceeding exponential growth, whereas (15) diverges only in infinite time. The limit to the growth rate, here, is kinematic, not one brought about by resource limitations, as one would expect in the case of a logistic curve. In other words, the model posed here is of an open system whose limitations are primarily self-induced, not a Malthusian model.

4. Population limit and the number of people who ever lived
The expression for growth (6) indicates a limit for world population

(16)

in the foreseeable future. The 90% point of the asymptotic limit is reached by the year 2138 in Model III, or in 3t years after t₀ = 2007. In Model X, the 90% point is reached by the year 2105, likewise 3t years after t₀ = 1989.

An important question that naturally arises in human evolution and population studies is how many people have ever lived up to a certain point of time. In actual fact, however, it is more natural to consider the total number of years that have been lived, measured in units of people-years. Given the rough parallel between the apparent speed perceived in historical progress and the size of the world population at a given time, it is also natural to identify this quantity as none other than a measure of the total quantity of history up to a given time. The unit people-years, which is the same unit that occurs with the model parameter C, is then the unit of history, itself.

In the framework of the model this total, H, ranging from the beginning point to the transition time may be determined by integrating y(t) between the times t_- and t₀, using the phase A and C estimates on either side of the imtermediate transition time t_*

(17)

and for Model III this comes out to H = K²ln K = 50 billion people-years.

To arrive at an estimate of the total number of people, y_tot who have lived requires a second input: the life-expectancy curve. To arrive at this estimate, in turn, we note (for model III), that ln K = 11.1, and assume that the average life time is around t/2 = 21 years, so that y_tot = 2H = 100 billion people-years. The assumption about life span matches that made in calculations by Weiss²⁰ and Keyfitz²¹. In these cases growth over the ages is described by a sequence of exponents and have led to y_tot from 80 to 150 billion. For different models y_tot = (C/21) ln K is practically independent of Table 2. However, for Model X, it only comes out to a figure of y_tot = 65 billion people.

The total is split almost evenly on either side of the crossing point t_* = -K^1/2, so that roughly half of history, by this measure in these models, has expired by the time we reach the time t_* = t₀ - tK^1/2 = 9000 BC. Likewise, assuming the rough constancy of the life-expectancy curve over most of this period, half of all people have already lived by this time. Using the cutoff time t_AB = t_- + K (about 2 million BC) for the transition from Epoch A to Epoch B, the totals H_A and y_A for Epoch A are respectively

(18)

It is best to present the growth process of the human system as a plot of log y vs. log t. This is not only a matter of convenience, with the variables changing over 10 orders of magnitude, but renders the self-similar growth of the power law in the intermediate phase as a straight line. The appearance of a straight line is a graphic indication of the constancy of the relative growth rate, of scaling. Later we shall see that a logarithmic time chart leads to greater insight into the meaning of time for the development of the global human population system.

5. Population growth and the model
By comparing the model with data of paleoanthropology and paleodemography a description of human development over a vast period is possible. The initial epoch A began 4.4 million years ago and lasted dt_A = Kt = 2.8 million years. Thus the model describes in general terms the initial period of development, an extensive interval that may be identified with the time when, according to modern data some 4.5 millin years ago hominids began to be separated from the hominoids⁵. By the end of Epoch A Homo Habilis appeared and the number of these primaeval hominids would have reached t_A,B = K tan t/K = K tan 1 = 104,000. This number corresponds well with the estimate ~ 100,000 suggested by Coppens for this decisive moment in the development of humankind²³. It was then in Africa at the beginning of Paleolithic that the toolmaking humanid first appeared. What matters is that a reasonable estimate can be provided for the time and numbers matching the data of anthropology, when some 1.6 million years ago a critical step in human development occurred. It was then that the cooperative social and technological pattern of self-accelerating development began, and since that time humankind started to spread throughout the world and grow in numbers, well beyond any other comparable creatures. This initial process of differentiation could be accompanied by fluctuations and the appearance of different parallel evolutionary lines. These events are beyond the scope of the model ^5,24,25 (Fig. 6).

Epoch B encompasses the Paleolithic, Neolithic and historical periods. During the 1.6 million year interval the number of people once more increased K times. By the onset of the population transition in t₀ - t = 1965 the global population had reached pK²/4 = 3.5 billion. Growth and development mainly took place on the Eurasian continent. Across these vast spaces, tribes migrated, languages and cultures developed, civilisations grew and vanished developing in a systematic, although turbulent pattern of growth. During the last millennium the Silk Road, connecting the civilisations of East and West, of China and Europe with an input from India was active. Along this main trade route world religions spread, technical and social information diffused. At first slowly, then faster and faster, the human system did develop, incessantly growing in numbers.

Most of population data to the extent that it is known within reasonable limits, fits the model. Although the further we go into the past, the accuracy of demographic data rapidly decreases. In general, both in history and anthropology dates are known much better than population numbers. For example, for the time of the 1st century onwards paleodemography provides estimates from 100 to 250 million. The model indicates 100 million, that corresponds to the number mentioned incidentally in the joint statement of 50 Academies of Sciences on demography¹⁵. Since 16th century, after the great geographic discoveries the quality of world demographic data improves. It was then that the world population system rapidly became more and more interconnected, although in the past this interdependence was always sustained and the gross features of our past are reasonably well described by the model.

Of interest is to compare the results of modelling with projections of demography into the foreseeable future. The model indicates a rapid transition to an asymptotic limit of 14 billion, reaching the 90% point of 12.5 billion by 2135. These numbers may be compared with projections of UN²⁷ and IIASA²⁸ (Fig. 7).

The present figures now clearly exclude all by the lowest projections, unbeknownst to the original author.

The projections of UN are based on summing up scenarios for fertility and mortality for nine regions of the world and are extended to 2150. By the optimal UN scenario the world population will reach a constant level of 11.6 billion, then extrapolated to 2200. Projections of IIASA cover a shorter time horizon - up to 2100 and are based on six regions with ten different patterns of growth. The optimal scenario -- the low birth rate decrease scenario, is where the projections of UN and IIASA practically coincide. Both modelling and demographic projections indicate a levelling off of the world population (Table 3).

An important proviso in these demographic extrapolations is that they are computationally unstable with respect to small changes in the scenario, as can already be seen from the graph alone. A shift of 2-3 years in assuming a change of fertility or mortality will result in quickly diverging scenarios. That is why such projections have, at best, a short time window. According to Sadic²⁹, most projections of demography have over the last decades been systematically revised upward.

Ironically, nearly all of them have to be revised downward in recent years. The source of this discrepancy, in large measure, is the distortion brought about by World War II and its aftermath. Even the relatively small dent this war left behind (and note the use of the word "relatively") leads to rapidly diverging results if incorporated into any projection. A second source of the overestimate is that the population curve, in fact, has dropped off into different tracks corresponding to the phase changes, called by Toffler "waves", that has swept over the world in the intervening period. If one attempts to fit the population curve for all of human history, for instance, to a logistic (or to a Kapitza arctangent-tangent curve), the result shows a huge swing all the way up to 15-20 billion or more. However, one can clearly distinguish an Industrial Era population curve, which closely follows (in fact) a logistic curve over the period 1850-onward, and tops off at around 9 billion. Even more, though, another curve is clearly distinguishable from around the cresting point of the Post-Industrial Revolution around the mid 1970's. This, too, is a near-exact logistic, but only tops off at 7.8 billion.

6. The demographic transition and population stabilisation
For the world population the demographic transition is a very special event, a phenomenon of great interest and complexity. The time window for this transition ranges over the period t₀ ± t = 1965 to 2049. It is during this time that the world population will on one hand increase its numbers three times and on the other hand undergo a most profound transformation. To describe the population transition Chesnais³ introduced the transition multiplier that for the model is

(19)

In this case the beginning of the transition is at the point of most rapid increase and its end - at the point of most rapid decrease of the growth rate. Chesnais provides the following data for M: China: 2.46, India: 3.67 and the world: 2.95, numbers that compare well with (19), although in some cases as France: 1.67 or Mexico: 7 the discrepancy may be large. What matters are not these special cases, that to a certain extent are dependent on when the beginning and end of the transitional period is determined, but the overall correspondence of modelling and the value of M for the largest contributors to the world population.

On the scale of history the transition is remarkably short - 1/50000^th of the time of growth, although 1/10^th of all people who ever lived are to experience this special period. This can be attributed to the strong interaction existing nowadays between countries and regions of the world. The world population definitely does behave as a system and the synchronisation of all the partial transitions is a measure of this interaction. Synchronisation and narrowing is a well-known phenomenon in nonlinear systems and points to the systemic nature of the global demographic transition.

The rate of the transition with a characteristic time of 42 years is in fact shorter than the life expectancy of 70 years in developed countries and practically equal to the world average of 40 years. (Since the original author wrote this, life expectancies for the nations of the world have icnreased substantially. Few nations are even under 40 years anymore. Most are in the 50's and 60's ranges, many up into the 70's and even 80's now.) This rapid non-equilibrium transition leads to the break-up and disruption of traditions and customs long established in human societies, factors that to a large extent stabilised the life-style in the past, setting up long-term correlations between generations. This is none other than Future Shock. Today it is customary to say that the connections between generations are severed and many see in this one of the reasons for the strife and stress of modern life, a factor that should be attributed to the transition through which we are passing.

The concept of a future stabilised world population is of significance. This is an immediate result of the model that describes the transition from quadratic growth during Epoch B culminating in the population explosion and a basic change in its growth after the transition (Fig. 8). Recalling Adam Smith, the invisible hand of self-organisation is seen as the collective agent that changes the paradigm of our growth. The change is due to a systemic crisis and is described by the change in the asymptotic behaviour of the human population system. Its fundamental reason may be connected with the analytic properties of the function y(t), its singularities and asymptotics. This point needs further consideration, as it could pave the way to greater insights into the foreseeable future of humankind. If we had only to extrapolate an established pattern of growth without this qualitative change, it would be much more simple to make this forecast, but its significance would be correspondingly less.

The systemic transformation of a country passing through the demographic transition has been graphically depicted by Vishnevsky³⁰, who introduced a phase plane to describe this process when the system rapidly moves from one attractor to another, indicating the stability of motion at each of these states of equilibrium (Fig. 9).

Chesnais in his detailed description of the demographic transition finally comes to the conclusion that it is impossible to explain this complex phenomenon in terms of cause and effect, of linear modelling for only an inherently nonlinear model can describe this transformation.

For a physicist accustomed to concepts of statistical physics and systemic behaviour, the demographic transition displays many features of a phase transition. It is the age distribution of the population that changes during the transition and this is the most important thing happening. In the framework of the model, the age distributions are not taken into account. An extension of the model could treat the changes in the age distribution, that were fairly constant during Epoch B and then rapidly change to a new pattern. These transformations can be described by the standard methods of demography, when this change is postulated, rather than as it appears in a natural way from a model (Fig. 10).

7. Transformation of growth with time
A significant corollary of the model is the foreshortening of time and speeding up of history approaching the moment of the transition. The time compression nearing the critical time t₀ is best seen on a logarithmic scale. Mathematically the change in the effective time scale is best described by considering the instantaneous exponential time

(20)

of growth is introduced (see (6)). For the past of epoch B

(21)

The present is very close to t₀ and t_e simply equals the time before present (B.P.). For example, 1.6 million years ago with the onset of Epoch B in the lower Paleolithic marked change could happen only in a million years. The very slow pace of historical change is a phenomenon well known in anthropology although a plausible explanation has yet to be offered for it.

By the end of the Stone Age a marked change is associated with the Neolithic Revolution, a.k.a. Agricultural Revolution. This is the First Wave of Toffler and is the major watershed that marks the transition from a hunter-gatherer society to an agricultural one. The time of this transition is close to t_* = 11000 B.P., which on a logarithmic scale is near the halfway point. As the average pattern of growth is described by the model, there is no discontinuity in the world population. But by that time the growth rate is 10,000 times greater than at the beginning of the Paleolithic and the world population was approaching the 15 million, a number that is close to most estimates⁵.

The speedup of history is a kinematic feature of model, a directly following from the near-hyperbolic growth in Epoch B. As the quasi-singularity at t₀ is approached t_e(t) is no longer linear in t, but instead satisfies equation (20). By 1989 t_e passes through its minimum value of 58 years, corresponding to the relative growth of 1.7% per year or a doubling time of t₂ = 40 years. This marks the true speed of the transition and that afterwards an extended exponential growth is no longer kept up. As population stabilises, t_e rapidly settles down to the asymptotic form t_e ~ (t - t₀)²/t for t > t₀ (Fig. 11).

A more revealing perspective comes from placing the large-scale historical events on this time scale (Table 4). For example, the history of ancient Egypt spans 3000 years and ended 2 years B.P. According to Gibbon the decline and fall of Roman Empire took 1500 years³³; nowadays empires built in centuries collapse in decades, if not less.

This transformation of time is best seen if the main periods designated both in history and anthropology are plotted on a logarithmic scale. The tradition of history and observations of anthropologists identify major periods that on a logarithmic scale are more or less equally spaced from t_{_}= million years B.P. to t₀ = 2007. At both limits, the historical period and the more remote past, the periods seen by archaeologists cannot be established with any great accuracy as they are defined not by changes in population growth, but by the less objective, although meaningful criteria of technological and social changes. With an accuracy as best as one could expect each cycle is 2.5 to 3 times shorter than the previous one, each having a corresponding growth in population. This cyclic pattern can be described by a sequence of intervals of a geometric progression

(22)

where q is the integer part of ln |t - t '| - the number for the cycle from q = 0 to q = 11 = ln K, assuming that each cycle is shorter by e = 2.718 times. Then the whole duration of human development from t_- to t₀. is

(23)

This estimate is very close to t₀ - t_- = pK/2 = 1.571 Kt (Eq. 13). The slight discrepancy should be ascribed to a different way of determining the initial Epoch A dt₀ = 2.8 million years of the 0^th cycle.

During epoch B, In K = 11 cycles took place and through each cycle dy = 2K² = 9 billion people lived, as the duration of each period changed from 1 million years to 42 years. The invariant population step dy = 9 billion appears as a constant for this periodic sequence of major cycles. As a conjecture this periodicity could be ascribed to the inherent periodicity of the logarithmic function for a complex variable, but in that case a phase has to be ascribed to y_B(t - t_-) = K² ln (t - t_-) of unspecified origin; or these exponentially sequenced cycles could be bifurcations of a more complicated set of equations.

These global transitions happen more or less synchronously on a world-wide scale and the whole subject of the simultaneity of major periods in history has been extensively discussed by historians³⁴. Recently D'yakonov in a review of human history³⁵ indicates that the sequence of gross features of our past explicitly follow a geometric progression, culminating in a singularity, but offers, apart from this insightful observation, no explanation for this phenomenon.

The simultaneity of transitions is due to the interactions in the world population system and this is indicative of its systemic behaviour over very extensive epochs. The time of transition is usually marked by the first appearance of a new technology or social changes, which then spreads throughout the world. Probably Kondratieff was the first to indicate such cycles³⁶. The hyperbolic periodicity is seen only on a logarithmic scale, but it extends throughout our history.

From the systemic process of global population growth it follows that any long term breakup of the global community will lead to a slowing down of development in any lesser part of the greater system. Even today in isolated communities societies at a neolithic or even Paleolithic stage of development can be found. Probably the most instructive case is that of the separation of the Americas from the Eurasian continent. This led to a slower rate of development for the pre-Columbian civilisations and came to an abrupt and tragic end with the European discovery of the New World.

8. Stability of growth and development
Of importance is the stability of growth. If the standard Lyapunoff criteria of systemic stability is applied to the human system, then a variation dy will grow as dy = dy₀ e^lt. For the Lyapunoff exponent l the following expression

(24)

is obtained by differentiating (14). The instability reaches its maximum value between t_- and t₀, y = pK/4 at t = -1 in 1965 when l_max = 1, halfway to y_max. Only after y₀ does change sign and systemically stable development if possible. In spite of this instability world population growth is in general stable and the reasons for this have to be understood.

The trend towards stable development is demonstrated by the large scale behaviour of the human population system. The system was destabilised by at least two global events - the Plague in the 14^th century and World Wars I and II in the 20^th century. Demographic data show that after these major disasters, when in Europe from 30 to 40% of the population died and with a 10% loss of the global population during the World Wars, the global system rapidly regained its losses and a generally stable pattern of development is sustained.

In The Economic Consequences of the Peace³⁶ Keynes cites as one of the reasons for WWI the rapid buildup of population in the warring countries. The collapse of global security then showed many signs of a systemic loss of stability. It may be simplistic to interpret history in such mechanistic terms, but a predisposition to a loss of stability, as indicated above, should be kept in mind. At present the developing world is now in a similar stage, with 2-digit growth figures for the economy of China and the 6 to 7% growth of India, indicating that these countries passing through the demographic transition may become a source of global insecurity.

Major instabilities cannot be predicted (in a straightforward fashion), but a critical predisposition should not pass unnoticed. In the developing world of today, changes are happening twice as fast as in the developed world at a similar stage of growth and the population involved is 15 times greater. Probably of all global problems that of security is of the greatest importance, while both an understanding of these threats and an effort to avert general instability should be uppermost on the global agenda, for no one can afford a world war of a scale up magnitude of the previous 20^th century conflicts. In considering military, economic and ecological factors of global security, the demographic factor is the first to be assessed. In this case not only quantitative factors, but qualitative, including ethnic ones, should be accounted for.

An estimate of fluctuations to be expected in the global population system for the present world population were conjectured in⁹. In relative terms the fluctuations were largest at the beginning of quadratic growth a million years ago.

(25)

For the stability of the world population system the spatial distribution should be taken into account. If diffusion is introduced into the kinetic equation one can expect a damping of systemic instabilities, as the eigenvalues of the Laplacian Ѳy are negative ¹⁷. This will require a more detailed analysis of fluctuations and instabilities of the solutions of partial differential equations describing changes in space and time.

It is well known that the distribution of populations in space is far from uniform, whether on a regional scale or of nations and cities. The distribution of the population of France has recently been studied by Le Bras, using the multifractal method.³⁷. The urban concentration of people in cities can be described by a hyperbolic distribution U(R)

(26)

where R is the rank of a city with U people, in spite of the difficulty in defining the population of a city in the present or past. The fractal distribution is well established for R > ln U₀. This expression is valid from U₀ - the population of the largest city in the world downwards to U_min = 1 as the lower limit (Fig. 13). By integrating (26) from R = 0 to R_max = U₀ ln U₀ at U_min = 1 the population of the world is normalised y = U₀ ln² U₀, so U₀ and U(R) can be found from y.

The distribution of cities is described on a global scale without introducing any new parameters. For example, at the start of the 1^st century y ~ 200 million for ancient Rome, where, according to historic data the Coliseum could seat 50,000, we get U₀ ~ 1 million. It should be noted that U₀/y = 1/ln2 U₀ ~ 0.4% does not on the average vary for the model. For obvious reasons (26) is not applicable to a separate country, where locally fractal distributions are valid only asymptotically, with large cities excluded³⁹. The way U₀ is brought in may be conjectured, rather than proven by assuming ln U₀ as the natural inherent unit of scaling for ranking cities and justifying the statistical approach to the global system⁵¹.

Incomes in a society also follow a power law - Pareto's law - indicating the non-equilibrium pattern of the distribution of wealth. These fractal laws should be taken into account when future projections for humankind are considered. The underlying reason for these distributions may be seen in the correlations, both spatial and temporal that are established in a developing system, as indicated by Scarrot³⁹. In the global population system a number of instabilities may develop and various stabilising factors, migration in the (26) first place operate. This is best demonstrated by the large scale cycles observed in the human population system, although a whole hierarchy of instabilities of a lesser scale do develop.

9. Influence of the enviroment
From the cyclic pattern of our growth, even without going into the mathematical details of modelling, it follows that a major period in the population growth and development has now come to an end. On the other hand, we see that all through the ages the human population system can be seen as a single and open entity. That the system is open means that the external resources do not directly affect growth, have not limited this growth in the past and should not in the foreseeable future.

The approaching population limit is determined by internal systemic factors, factors operating at all times for more than a million years, over many cycles of our development. At all stages of growth humankind always had enough resources for sustained development. If at any point, resources were lacking, then by migrating to other lands new resources were brought into play and growth continued unabated. During each cycle of development fewer and fewer people were engaged in feeding the population, so that today in advanced societies 2-3% can feed the whole country. According to estimates by FAO in principle the world at present could feed 20-25 million people, roughly twice the expected limit.

Conventional Malthusian wisdom says that demographic transitions are due to the depletion of resources. But a growing body of evidence points to the opposite and only by treating the human population as an evolving system can we obtain some insight into the reasons for the demographic transition⁴². Recently, Le Bras illustrated this point in a number of cases, showing that a naive Malthusian limit of resources cannot explain the transition⁴³.

Moreover, in the future it seems that such resources will be available and in principle allow for a crossover unhindered by resources of humankind through the global demographic transition. In this case the population from 5.7 billion at the time of writing (6 billion in 1999) will grow by less than 2.5 times to reach the limit of 14 billion in Model III, which is only double in the next hundred years; or just 10 billion in Model X, which is less than 2 times the population.

In other words the model shows that the world population growth in the first approximation is independent of resources. If a cut off of resources should come, the first reaction of the global population system would be a much more uniform worldwide distribution of people, signaled by a change in constants, that describe growth. The independence from global resources may be stated as the principle of the demographic imperative, as the expression of the fundamental independence of growth in an open model.

The non-uniformity of the use of resources is in a sense amazing. A good example is the comparison of India and Argentina. If the area of the subcontinent is only 30% larger, the population of India is 30 times greater than in Argentina. There conditions are such that, if modern agricultural methods were used, by exploiting the resources to the limit it could feed the whole world. The non-uniform distribution of population and resources existed also in the past, for at the height of the Roman Empire more than a million people lived in Rome.

The concepts of the systemic model contradict most of the assumptions of "Limits to Growth"⁴⁴. In the last publication of Medows, special importance is ascribed to exponential growth. Data and modelling show that exponential growth hardly ever finds its expression in a pattern of development for any length of time. It should be noted that the Club of Rome in its last and very significant report "The First Global Revolution" by King and Schneider has definitely parted with the initial Malthusian reductionist pronouncements of the Club, and moved towards a more synthetic and holistic approach to describing the destiny of humankind^45,46.

On the agenda of these studies for the foreseeable future an important issue will be the new criteria of growth, that are to appear in a stabilised world population. Will this state be stable and what will be the measure of growth? With a pronounced change in the age structure of society a profound change in values and connections between generations is to be expected. Greater social security expenditures and educational outlays can be anticipated. Of significance is the distribution of wealth and its global evolution. These issues are beyond the scope of the present model, but the general change in the pattern of growth will set the scene for this emerging new world of a stabilised global population, if and when it is to happen.

10. Conclusions
For the population of the world the model provides a description of the gross features of growth and evolution for humankind. The invariance of the self-similar pattern of growth, where inherent limits are set, is the main feature and result of the modelling, when a bare minimal number of constants are introduced.

The model provides for a phenomenological, macroscopic treatment but does not profess to explain in detail the processes leading to growth. The model is an open one and growth is not explicitly dependent on resources as it is the result of cooperative interactions of all relevant forms of human activities. This interaction, summing up all contributions to development that can be seen as the ultimate mechanism of growth.

The model is justified not only by the extent to which the results of modelling correspond to the facts of life, but also by the fundamental principles of systemic growth that it is based on. The concept of self-similar growth is an expression of systemic dynamics, initially developed by Haken and Prigogine and now applied to the description of global population growth. Describing the overall process of development by an essentially nonlinear model it should be kept in mind that it cannot be directly applied to local or regional growth. But the global process of development does definitely influence any of its parts by the connections and interactions implied in the world model.

The transformation of the effective time scale is a significant result of the theory, a kinematic consequence of self-accelerated growth. The moment from which time should be measured is set and the scale changes as we go into the past, corresponding to the intuitive insight of historians and anthropology on past cycles. The model indicates that humankind is now rapidly passing through a critical period. A fundamental change in the growth paradigm is occurring, a change never experienced before. Some historians have pronounced the end of history⁴⁸. Today we are witnessing a much more profound transformation, a critical period compressed into a remarkably short time of drastic global changes⁴⁹.

In this study the demographic factor is taken to be decisive. Until recently the approach through demography to global problems was to a great extent blocked by some parties and excluded from most of the international debate. Now this has hopefully changed⁵⁰. As these discussions excite high feelings on matters concerning our common future and the issues at stake are great, it is most necessary to develop and foster interdisciplinary research, following different intellectual traditions. In these studies mathematical modelling should be seen not only as a useful tool, but hopefully as a theoretical basis for analysis and projections of world population growth and the consequences it may have for sustainable development and global security.

In the foreseeable future we can hardly expect to significantly change and influence the overall growth pattern. The sheer size of the world population and given the pace of events it is difficult to imagine how the world community can have a major effect on global population growth. The fundamental understanding of growth is still rather limited and definitive advice for action is hard to provide, apart from very general recommendations. Probably the most important issue is by all means to ensure the stability and security for the world to be, as the prerequisite for resolving global problems.

The original author's statements have to be modified in light of the latest developments throughout the world. In fact, what originally may have looked like a growth up to 14 billion, now shows itself, given the new data provided, to be a growth only up to 10 billion. The growth rates in recent years have come down substantially, far in excess of what the UN initially estimated. This development may seem like a mystery upon first sight, but the mystery is resolved when it's juxtaposed alongside another, equally significant, development.

In recent years, throughout the world, the proportion of college-level students that are female has risen substantially, not just up to 50%, but in many places both in the developed world and elsewhere, well beyond 50%, into the 60's and even 70's. Part of the reason for this occurrence is tied to a progressive loss of direction and focus of the male half of the human race, tied to the rise of Cabin Fever, manifesting in the form of the Rapa Nui Symdrome, particularly since the closing of the age of exploration. As a symptom, it points to the growing restlessness of the human race, and its surging desire for a new era of exploration.

But another, equally substantial, part of the story is found by looking at the actual numbers of students attending colleges over time throughout the world. It's not so much that males are dropping out. They are still attending colleges at roughly the same rate as before (and are, therefore, also stagnating in a world that has become increasingly knowledge-dependent). It's that the number of females attending colleges has skyrocketed.

What does this have to do with anything, especially the population issue? In retrospect the connection is obvious and should have been foreseen long ago. Women who are going to college are deferring childbirth. Nations with an increased number of females attending college also have a birth rate that is dropping through the floor. Even places like Egypt, India, Pakistan, once major centers of population growth, the growth rate is down under 1%/year. As true as it is that men have lost their classical birthright in recent years -- the directive to go forth and explore new horizons -- these developments show clearly that women too have lost their classical birthright -- bringing new life into an increasingly overcrowded world -- and that both developments are tied directly to the population curve. In the case of women, with the prospect of childrearing becoming less accessible they are compensating by doing the only other thing readily available to them: going to college and entering the ranks of the professonal elite. Whereas all the male energies that drive forth the directive to explore have been thwarted and are now manifesting in forms that are the classical red-flag pattern of the Rapa Nui Syndrome, the female energies, having been released by loss of the child rearing prerogative, are being diverted into education and the professions, with the surge of all these energies will overwhelming these sectors in a tidal wave of estrogen.

This development has engineered a downward jolt in the entire evolution of the population curve. Birthrates are dropping everywhere, and the result is that the population explosion is coming to a screeching halt. It will not get much above 10 billion.

The original author stated that if the model is to be supported by further research (as has just been done here), and the insight it provides is valid, then it may help to lead to greater understanding of the present state of affairs. Indeed, the growing contrast between the model and what's actually occurring puts the other developments just described in a whole new light. The curve, itself, must still stand against its greatest challenge: the rival assumption of a logistic curve. It turns out, when fitting the recent population data⁵¹, 1950-2004 to a logistic curve, that no logistic curve starting at 0 will fit well. Instead, the best fitting curve requires a positive starting value of around 1 billion set in recent time, around the start of the Industrial Revolution. Correspondingly, the asymptotic value will be reduced to 9 billion. The resulting curve fits the recent history extremely well and shows that the population explosion of the past 150 years should really be considered as an entirely separate development, corollary to the Industrial Revolution, rather than a continuation of the population increases that took place before. Trying to fit something to the curve on both sides of the Industrial Revolution will achieve limited results, at best.

The difference between the best-fitting logistic curve and the one posed by the original author will become readily apparent as time progresses, and one will be able to decide which model is more appropriate probably by 2010 or earlier still.

The original author continues on to state the following: the model posed here can offer a common frame of reference for anthropology and history, demography and sociology, for studies in human evolution and genetics. For doctors and politicians it may elicit an understanding of the sources of stress and tensions in this transient period, so unique in human development, both for an individual and at a broader societal level. In this case interdisciplinary studies and experiences are worth the effort in developing the model and the promise it can provide in facing the predicament of humankind.

Indeed, a proper understanding of the population curve, and its relation to the threshold phenomena corresponding to what the futurist Toffler termed "waves", provides an underpinning to the sequel to the Tofflerian trilogy currently being developed: the Federation Series. It is the focal point that ties together the central developments that each volume covers. The cabin fever underlying the decline of males is the central theme of the second volume, The Fourth Wave. This has arisen as a consequence of the closing up of the world, which in turn may be seen as a direct consequence of the demographic transition the world has been undergoing during this time. The rise of women within the ranks of the educated and professional elite, particularly when seen in conjunction with the decline of males, is the central theme of the first volume, The Fall Of Mankind. Finally, the outcomes of these developments, the emergence of a world federation and of a spacefaring civilization spanning the inner solar system, is the central focus of the third volume, Progeny.

This work would not have been possible without the valuable discussions and support of many colleagues. I would specially like to thank G Barenblatt, Y Coppens, D Crighton, G Friedlander, A Gaponov-Grekhov, O Gazenko, V Ginzburg, V Goldin, A Gonchar, B Hess, B Kadomtzev, N Keyfitz, S Kurdiumov, H Le Bras, P Morrison, L Pitaevsky, I Prigogine, A Vishnevskii, N Vorontsov and V Weisskopf for inspiration and advice. The support by UNESCO, the Royal Society of London, of the INTAS and Soros Foundations is gratefully acknowledged.

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