Diagram 0.8. Block Scheme of the Nonlinear Second Order Positive
Feedback between Technological Development
and Demographic Growth (version 1)
In fact, this positive feedback can be graphed even more succinctly (see Diagram 0.9a):
Diagram 0.9a. Block Scheme of the Nonlinear Second Order Positive
Feedback between Technological Development
and Demographic Growth (version 2)
Note that the relationship between technological development and demographic growth cannot be analyzed through any simple causeandeffect model, as we observe a true dynamic relationship between these two processes – each of them is both the cause and the effect of the other.
It is remarkable that Kremer's model suggests ways to answer one of the main objections raised against the hyperbolic models of the world's population growth. Indeed, at present the mathematical models of world population growth as hyperbolic have not been accepted by the academic social science community [The title of the most recent article by a social scientist discussing Kapitza's model, "Demographic Adventures of a Physicist" (Shishkov 2005), is rather telling in this respect]. We believe that there are substantial reasons for such a position, and that the authors of the respective models are as much to blame for this rejection as are social scientists.
Indeed, all these models are based on an assumption that world population can be treated as having been an integrated system for many centuries, if not millennia, before 1492. Already in 1960, von Foerster, Mora, and Amiot spelled out this assumption in a rather explicit way:
"However, what may be true for elements which, because of lack of adequate communication among each other, have to resort to a competitive, (almost) zerosum multiperson game may be false for elements that possess a system of communication which enables them to form coalitions until all elements are so strongly linked that the population as a whole can be considered from a gametheoretical point of view as a single person playing a twoperson game with nature as its opponent" (von Foerster, Mora, and Amiot 1960: 1292).
However, did, e.g., in 1–1500 CE, the inhabitants of, say, Central Asia, Tasmania, Hawaii, Terra del Fuego, the Kalahari etc. (that is, just the world population) really have "adequate communication" to make "all elements… so strongly linked that the population as a whole can be considered from a gametheoretical point of view as a single person playing a twoperson game with nature as its opponent"? For any historically minded social scientist the answer to this question is perfectly clear and, of course, it is squarely negative. Against this background it is hardly surprising that those social scientists who have happened to come across hyperbolic models for world population growth have tended to treat them merely as "demographic adventures of physicists" (note that indeed, nine out of eleven currently known authors of such models are physicists); none of the respective authors (von Foerster, Mora, and Amiot 1960; von Hoerner 1975; Kapitza 1992, 1999; Kremer 1993; Cohen 1995; Podlazov 2000, 2001, 2002, 2004; Johansen and Sornette 2001; Tsirel 2004), after all, has provided any convincing answer to the question above.
However, it is not so difficult to provide such an answer.
The hyperbolic trend observed for the world population growth after 10000 BCE does appear to be primarily a product of the growth of quite a real system, a system that seems to have originated in West Asia around that time in direct connection with the Neolithic Revolution. With Andre Gunder Frank (1990, 1993; Frank and Gills 1994), we denote this system as "the World System" (see also, e.g., Modelski 2000, 2003; Devezas and Modelski 2003). The presence of the hyperbolic trend itself indicates that the major part of the entity in question had some systemic unity, and the evidence for this unity is readily available. Indeed, we have evidence for the systematic spread of major innovations (domesticated cereals, cattle, sheep, goats, horses, plow, wheel, copper, bronze, and later iron technology, and so on) throughout the whole North African – Eurasian Oikumene for a few millennia BCE (see, e.g., Chubarov 1991, or Diamond 1999 for a synthesis of such evidence). As a result, the evolution of societies of this part of the world already at this time cannot be regarded as truly independent. By the end of the 1^{st} millennium BCE we observe a belt of cultures, stretching from the Atlantic to the Pacific, with an astonishingly similar level of cultural complexity characterized by agricultural production of wheat and other specific cereals, the breeding of cattle, sheep, and goats; use of the plow, iron metallurgy, and wheeled transport; development of professional armies and cavalries deploying rather similar weapons; elaborate bureaucracies, and Axial Age ideologies, and so on – this list could be extended for pages). A few millennia before, we would find another belt of societies strikingly similar in level and character of cultural complexity, stretching from the Balkans up to the Indus Valley outskirts (Peregrine and Ember 2001: vols. 4 and 8; Peregrine 2003). Note that in both cases, the respective entities included the major part of the contemporary world's population (see, e.g. McEvedy and Jones 1978; Durand 1977 etc.). We would interpret this as a tangible result of the World System's functioning. The alternative explanations would involve a sort of miraculous scenario – that these cultures with strikingly similar levels and character of complexity somehow developed independently of one another in a very large but continuous zone, while for some reason nothing comparable to them appeared elsewhere in the other parts of the world, which were not parts of the World System. We find such an alternative explanation highly implausible.
Thus, we would tend to treat the world population's hyperbolic growth pattern as reflecting the growth of quite a real entity, the World System.
A few other points seem to be relevant here. Of course there would be no grounds for speaking about a World System stretching from the Atlantic to the Pacific, even at the beginning of the 1^{st} millennium CE, if we applied the "bulkgood" criterion suggested by Wallerstein (1974, 1987, 2004), as there was no movement of bulk goods at all between, say, China and Europe at this time (as we have no reason to disagree with Wallerstein in his classification of the 1^{st} century Chinese silk reaching Europe as a luxury rather than a bulk good). However, the 1^{st} century CE (and even the 1^{st} millennium BCE) World System definitely qualifies as such if we apply the "softer" informationnetwork criterion suggested by ChaseDunn and Hall (1997). Note that at our level of analysis the presence of an information network covering the whole World System is a perfectly sufficient condition, which makes it possible to consider this system as a single evolving entity. Yes, in the 1^{st} millennium BCE any bulk goods could hardly penetrate from the Pacific coast of Eurasia to its Atlantic coast. However, the World System had reached by that time such a level of integration that iron metallurgy could spread through the whole of the World System within a few centuries.
Yes, in the millennia preceding the European colonization of Tasmania its population dynamics – oscillating around the 4000 level (e.g., Diamond 1999) – were not influenced by World System population dynamics and did not influence it at all. However, such facts just suggest that since the 10^{th} millennium BCE the dynamics of the world population reflect very closely just the dynamics of the World System population.
On the basis of Kremer's model we (Korotayev, Malkov, and Khaltourina 2006a: 34–66) have developed a mathematical model that describes not only the hyperbolic world population growth, but also the macrodynamics of the world GDP production up to 1973:
,

(0.11)

,

(0.13)

,

(0.12)

where G is the world GDP, T is the World System technological level, N is population, and S is the surplus produced, per person, over the amount (m) minimally necessary to reproduce the population with a zero growth rate in a Malthusian system (thus, S = g – m, where g denotes per capita GDP); k_{1}, k_{2}, k_{3}, and α (0 < α < 1) are parameters.
We have also shown (Korotayev, Malkov, and Khaltourina 2006a: 34–66) that this model can be further simplified to the following form:
,

(0.13)

,

(0.14)

while the world GDP (G) can be calculated using the following equation:
Note that the mathematical analysis of the basic model (0.11)(0.13)(0.12) suggests that during the "MalthusianKuznetsian" macroperiod of human history (that is, up to the 1960s) the amount of S (per capita surplus produced at the given level of World System development) should be proportional, in the long run, to the World System's population: S = kN. Our statistical analysis of available empirical data has confirmed this theoretical proportionality (Korotayev, Malkov, and Khaltourina 2006a: 49–50). Thus, in the righthand side of equation (0.13) S can be replaced with kN, and as a result we arrive at the following equation:

(0.9)^{25}

As we remember, the solution of this type of differential equations is
,

(0.1)

and this produces simply a hyperbolic curve.
As, according to our model, S can be approximated as kN, its longterm dynamics can be approximated with the following equation:
.

(0.16)

Thus, the longterm dynamics of the most dynamic component of the world GDP, SN, "the world surplus product ", can be approximated as follows:
.

(0.17)

Of course, this suggests that the longterm world GDP dynamics up to the early 1970s must be approximated better by a quadratic hyperbola than by a simple one; and, as we could see above (see Diagram 0.7), this approximation works very effectively indeed.
Thus, up to the 1970s the hyperbolic growth of the world population was accompanied by the quadratichyperbolic growth of the world GDP, just as is suggested by our model. Note that the hyperbolic growth of the world population and the quadratic hyperbolic growth of the world GDP are very tightly connected processes, actually two sides of the same coin, two dimensions of one process propelled by the nonlinear second order positive feedback loops between the technological development and demographic growth (see Diagram 0.9b):
Diagram 0.9b. Block Scheme of the Nonlinear Second Order Positive
Feedback between Technological Development
and Demographic Growth (version 3)
We have also demonstrated (Korotayev, Malkov, and Khaltourina 2006a: 67–80) that the World System population's literacy (l) dynamics are rather accurately described by the following differential equation:
,

(0.18)

where l is the proportion of the population that is literate, S is per capita surplus, and a is a constant. In fact, this is a version of the autocatalytic model. It has the following sense: the literacy growth is proportional to the fraction of the population that is literate, l (potential teachers), to the fraction of the population that is illiterate, (1 – l) (potential pupils), and to the amount of per capita surplus S, since it can be used to support educational programs (in addition to this, S reflects the technological level T that implies, among other things, the level of development of educational technologies). Note that, from a mathematical point of view, equation (0.18) can be regarded as logistic where saturation is reached at literacy level l = 1, and S is responsible for the speed with which this level is being approached.
It is important to stress that with low values of l (which would correspond to most of human history, with recent decades being the exception), the rate of increase in world literacy generated by this model (against the background of hyperbolic growth of S) can be approximated rather accurately as hyperbolic (see Diagram 0.10):
Diagram 0.10. World Literacy Dynamics, 1 – 1980 CE (%%):
the fit between predictions of the hyperbolic model
and the observed data
NOTE: R = 0.997, R^{2} = 0.994, p << 0.0001. Black dots correspond to UNESCO/World Bank (2005) estimates for the period since 1970, and to Meliantsev's (1996, 2003, 2004a, 2004b) estimates for the earlier period. The grey solid line has been generated by the following equation:
.
The bestfit values of parameters С (3769.264) and t_{0} (2040) have been calculated with the least squares method.
The overall number of literate people is proportional both to the literacy level and to the overall population. As both of these variables experienced hyperbolic growth until the 1960s/1970s, one has sufficient grounds to expect that until recently the overall number of literate people in the world (L)^{26} was growing not just hyperbolically, but rather in a quadratichyperbolic way (as was world GDP). Our empirical test has confirmed this – the quadratichyperbolic model describes the growth of the literate population of this planet with an extremely good fit indeed (see Diagram 0.11):
Diagram 0.11. World Literate Population Dynamics, 1 – 1980 CE (L, millions): the fit between predictions of the quadratichyperbolic model and the observed data
NOTE: R = 0.9997, R^{2} = 0.9994, p << 0.0001. The black dots correspond to UNESCO/World Bank (2006) estimates for the period since 1970, and to Meliantsev's (1996, 2003, 2004a, 2004b) estimates for the earlier period; we have also taken into account the changes of age structure on the basis of UN Population Division (2006) data. The grey solid line has been generated by the following equation:
.
The bestfit values of parameters С (4958551) and t_{0} (2033) have been calculated with the least squares method.
Similar processes are observed with respect to world urbanization, the macrodynamics of which appear to be described by the differential equation:
,

(0.19)

where u is the proportion of the population that is urban, S is per capita surplus produced with the given level of the World System's technological development, b is a constant, and u_{lim} is the maximum possible proportion of the population that can be urban. Note that this model implies that during the "MalthusianKuznetsian" era of the blowup regime, the hyperbolic growth of world urbanization must have been accompanied by a quadratichyperbolic growth of the urban population of the world, which is supported by our empirical tests (see Diagrams 0.12–13):
Diagram 0.12. World Megaurbanization Dynamics (% of the world population living in cities with > 250 thousand inhabitants), 10000 BCE – 1960 CE: the fit between predictions of the hyperbolic model and empirical estimates
NOTE: R = 0.987, R^{2} = 0.974, p << 0.0001. The black dots correspond to estimates of Chandler (1987), UN Population Division (2005), and White et al. (2006). The grey solid line has been generated by the following equation:
.
The bestfit values of parameters С (403.012) and t_{0} (1990) have been calculated with the least squares method. For a comparison, the best fit (R^{2}) obtained here for the exponential model is 0.492.
Diagram 0.13. Dynamics of World Urban Population Living in Cities with > 250000 Inhabitants (mlns.), 10000 BCE – 1960 CE: the fit between predictions of the quadratichyperbolic model and the observed data

NOTE: R = 0.998, R^{2} = 0.996, p << 0.0001. The black markers correspond to estimates of Chandler (1987), UN Population Division (2005), and White et al. (2006). The grey solid line has been generated by the following equation:
.
The bestfit values of parameters С (912057.9) and t_{0} (2008) have been calculated with the least squares method. For a comparison, the best fit (R^{2}) obtained here for the exponential model is 0.637.

Within this context it is hardly surprising to find that the general macrodynamics of the size of the largest settlement within the World System are also quadratichyperbolic (see Diagram 0.14):
Diagram 0.14. Dynamics of Size of the Largest Settlement of the World (thousands of inhabitants), 10000 BCE – 1950 CE: the fit between predictions of the quadratichyperbolic model and the observed data

NOTE: R = 0.992, R^{2} = 0.984, p << 0.0001. The black markers correspond to estimates of Modelski (2003) and Chandler (1987). The grey solid line has been generated by the following equation:
.
The bestfit values of parameters С (104020618,5) and t_{0} (2040) have been calculated with the least squares method. For a comparison, the best fit (R^{2}) obtained here for the exponential model is 0.747.

As has been demonstrated by crosscultural anthropologists (see, e.g., Naroll and Divale 1976; Levinson and Malone 1980: 34), for preagrarian, agrarian, and early industrial cultures the size of the largest settlement is a rather effective indicator of the general sociocultural complexity of a social system. This, of course, suggests that the World System's general sociocultural complexity also grew, in the "MalthusianKuznetsian" era, in a generally quadratichyperbolic way.
It is world literacy for which it is most evident that its hyperbolic growth could not continue, for any significant period, after the mid1960s; after all, the literacy rate by definition cannot exceed 100 per cent just by definition. What is more, since the 1970s the saturation effect^{27} described by our model started being felt more and more strongly and the rate of world literacy's growth began to slow (see Diagram 0.15):
Diagram 0.15. World Literacy Growth Dynamics, 1975 – 1995, the increase in percentage of adult literate population, by fiveyear periods
However, already before this, the hyperbolic growth of world literacy and of the other indicators of the human capital development had launched the process of diverging from the blowup regime, signaling the end of the era of hyperbolic growth. As has been shown by us earlier (Korotayev, Malkov, and Khaltourina 2006a: 67–86), hyperbolic growth of population (as well as of cities, schools etc.) is only observed at relatively low (< 0.5, i.e., < 50%) levels of world literacy. In order to describe the World System's demographic dynamics in the last decades (as well as in the near future), it has turned out to be necessary to extend the equation system (0.13)(0.14) by adding to it equation (0.21), and by adding to equation (0.13) the multiplier (1 – l), which results in equation (0.20), and produces a mathematical model that describes not only the hyperbolic development of the World System up to the 1960s/1970s, but also its withdrawal from the blowup regime afterwards:
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