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Fig. 3. Cognitive scheme of the nonlinear second order positive feedback be-tween technological development and demographic growth
Note that the relationship between technological development and demographic growth cannot be analyzed through any simple cause-and-effect 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.
Note also that the process discussed above should be identified with the process of collective learning (on the notion of ‘collective learning’ see first of all Christian 2005: 146–148; see also David Christian's and David Baker's contributions to the present volume). Respectively, the mathematical models of the World System development discussed in this article can be interpreted as mathematical models of the influence of collective learning on the global social evolution. Thus, a rather peculiar hyper-bolic shape of the acceleration of the global development observed prior to the early 1970s may be regarded just as a product of the global collec-tive learning. Elsewhere we have also shown (Korotayev, Malkov, and
196 Mathematical Modeling of Big History Phases
Khaltourina 2006a: 34–66) that for the period prior to the 1970s the World System economic and demographic macrodynamics driven by the above mentioned positive feedback loops can be described mathematically in a rather accurate way with the following extremely simple mathematical model:

dN	aSN ,	(5)
dt
dS	bNS ,	(6)
dt

while the world GDP (G) can be calculated using the following equa-tion:

G = mN + SN,

(7)

where G is the world GDP, N is population, and S is the produced sur-plus per capita, over the subsistence amount (m) that is minimally nec-essary to reproduce the population with a zero growth rate in a Malthu-sian system (thus, S = g – m, where g denotes per capita GDP); a and b are parameters.

Note that the mathematical analysis of the basic model (not pre-sented here) suggests that up to the 1970s 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 popula-tion: S = kN. Our statistical analysis of the available empirical data has confirmed this theoretical proportionality (Korotayev, Malkov, and Khaltourina 2006a: 49–50). Thus, in the right-hand side of equation (6) S can be replaced with kN, and as a result we arrive at the following equation:

dN	kaN ^2 .	(3)
dt

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

Nt		C	,	(1)
	(t	t)
	0

and this produces simply a hyperbolic curve.

As, according to our model, S can be approximated as kN, its long-term dynamics can be approximated with the following equation:

S	kC	.
	t0 t

Thus, the long-term dynamics of the most dynamic component of the world GDP, SN, ‘the world surplus product’, can be approximated as follows:

Andrey V. Korotayev and Alexander V. Markov			197
	kC ^2
SN		.	(8)
	t0 t ^2

Of course, this suggests that the long-term 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 below (see Fig. 4), this ap-proximation works very effectively indeed:
18000

16000
14000

12000
10000
8000
6000
4000

2000	predicted

0								observed
0	250	500	750	1000	1250	1500	1750	2000

Fig. 4. World GDP Dynamics, 1–1973 CE (in billions of 1990 international dol-lars, PPP): the fit between predictions of a quadratic-hyperbolic model and the observed data
Note: R = .9993, R² = .9986, p << .0001. The black markers correspond to Maddi-son's (2001) estimates (Maddison's estimates of the world per capita GDP for 1000 CE has been corrected on the basis of [Meliantsev 2004]). The grey solid line has been generated by the following equation:

G	17749573.1		.
		( 2006t ) ^2

Thus, up to the 1970s the hyperbolic growth of the world population was accompanied by the quadratic-hyperbolic growth of the world GDP, just as our model suggests. Note that the hyperbolic growth of the world population and the quadratic-hyperbolic growth of the world GDP are tightly interconnected processes, actually two sides of the same coin, two dimensions of one process propelled by the nonlinear second

198 Mathematical Modeling of Big History Phases
order positive feedback loops between the technological development and demographic growth (see Fig. 5).



Fig. 5. Cognitive Scheme of the Generation of Quadratic-Hyperbolic Trend of the World Economic Growth by the Nonlinear Second Order Positive Feedback between Technological Development and Demographic Growth
We have also demonstrated (Korotayev, Malkov, and Khaltourina 2006a: 67–80) that the dynamics of the World System population's literacy (l) is rather accurately described by the following differential equation:

dl	aSl (1	l ),	(9)
dt

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 increasing literacy 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 educa-tional 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 9 can be regarded logistic where saturation is reached at literacy level l = 1, and S is responsible for the speed with which this level is approached.

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It is important to emphasize that with low values of l (which corre-spond to most part of human history except for the recent decades), the increasing rate of the world literacy generated by this model (against the background of hyperbolic growth of S) can be approximated rather accurately as hyperbolic (see Fig. 6).

70
60	observed
50	predicted
40
30
20
10
0
0	500	1000	1500	2000

Fig. 6. World Literacy Dynamics, 1 – 1980 CE (%%): the fit between predictions of the hyperbolic model and the observed data
Note: R = 0.997, R² = 0.994, p << 0.0001. Black dots correspond to UNESCO/World Bank (2014) estimates for the period after 1970, and to Meli-antsev's (2004) estimates for the earlier period. The grey solid line has been gen-erated by the following equation:

lt		3769.264		.
		(2040	t ) ^2

The best -fit values of parameters С (3769.264) and t0 (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 experi-enced a hyperbolic growth until the 1960s/1970s, one has sufficient grounds to expect that until recently the overall number of literate peo-ple in the world (L)⁹ grew not just hyperbolically, but rather in a quad-ratic-hyperbolic way (as the world GDP did). Our empirical test has confirmed this – the quadratic-hyperbolic model describes the growth of

9. 
   Since literacy appeared, almost all of the Earth's literate population has lived within the World System; hence, the literate population of the Earth and the literate population of the World System have been almost perfectly synonymous.
   

200 Mathematical Modeling of Big History Phases

the literate population of the planet with an extremely good fit indeed (see Fig. 7).

1800
1600	observed
1400	predicted
1200
1000
800
600
400
200
0
0	500	1000	1500	2000

Fig. 7. World Literate Population Dynamics, 1–1980 CE (L, millions): the fit be-tween predictions of the quadratic-hyperbolic model and the observed data
Note: R = 0.9997, R² = 0.9994, p << 0.0001. The black dots correspond to UNESCO/World Bank (2014) estimates for the period since 1970, and to Meliantsev's (2004) estimates for the earlier period; we have also taken into account the changes of age structure on the basis of UN Population Division (2014) data. The grey solid line has been generated by the following equation:

^Lt		4958551		.
		(2033	t)^2

The best-fit values of parameters С (4958551) and t0 (2033) have been calculated with the least squares method.
Similar processes are observed with respect to world urbanization, whose macro dynamics appears to be described by the differential equa-tion:
^du bSu (u _lim u ) , dt
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 techno-logical development, b is a constant, and ulim is the maximum possible proportion of the urban population. Note that this model implies that during the blow-up regime of the ‘Malthusian-Kuznetsian’ era, the hy-perbolic growth of world urbanization must have been accompanied by a quadratic-hyperbolic growth of the urban population of the world, which is supported by our empirical tests (see Figs 8–9).

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