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Mathematical Modeling of Biological and Social
Phases of Big History

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Mathematical Modeling of Biological and Social
Phases of Big History
Andrey V. Korotayev and Alexander V. Markov

Abstract
The present article demonstrates that changes in biodiversity through the Phanerozoic correlate with a hyperbolic model (widely used in demography and macrosociology) much more strongly than with exponential and logistic models (traditionally used in population biology and extensively applied to fossil biodi-versity as well). The latter models imply that changes in diversity are guided by a first-order positive feedback (more ancestors – more descendants) and/or a negative feedback arising from resource limitation. The hyperbolic model im-plies a second-order positive feedback. The authors demonstrate that the hyper-bolic pattern of the world population growth arises from a second-order positive feedback between the population size and the rate of technological growth (this can also be identified with the collective learning mechanism). The feedback between the diversity and community structure complexity can also contribute to the hyperbolic character of biodiversity. This suggests that some mechanisms vaguely resembling the collective learning might have operated throughout the biological phase of Big History. Our findings suggest that we can trace rather similar macropatterns within both the biological and social phases of Big History which one can describe in a rather accurate way with very simple mathematical models.
Keywords: biological phase of Big History, social phase of Big History, mathe-matical modeling, collective learning, positive feedback, biodiversity, demogra-phy, sociology, paleontology, geology, hyperbolic growth.
In 2005, in the town of Dubna, near Moscow, at what seems to have been the first ever international conference devoted specifically to Big History studies, the two authors of the present article – sociolo-gist/anthropologist Andrey Korotayev and biologist/paleontologist Alexander Markov – one after another demonstrated two diagrams.¹ One of those diagrams illustrated the dynamics of the population of China between 700 BCE and 1851 CE, the other illustrated the dynamics

1. 
   We would like to emphasize that we saw each other at that session for the first time, so we had no chance to arrange in advance the demonstration of those two diagrams.
   

Teaching & Researching Big History 188–219
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of marine Phanerozoic biodiversity during the last 542 million years (see Fig. 1):

a

b
Fig. 1. Similarity of the dynamics of Phanerozoic marine biodiversity and long-term population dynamics of China: а – Population dynamics of China (million people, 700 BCE – 1851 CE), based on estimates in Korotayev, Malkov, and Khaltourina (2006b: 47–88); b – Global change in marine biodiversity (number of genera, N ) through the Phanerozoic based on empirical data surveyed in Markov and Korotayev (2007a)
190 Mathematical Modeling of Big History Phases
Nevertheless, one can hardly ignore the striking similarity between two diagrams depicting the development of rather different systems (human population, on the one hand, and biota, on the other) at different time scales (hundreds of years, on the one hand, and millions of years, on the other) studied by different sciences (Historical Demography, on the one hand, and Paleontology, on the other) using different sources (demo-graphic estimates, on the one hand, and paleontological chronicles, on the other hand). What are the causes of this similarity in the develop-ment dynamics of rather different systems?
* * *
In 1960, von Foerster, Mora, and Amiot published a striking discov-ery in the journal Science. They showed that between 1 and 1958 CE the world's population (N) dynamics can be described in an extremely accu-rate way with an astonishingly simple equation:²

N t		C	,	(1)
	t0	t

where Nt is the world population at time t, and C and t0 are constants, with t0 corresponding to an absolute limit (‘singularity’ point) at which N would become infinite.
Of course, von Foerster and his colleagues did not imply that one day the world population would actually become infinite. The real im-plication was that prior to 1960 the world population growth for many centuries had followed a pattern which was about to come to an end and to transform into a radically different pattern. Note that this predic-tion started to come true only a few years after the ‘Doomsday’ paper had been published, because after the early 1970s the World System growth in general (and world population growth in particular) began to diverge more and more from the blow-up regime, and now it is not hyperbolic any more with its pattern being closer to a logistic one (see, e.g., Korotayev, Malkov, and Khaltourina 2006a, where we present a compact mathematical model that describes both the hyperbolic development of the World System in the period prior to the early 1970s, and its withdrawal from the blow-up regime in the subsequent period; see also Korotayev 2009).

² To be exact, the equation proposed by von Foerster and his colleagues looked as follows:

Nt		C		. However, as von Hoerner (1975) and Kapitza (1999) showed, it can be
	(t0	t)0.99
simplified as			N t		C		.
					t0	t

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Parameter t0 was estimated by von Foerster and his colleagues as 2026.87, which corresponded to November 13, 2006; this allowed them to give their article an attractive and remarkable title – ‘Doomsday: Fri-day, 13 November, A.D. 2026’.

The overall correlation between the curve generated by the von Fo-erster equation and the most detailed series of empirical estimates looks as follows (see Fig. 2).



Fig. 2. Correlation between empirical estimates of world population (in millions, AD 1000–1970) and the curve generated by the von Foerster equation
Note: black markers correspond to empirical estimates of the world population by McEvedy and Jones (1978) for the interval between 1000 and 1950 and the U.S. Bureau of the Census (2014) for 1950–1970. The grey curve has been generated by the von Foerster equation (1).
The formal characteristics are as follows: R = 0.998; R² = 0.996; p = = 9.4 × × 10^-17 ≈ 1 × 10^–16. For readers unfamiliar with mathematical sta-tistics we can explain that R² can be regarded as a measure of the fit be-tween the dynamics generated by a mathematical model and the em-pirically observed situation, and can be interpreted as the proportion of the variation accounted for by the respective equation. Note that 0.996 also can be expressed as 99.6 per cent.³ Thus, the von Foerster equation

3. 
   The second characteristic (p, standing for ‘probability’) is a measure of the correlation's statistical significance. A bit counter-intuitively, the lower the value of p, the higher the sta-tistical significance of the respective correlation. This is because p indicates the probabil-ity that the observed correlation could be accounted solely by chance. Thus, p = 0.99
   

192 Mathematical Modeling of Big History Phases

accounts for an astonishing 99.6 per cent of all the macrovariation in the world population, from 1000 CE through 1970, as estimated by McEvedy and Jones (1978) and the U.S. Bureau of the Census (2014).⁴
Note also that the empirical estimates of world population align in an extremely accurate way along the hyperbolic curve, which convinc-ingly justifies the designation of the pre-1970s world population growth pattern as ‘hyperbolic’.

To start with, the von Foerster equation	N t	C		is just a solution
		^t 0	t

of the following differential equation (see, e.g., Korotayev, Malkov, Khaltourina 2006a: 119–20):

dN		N	2		(2)
dt		C .

This equation can be also written as:

		dN	aN ^2	,		(3)
		dt
where a	1	.
	C				dN
What is the meaning of this mathematical expression,						aN ^2 ? In
					dt

our case, dN/dt denotes an absolute population growth rate at a certain moment of time. Thus, this equation shows that at any moment of time an absolute population growth rate should be proportional to the square of population at this moment.

Note that this significantly demystifies the problem of the world population hyperbolic growth. Now to explain this hyperbolic growth, we should just explain why for many millennia the absolute rate

indicates an extremely low statistical significance, as it means that there are 99 chances out of 100 that the observed correlation is the result of a coincidence, and, thus, we can be quite confident that there is no systematic relationship (at least, of the kind that we study) between the two respective variables. On the other hand, p = 1 × 10^–16 indicates an extremely high statistical significance for the correlation, as it means that there is only one chance out of 10000000000000000 that the observed correlation is the result of pure coincidence (in fact, a correlation is usually considered as statistically significant with p < 0.05).

4. 
   In fact, with slightly different parameters (С = 164890.45; t0 = 2014) the fit (R²) between the dynamics generated by the von Foerster equation and the macrovariation of world population for CE 1000–1970 as estimated by McEvedy and Jones (1978) and the U.S. Bu-reau of the Census (2014) reaches 0.9992 (99.92 per cent), whereas for 500 BCE – 1970 CE this fit increases to 0.9993 (99.93 per cent) (with the following parameters: С = 171042.78; t0 = 2016).
   

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of world population growth tended to be proportional to the square of population.

The main mathematical models of the hyperbolic pattern of the world's population growth (Taagapera 1976, 1979; Kremer 1993; Cohen 1995; Podlazov 2004; Tsirel 2004; Korotayev 2005, 2007, 2008, 2009, 2012; Korotayev, Malkov, and Khaltourina 2006a: 21–36; Khaltourina, Malkov, and Korotayev 2006; Golosovsky 2010; Korotayev and Malkov 2012) are based on the following two assumptions:

1. 
   ‘the Malthusian (1978 [1798]) assumption that population is lim-ited by the available technology, so that the growth rate of population is proportional to the growth rate of technology’ (Kremer 1993: 681–682).⁵ This statement seems rather convincing. Indeed, throughout most of human history the world population was limited by the technologically determined ceiling of land carrying capacity. For example, with forag-ing subsistence technologies the Earth could hardly support more than 8 million people, because the amount of naturally available useful bio-mass on the planet is limited, and the world population could overgrow this limit only when people started to apply various means to artificially increase the amount of available biomass, that is with a transition from foraging to food production. However, the extensive agriculture can only support a limited number of people, and world population further growth became possible only with the intensification of agriculture and other technological improvements (see, e.g., Turchin 2003; Korotayev, Malkov, and Khaltourina 2006a, 2006b; Korotayev and Khaltourina 2006).
   

However, it is well known that the technological level is not a con-stant, but a variable (see, e.g., Grinin 2007a, 2007b, 2012). And in order to describe its dynamics the second basic assumption is employed:

2. 
   ‘High population spurs technological change because it increases the number of potential inventors…⁶ In a larger population there will be proportionally more people lucky or smart enough to come up with new ideas’ (Kremer 1993: 685), thus, ‘the growth rate of technology is proportional to total population’.⁷ In fact, here Kremer uses the main
   

5. 
   In addition to this, the absolute growth rate is proportional to the population number – with a given relative growth rate a larger population will increase more in absolute numbers than a smaller one.
   

5. 
   ‘This implication flows naturally from the non-rivalry of technology… The cost of inventing a new technology is independent of the number of people who use it. Thus, holding constant the share of resources devoted to research, an increase in population leads to an increase in the probability of technological change’ (Kremer 1993: 681); note that in the framework proposed by David Christian (2005) this corresponds precisely to the pattern of collective learning.
   

5. 
   Note that ‘the growth rate of technology’ means here the relative growth rate (i.e. the level to which technology will grow in a given unit of time in proportion to the level observed at
   

194 Mathematical Modeling of Big History Phases

assumption of the Endogenous Technological Growth theory (Kuznets 1960; Grossman and Helpman 1991; Aghion and Howitt 1998; Simon 1977, 2000; Komlos and Nefedov 2002; Jones 1995, 2005, etc.). To our knowledge, this supposition was first put forward by Simon Kuznets (1960), so we will denote a corresponding type of dynamics as ‘Kuznet-sian’, while the systems in which the ‘Kuznetsian’ population-technological dynamics combines with the ‘Malthusian’ demographic one will be denoted as ‘Malthusian-Kuznetsian’. In general, we find this assumption rather plausible – in fact, it is quite probable that, ceteris paribus, within a given period of time, a billion people will make ap-proximately a thousand times more inventions than a million people.
This assumption was expressed by Kremer mathematically in the following way:

dT	(4)
dt	kNT .

Actually, this equation just says that the absolute technological growth rate at a given moment of time (dT/dt) is proportional to the technological level (T) observed at this moment (the wider is the techno-logical base, the more inventions could be made on its basis), and, on the other hand, it is proportional to the population (N) (the larger the population, the larger the number of potential inventors).⁸
The resultant models provide a rather convincing explanation of why throughout most of human history the world population followed the hy-perbolic pattern with an absolute population growth rate tending to be proportional to N². For example, why would the growth of population from, say, 10 million to 100 million, result in the hundredfold growth of dN/dt ? The above mentioned models explain this rather convincingly. The point is that the growth of world population from ten to a hundred mil-lion implies that human subsistence technologies also grew approxi-mately ten times (given that it will prove, after all, to be able to support a ten times larger population). On the other hand, the tenfold popula-tion growth also implies a tenfold growth of the number of potential inventors, and, consequently, a tenfold increase in a relative technologi-cal growth rate. Hence, the absolute technological growth rate would grow 10 × 10 = 100 times (as Equation 4 shows that an order of magni-tude larger number of people with an order of magnitude broader tech-nological basis would likely make two orders of magnitude more inven-
the beginning of this period).

8. 
   Kremer did not test this hypothesis empirically in a direct way. Note, however, that our own empirical test of this hypothesis has supported it (see Korotayev, Malkov, Khal-tourina 2006b: 141–146).
   

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tions). And as throughout the Malthusian epoch the world population (N) tended to the technologically determined carrying capacity ceiling of the Earth, we have good reason to expect that dN/dt will also grow just about 100 times.

In fact, one can demonstrate (see, e.g., Korotayev, Malkov, and Khal-tourina 2006a, 2006b; Korotayev and Khaltourina 2006) that the hyper-bolic pattern of the world's population growth can be explained by the nonlinear second order positive feedback mechanism that was shown long ago to generate just the hyperbolic growth, known also as the ‘blow-up regime’(see, e.g., Kurdyumov 1999). In our case this nonlinear second order positive feedback looks as follows: more people – more potential inventors – a faster technological growth – a faster growth of the Earth's carrying capacity – a faster population growth – with more people you also have more potential inventors – hence, faster techno-logical growth, and so on (see Fig. 3).