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In the first part of this article we survey general similarities and differences between biological and social macroevolution. In the second (and main) part, we consider a concrete mathematical model capable of describing important features of both biological and social macroevolution. In mathematical models of historical macrodynamics, a hyperbolic pattern of world population growth arises from non-linear, second-order positive feedback between demographic growth and technological development. This is more or less identical with the working of the collective learning mechanism. Based on diverse paleontological data and an analogy with macrosociological models, we suggest that the hyperbolic character of biodiversity growth can be similarly accounted for by non-linear, second-order positive feedback between diversity growth and the complexity of community structure, suggesting the presence within the biosphere of a certain analogue of the collective learning mechanism. We discuss how such positive feedback mechanisms can be modelled mathematically.
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Cliodynamics: The Journal of Theoretical
and Mathematical History
UC Riverside
Peer Reviewed
Title:
On Similarities between Biological and Social Evolutionary Mechanisms: Mathematical Modeling
Journal Issue:
Cliodynamics, 4(2)
Author:
Grinin, Leonid, Center of Big History and System Forecasting, Institute of Oriental Studies,
Russian Academy of Sciences and Faculty of Global Studies, Moscow State University
Markov, Alexander, Institute of Paleontology, Russian Academy of Sciences
Korotayev, Andrey, Center of Big History and System Forecasting, Institute of Oriental Studies,
Russian Academy of Sciences and Faculty of Global Studies, Moscow State University
Publication Date:
2013
Publication Info:
Cliodynamics, The Institute for Research on World-Systems, UC Riverside
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http://escholarship.org/uc/item/93b5m968
Keywords:
social evolution, mathematical model, world population growth
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irows_cliodynamics_21334
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the abstract is included in the main body of the paper
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Cliodynamics: the Journal of Theoretical and Mathematical History
-mail: akorotayev@gmail.com
Citation: Grinin, Leonid; Markov, Alexander; Korotoyev, Andrey. 2013. On Similarities
between Biological and Social Evolutionary Mechanisms: Mathematical Modeling.
Cliodynamics 4: 185228.
On Similarities between Biological and Social
Evolutionary Mechanisms: Mathematical
Modeling
Leonid Grinin
Center of Big History and System Forecasting, Institute of Oriental Studies,
Russian Academy of Sciences and Faculty of Global Studies, Moscow State
University
Alexander Markov
Institute of Paleontology, Russian Academy of Sciences
Andrey Korotayev
Center of Big History and System Forecasting, Institute of Oriental Studies,
Russian Academy of Sciences and Faculty of Global Studies, Moscow State
University
In the first part of this article we survey general similarities and
differences between biological and social macroevolution. In the
second (and main) part, we consider a concrete mathematical
model capable of describing important features of both biological
and social macroevolution. In mathematical models of historical
macrodynamics, a hyperbolic pattern of world population growth
arises from non-linear, second-order positive feedback between
demographic growth and technological development. This is more
or less identical with the working of the collective learning
mechanism. Based on diverse paleontological data and an analogy
with macrosociological models, we suggest that the hyperbolic
character of biodiversity growth can be similarly accounted for by
non-linear, second-order positive feedback between diversity
growth and the complexity of community structure, suggesting
the presence within the biosphere of a certain analogue of the
collective learning mechanism. We discuss how such positive
feedback mechanisms can be modelled mathematically.
Introduction
The present article represents an attempt to move further in our research on
the similarities and differences between social and biological evolution (see
Grinin, Markov, et al. 2008, 2009a, 2009b, 2011, 2012). We have endeavored
to make a systematic comparison between biological and social evolution at
different levels of analysis and in various aspects. We have formulated a
Grinin et al: Biological and Social Evolution. Cliodynamics 4.2 (2013)
186
considerable number of general principles and rules of evolution, and worked
to develop a common terminology to describe some key processes in biological
and social evolution. In particular, we have introduced the notion of ‘social
aromorphosis’ to describe the process of widely diffused social innovation that
enhances the complexity, adaptability, integrity, and interconnectedness of a
society or social system (Grinin, Markov, et al. 2008, 2009a, 2009b). This
work has convinced us that it might be possible to find mathematical models
that can describe important features of both biological and social
macroevolution. In the first part of this article we survey general similarities
and differences between the two types of macroevolution. In the second (and
main) part, we consider a concrete mathematical model that we deem capable
of describing important features of both biological and social macroevolution.
Introductory Remarks
The comparison of biological and social evolution is an important but
(unfortunately) understudied subject. Students of culture still vigorously
debate the applicability of Darwinian evolutionary theory to social/cultural
evolution. Unfortunately, the result is largely a polarization of views. On one
hand is a total rejection of Darwin's theory of social evolution (see, e.g.,
Hallpike 1986). On the other, are arguments that cultural evolution
demonstrates all of the key characteristics of Darwinian evolution (Mesoudi et
al. 2006).
We believe that, instead of following the outdated objectivist principle of
‘either or’, we should concentrate on the search for methods that could allow
us to apply the achievements of evolutionary biology to understanding social
evolution and vice versa. In other words, we should search for productive
generalizations and analogies for the analysis of evolutionary mechanisms in
both contexts. The Universal Evolution approach aims for the inclusion of all
mega-evolution within a single paradigm (discussed in Grinin, Carneiro, et al.
2011). Thus, this approach provides an effective means by which to address the
above-mentioned task.
It is not only systems that evolve, but also mechanisms of evolution (see
Grinin, Markov, et al. 2008). Each sequential phase of macroevolution is
accompanied by the emergence of new evolutionary mechanisms. Certain
prerequisites and preadaptations can, therefore, be detected within the
previous phase, and the development of new mechanisms does not invalidate
the evolutionary mechanisms that were active during earlier phases. As a
result, one can observe the emergence of a complex system of interaction
composed of the forces and mechanisms that work together to shape the
evolution of new forms.
Biological organisms operate in the framework of certain physical, chemical
and geological laws. Likewise, the behaviors of social systems and people have
certain biological limitations (naturally, in addition to various social-
Grinin et al: Biological and Social Evolution. Cliodynamics 4.2 (2013)
187
structural, historical, and infrastructural limitations). From the standpoint of
Universal Evolution, new forms of evolution that determine phase transitions
may result from activities going in different directions. Some forms that are
similar in principle may emerge at breakthrough points, but may also result in
evolutionary dead-ends. For example, social forms of life emerged among
many biological phyla and classes, including bacteria, insects, birds, and
mammals. Among insects, in particular, one finds rather highly developed
forms of socialization (see, e.g., Robson and Traniello 2002; Ryabko and
Reznikova 2009; Reznikova 2011). Yet, despite the seemingly common
trajectory and interrelation of social behaviors among these various life forms,
the impacts that each have had on the Earth are remarkably different.
Further, regarding information transmission mechanisms, it appears
possible to speak about certain ‘evolutionary freaks’. Some of these
mechanisms were relatively widespread in the biological evolution of simple
organisms, but later became less so. Consider, for example, the horizontal
exchange of genetic information (genes) among microorganisms, which makes
many useful genetic ‘inventions’ available in a sort of ‘commons’ for microbe
communities. Among bacteria, the horizontal transmission of genes
contributes to the rapid development of antibiotic resistance (e.g., Markov and
Naymark 2009). By contrast, this mechanism of information transmission
became obsolete or was transformed into highly specialized mechanisms (e.g.,
sexual reproduction) in the evolution of more complex organisms. Today,
horizontal transmission is mostly confined to the simplest forms of life.
These examples suggest that an analysis of the similarities and differences
between the mechanisms of biological and social evolution may help us to
understand the general principles of megaevolution
1
in a much fuller way.
These similarities and differences may also reveal the driving forces and supra-
phase mechanisms (i.e., mechanisms that operate in two or more phases) of
megaevolution. One of our previous articles was devoted to the analysis of one
such mechanism: aromorphosis, the process of widely diffused social
innovation that enhances the complexity, adaptability, integrity, and
interconnectedness of a society or social system (Grinin, Markov, et al. 2011;
see also Grinin and Korotayev 2008, 2009a, 2009b; Grinin, Markov, et al.
2009a, 2009b).
It is important to carefully compare the two types of macroevolution (i.e.,
biological and social) at various levels and in various aspects. This is necessary
because such comparisons often tend to be incomplete and deformed by
conceptual extremes. These limitations are evident, for example, in the above-
referenced paper by Mesoudi et al. (2006), which attempts to apply a
1
We denote as megaevolution all the process of evolution throughout the whole of Big
History, whereas we denote as macroevolution the process of evolution during one of
its particular phases.
Grinin et al: Biological and Social Evolution. Cliodynamics 4.2 (2013)
188
Darwinian method to the study of social evolution. Unfortunately, a failure to
recognize or accept important differences between biological and social
evolution reduces the overall value of the method that these authors propose.
Christopher Hallpike’s rather thorough monograph, Principles of Social
Evolution (1986), provides another illustration of these limitations. Here,
Hallpike offers a fairly complete analysis of the similarities and differences
between social and biological organisms, but does not provide a clear and
systematic comparison between social and biological evolution. In what
follows, we hope to avoid similar pitfalls.
Biological and Social Evolution: A Comparison at
Various Levels
There are a few important differences between biological and social
macroevolution. Nonetheless, it is possible to identify a number of
fundamental similarities, including at least three basic sets of shared factors.
First, we are discussing very complex, non-equilibrium, but stable systems
whose function and evolution can be described by General Systems Theory, as
well as by a number of cybernetic principles and laws. Second, we are not
dealing with isolated systems, but with the complex interactions between
organisms and their external environments. As a result, the reactions of
systems to ‘external’ challenges can be described in terms of general principles
that express themselves within a biological reality and a social reality. Third
(and finally), a direct ‘genetic’ link exists between the two types of
macroevolution and their mutual influence.
We believe that the laws and forces driving the biological and social phases
of Big History can be comprehended more effectively if we apply the concept of
biological and social aromorphosis (Grinin, Markov, et al. 2011). There are
some important similarities between the evolutionary algorithms of biological
and social aromorphoses. Thus, it has been noticed that the basis of biological
aromorphosis
is usually formed by some partial evolutionary change that... creates
significant advantages for an organism, puts it in more favorable
conditions for reproduction, multiplies its numbers and its
changeability..., thus accelerating the speed of its further evolution. In
those favorable conditions, the total restructurization of the whole
organization takes place afterwards (Shmal'gauzen 1969: 410; see also
Severtsov 1987: 6476).
During the course of adaptive radiation, such changes in organization diffuse
more or less widely (frequently with significant variations).
Grinin et al: Biological and Social Evolution. Cliodynamics 4.2 (2013)
189
A similar pattern is observed within social macroevolution. An example is
the invention and diffusion of iron metallurgy. Iron production was practiced
sporadically in the 3rd millennium BCE, but regular production of low-grade
steel did not begin until the mid-2nd millennium BCE in Asia Minor (see, e.g.,
Chubarov 1991: 109). At this point, the Hittite kingdom guarded its monopoly
over the new technology. The diffusion of iron technology led to revolutionary
changes in different spheres of life, including a significant advancement in
plough agriculture and, consequently, in the agrarian system as a whole
(Grinin and Korotayev 2006); an intensive development of crafts; an increase
in urbanism; the formation of new types of militaries, armed with relatively
cheap but effective iron weapons; and the emergence of significantly more
developed systems of taxation, as well as information collection and processing
systems, that were necessary to support these armies (e.g., Grinin and
Korotayev 2007a, 2007b). Ironically, by introducing cheaply made weapons
and other tools into the hands of people who might resist the Hittite state, this
aromorphosis not only supported the growth of that kingdom, it also laid the
groundwork for historical phase shifts.
Considering such cases through the lens of aromorphosis has helped us to
detect a number of regularities and rules that appear to be common to
biological and social evolution (Grinin, Markov, Korotayev 2011). Such rules
and regularities (e.g., payment for arogenic progress, special conditions for the
emergence of aromorphosis, etc.) are similar for both biological and social
macroevolution. It is important to emphasize, however, that similarity between
the two types of macroevolution does not imply commonality. Rather,
significant similarities are frequently accompanied by enormous differences.
For example, the genomes of chimpanzees and the humans are 98 percent
similar, yet there are enormous intellectual and social differences between
chimpanzees and humans that arise from the apparently ‘insignificant’
variations between the two genomes (see Markov and Naymark 2009).
Despite its aforementioned limitations, it appears reasonable to continue
the comparison between the two types of macroevolution following the
analysis offered by Hallpike (1986). Therefore, it may prove useful to revisit
the pertinent observations of this analysis here. Table 1 summarizes the
similarities and differences that Hallpike (1986: 3334) finds between social
and biological organisms.
While we do not entirely agree with all of his observations for example,
the establishment of colonies could be seen as a kind of social reproduction
akin to organic reproduction we do feel that Hallpike comes to a sound
conclusion: that similarities between social and biological organisms are, in
general, determined by similarities in organization and structure (we would
say similarities between different types of systems). As a result, Hallpike
believes that one can use certain analogies in which institutions are similar to
some organs. In this way, cells may be regarded as similar to individuals,
Grinin et al: Biological and Social Evolution. Cliodynamics 4.2 (2013)
190
central government similar to the brain, and so on. Examples of this kind of
thinking can be found in the classic texts of social theory (see, e.g., Spencer
1898 and Durkheim 1991 [1893]), as well as in more recent work (see, e.g.,
Heylighen 2011).
Table 1. Similarities and differences between social and biological organisms,
as described by Hallpike (1986)
Similarities
Differences
Social institutions are interrelated in
a manner analogous to the organs of
the body.
Individual societies do not have
clear boundaries. For example, two
societies may be distinct politically,
but not culturally or religiously.
Despite changes in membership,
social institutions maintain
continuity, as do biological organs
when individual cells are replaced.
Unlike organic cells, the individuals
within a society have agency and are
capable of learning from experience.
The social division of labor is
analogous to the specialization of
organic functions.
Social structure and function are far
less closely related than in organic
structure and function.
Self-maintenance and feedback
processes characterize both kinds of
system.
Societies do not reproduce. Cultural
transmission between generations
cannot be distinguished from the
processes of system maintenance.
Adaptive responses to the physical
environment characterize both kinds
of system.
Societies are more mutable than
organisms, displaying a capacity for
metamorphosis only seen in organic
phylogeny.
The trade, communication, and
other transmission processes that
characterize social systems are
analogous to the processes that
transmit matter, energy, and
information in biological organisms.
Societies are not physical entities,
rather their individual members are
linked by information bonds.
When comparing biological species and societies, Hallpike (1986: 34)
singles out the following similarities:
1. that, like societies, species do not reproduce,
2. that both have phylogenies reflecting change over time, and
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191
3. that both are made up of individuals who compete against one
another.
Importantly, he also indicates the following difference: [S]ocieties are
organized systems, whereas species are simply collections of individual
organisms” (34).
Hallpike tries to demonstrate that, because of the differences between
biological and social organisms, the very idea of natural selection does not
appear to apply to social evolution. However, we do not find his proofs very
convincing on this account, although they do make sense in certain respects.
Further, his analysis is confined mainly to the level of the individual organism
and the individual society. He rarely considers interactions at the supra-
organism level (though he does, of course, discuss the evolution of species).
His desire to demonstrate the sterility of Darwinian theory to discussions of
social evolution notwithstanding, it seems that Hallpike involuntarily
highlights the similarity between biological and social evolution. As he,
himself, admits, the analogy between the biological organism and society is
quite noteworthy.
Just as he fails to discuss interactions and developments at the level of the
supra-organism in great detail, Hallpike does not take into account the point in
social evolution where new supra-societal developments emerge (up to the
level of the emergence of the World System [e.g., Korotyev 2005, 2007, 2008,
2012; Grinin, Korotyev 2009b]). We contend that it is very important to
consider not only evolution at the level of a society but also at the level above
individual societies, as well as the point at which both levels are
interconnected. The supra-organism level is very important to understanding
biological evolution, though the differences between organisms and societies
make the importance of this supra-level to understanding social evolution
unclear. Thus, it might be more productive to compare societies with
ecosystems rather than with organisms or species. However, this would
demand the development of special methods, as it would be necessary to
consider the society not as a social organism, but as a part of a wider system,
which includes the natural and social environment (cf., Lekevičius 2009,
2011).
In our own analysis, we seek to build on the observations of Hallpike while,
at the same time, providing a bit more nuance and different scales of analysis.
Viewing each as a process involving selection (natural, social, or both), we
identify the differences between social and biological evolution at the level of
the individual biological organism and individual society, as well as at the
supra-organismic and supra-societal level.
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192
Natural and Social Selection
Biological evolution is more additive (cumulative) than substitutive. Put
another way: the new is added to the old. By contrast, social evolution
(especially over the two recent centuries) is more substitutive than additive:
the new replaces the old (Grinin, Markov, et al. 2008, 2011).
Further, the mechanisms that control the emergence, fixation, and
diffusion of evolutionary breakthroughs (aromorphoses) differ between
biological and social evolution. These differences lead to long-term
restructuring in the size and complexity of social organisms. Unlike biological
evolution, where some growth of complexity is also observed, social
reorganization becomes continuous. In recent decades, societies that do not
experience a constant and significant evolution look inadequate and risk
extinction.
In addition, the size of societies (and systems of societies) tends to grow
constantly through more and more tightly integrative links (this trend has
become especially salient in recent millennia), whereas the trend towards
increase in the size of biological organisms in nature is rather limited and far
from general. At another level of analysis, one can observe the formation of
special suprasocietal systems that also tend to grow in size. This is one of the
results of social evolution and serves as a method of aromorphosis fixation and
diffusion.
The Individual Biological Organism and the Individual
Society
It is very important to note that, although though biological and social
organisms are significantly (actually ‘systemically’) similar, they are radically
different in their capacities to evolve. For example, as indicated by Hallpike
(see above), societies are capable rapid evolutionary metamorphoses that were
not observed in the pre-human organic world. In biological evolution, the
characteristics acquired by an individual are not inherited by its offspring;
thus, they do no not influence the very slow process of change.
There are critical differences in how biological and social information are
transmitted during the process of evolution. Social systems are not only
capable of rapid transformation, they are also able to borrow innovations and
new elements from other societies. Social systems may also be transformed
consciously and with a certain purpose. Such characteristics are absent in
natural biological evolution.
The biological organism does not evolve by itself: evolution may only take
place at a higher level (e.g., population, species, etc.). By contrast, social
evolution can often be traced at the level of the individual social organism (i.e.,
society). Moreover, it is frequently possible to trace the evolution of particular
institutions and subsystems within a social organism. In the process of social
Grinin et al: Biological and Social Evolution. Cliodynamics 4.2 (2013)
193
evolution the same social organism or institution may experience radical
transformation more than once.
The Supra-organic and Supra-societal Level
Given the above-mentioned differences, within the process of social evolution
we observe the formation of two types of special supra-societal entity: (1)
amalgamations of societies with varieties of complexity that have analogues in
biological evolution, and (2) elements and systems that do not belong to any
particular society and lack many analogues in biological evolution.
The first type of amalgamation is rather typical, not only in social but also
in biological evolution. There is, however, a major difference between the two
kinds of evolution. Any large society usually consists of a whole hierarchy of
social systems. For example, a typical agrarian empire might include nuclear
families, extended families, clan communities, village communities, primary
districts, secondary districts, and provinces, each operating with their own
rules of interaction but at the same time interconnected. This kind of supra-
societal amalgamation can hardly be compared with a single biological
organism (though both systems can still be compared functionally, as is
correctly noted by Hallpike [1986]). Within biological evolution,
amalgamations of organisms with more than one level of organization (as
found in a pack or herd) are usually very unstable and are especially unstable
among highly organized animals. Of course, analogues do exist within the
communities of some social animals (e.g., social insects, primates). Neither
should we forget that scale is important: while we might compare a society
with an individual biological organism, we must also consider groups of
organisms bound by cooperative relationships (see, e.g., Boyd and Richerson
1996; Reeve and Hölldobler 2007). Such groups are quite common among
bacteria and even among viruses. These caveats aside, it remains the case that
within social evolution, one observes the emergence of more and more levels:
from groups of small sociums to humankind as a whole.
The multiplication of these levels rapidly produces the second kind of
amalgamation. It is clear that the level of analysis is very important for
comparison of biological and social evolution. Which systems should be
compared? Analogues appear to be more frequent when a society (a social
organism) is compared to a biological organism or species. However, in many
cases, it may turn out to be more productive to compare societies with other
levels of the biota's systemic organization. This might entail comparisons with
populations, ecosystems and communities; with particular structural elements
or blocks of communities (e.g., with particular fragments of trophic networks
or with particular symbiotic complexes); with colonies; or with groups of
highly organized animals (e.g., cetaceans, primates, and other social mammals
or termites, ants, bees and other social insects).
Grinin et al: Biological and Social Evolution. Cliodynamics 4.2 (2013)
194
Thus, here we confront a rather complex and rarely studied methodological
problem: which levels of biological and social process are most congruent?
What are the levels whose comparison could produce the most interesting
results? In general, it seems clear that such an approach should not be a
mechanical equation of ‘social organism = biological organism’ at all times and
in every situation. The comparisons should be operational and instrumental.
This means that we should choose the scale and level of social and biological
phenomena, forms, and processes that are adequate for and appropriate to our
intended comparisons.
Again, it is sometimes more appropriate to compare a society with an
individual biological organism, whereas in other cases it could well be more
appropriate to compare the society with a community, a colony, a population,
or a species. At yet another scale, as we will see below, in some cases it appears
rather fruitful to compare the evolution of the biosphere with the evolution of
the anthroposphere.
Mathematical Modeling of Biological and Social
Macroevolution
The authors of this article met for the first time in 2005, in the town of Dubna
(near Moscow), at what seems to have been the first ever international
conference dedicated specifically to Big History studies. Without advance
knowledge of one another, we found ourselves in a single session. During the
course of the session, we presented two different diagrams. One illustrated
population dynamics in China between 700 BCE and 1851 CE, the other
illustrated the dynamics of marine Phanerozoic biodiversity over the past 542
million years (Fig. 1):
The similarity between the two diagrams was striking. This, despite the fact
that they depicted the development of very different systems (human
population vs. biota) at different time scales (hundreds of years vs. millions of
years), and had been generated using the methods of different sciences
(historical demography vs. paleontology) with different sources (demographic
estimates vs. paleontological data). What could have caused similarity of
developmental dynamics in very different systems?
* * *
In 1960, von Foerster et al. published a striking discovery in the journal
Science. They showed that between 1 and 1958 CE, the world's population (N)
dynamics could be described in an extremely accurate way with an
astonishingly simple equation
2
:
2
To be exact, the equation proposed by von Foerster and his colleagues looked as
follows:
  -

. However, as has been shown by von Hoerner (1975) and
Kapitza (1999), it can be written more succinctly as  
 .
Grinin et al: Biological and Social Evolution. Cliodynamics 4.2 (2013)
195
(1)
)( 0tt C
Nt
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. Parameter t0 was estimated by von Foerster and his colleagues
as 2026.87, which corresponds to November 13, 2026; this made it possible for
them to supply their article with a title that was a public-relations masterpiece:
“Doomsday: Friday, 13 November, A.D. 2026”.
Grinin et al: Biological and Social Evolution. Cliodynamics 4.2 (2013)
196
Of course, von Foerster and his colleagues did not imply that the world
population on that day could actually become infinite. The real implication was
that the world population growth pattern that operated for many centuries
prior to 1960 was about to end and be transformed into a radically different
pattern. This prediction began to be fulfilled only a few years after the
“Doomsday” paper was published as World System growth (and world
population growth in particular) began to diverge more and more from the
previous blow-up regime. Now no longer hyperbolic, the world population
growth pattern is closer to a logistic one (see, e.g., Korotayev, Malkov, et al.
2006a; Korotayev 2009).
Figure 2 presents the overall correlation between the curve generated
by von Foerster et al.'s equation and the most detailed series of empirical
estimates of world population (McEvedy and Jones 1978, for the period 1000
1950; U.S. Bureau of the Census 2013, for 19501970). The formal
characteristics are: R = 0.998; R2 = 0.996; p = 9.4 × 10-17 ≈ 1 × 10-16. For
readers unfamiliar with mathematical statistics: R2 can be regarded as a
measure of the fit between the dynamics generated by a mathematical model
and the empirically 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%.
3
Thus, von Foerster et al.'s equation
accounts for an astonishing 99.6% of all the macrovariation in world
population, from 1000 CE through 1970, as estimated by McEvedy and Jones
(1978) and the U.S. Bureau of the Census (2013).
4
Note also that the empirical
estimates of world population find themselves aligned in an extremely neat
way along the hyperbolic curve, which convincingly justifies the designation of
the pre-1970s world population growth pattern as ‘hyperbolic’.
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 statistical significance of the respective correlation. This is because p
indicates the probability that the observed correlation could be accounted solely by
chance. Thus, p = 0.99 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
10,000,000,000,000,000 that the observed correlation is the result of pure
coincidence (a correlation is usually considered statistically significant once p < 0.05).
4
In fact, with slightly different parameters ( = 164890.45; t0 = 2014) the fit (R2)
between the dynamics generated by von Foerster's equation and the macrovariation of
world population for 1000 1970 CE as estimated by McEvedy and Jones (1978) and
the U.S. Bureau of the Census (2013) reaches 0.9992 (99.92%); for 500 BCE 1970 CE
this fit increases to 0.9993 (99.93%) with the following parameters: = 171042.78; t0 =
2016.
Grinin et al: Biological and Social Evolution. Cliodynamics 4.2 (2013)
197
The von Foerster et al. equation,
 
 , is the solution for the
following differential equation (see, e.g., Korotayev, Malkov, et al. 2006a: 119
120):
(2)
C
N
dt
dN 2
This equation can be also written as:
(3)
2
aN
dt
dN
where   
.
What is the meaning of this mathematical expression? In our context,
dN/dt denotes the absolute population growth rate at a certain moment in
time. Hence, this equation states that the absolute population growth rate at
any moment in time should be proportional to the square of world population
at this moment. This significantly demystifies the problem of hyperbolic
growth. To explain this hyperbolic growth, one need only explain why for many
Figure 2. Correlation between Empirical Estimates of WorldPopulation (black,
in millions of people, 1000 1970) and the Curve Generated by von Foerster et
al.'s Equation (grey)
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millennia the world population's absolute growth rate tended to be
proportional to the square of the population.
The main mathematical models of hyperbolic growth in the world
population (Taagapera 1976, 1979; Kremer 1993; Cohen 1995; Podlazov 2004;
Tsirel 2004; Korotayev 2005, 2007, 2008, 2009, 2012; Korotayev, Malkov, et
al. 2006a: 2136; Golosovsky 2010; Korotayev and Malkov 2012) are based on
the following two assumptions:
(1) “the Malthusian (1978[1798]) assumption that population is limited by
the available technology, so that the growth rate of population is
proportional to the growth rate of technology” (Kremer 1993: 681
682)
5
, and
(2) the idea that “[h]igh population spurs technological change because it
increases the number of potential inventors… 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”.
6
Here Kremer uses the
main assumption of Endogenous Technological Growth theory (see, e.g.,
Kuznets 1960; Grossman and Helpman 1991; Aghion and Howitt 1998;
Simon 1977, 2000; Komlos and Nefedov 2002; Jones 1995, 2005).
The first assumption looks quite convincing. Indeed, throughout most of
human history the world population was limited by the technologically
determined ceiling of the carrying capacity of land. For example, with foraging
subsistence technologies the Earth could not support more than 8 million
people because the amount of naturally available useful biomass on this planet
is limited. The world population could only grow over this limit when people
started to apply various means to artificially increase the amount of available
biomass, that is with the transition from foraging to food production.
Extensive agriculture is also limited in terms of the number of people that it
can support. Thus, further growth of the world population only became
possible with the intensification of agriculture and other technological
improvements (see, e.g., Turchin 2003; Korotayev, Malkov, et al. 2006a,
2006b; Korotayev and Khaltourina 2006). However, as is well known, the
technological level is not constant, but variable (see, e.g., Grinin 2007a, 2007b,
2012), and in order to describe its dynamics the second basic assumption is
employed.
5
In addition to this, the absolute growth rate is proportional to the population itself.
With a given relative growth rate, a larger population will increase more in absolute
number than a smaller one.
6
Note that ‘the growth rate of technology’ here means 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 the beginning of this period).
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As this second supposition was, to our knowledge, first proposed by Simon
Kuznets (1960), we shall denote the corresponding type of dynamics as
‘Kuznetsian’. (The systems in which the Kuznetsian population-technological
dynamics are combined with Malthusian demographic dynamics will be
denoted as ‘Malthusian-Kuznetsian’.) In general, we find this assumption
rather plausible in fact, it is quite probable that, other things being equal,
within a given period of time, five million people will make approximately five
times more inventions than one million people.
This assumption was expressed mathematically by Kremer in the following
way:
(4)
kNT
dt
dT
This equation simply says that the absolute technological growth rate at a
given moment in time (dT/dt) is proportional to the technological level (T)
observed at this moment (the wider the technological base, the higher the
number of inventions that can be made on its basis). On the other hand, this
growth rate is also proportional to the population (N): the larger the
population, the larger the number of potential inventors.
7
When united in one system, Malthusian and Kuznetsian equations account
quite well for the hyperbolic growth of the world population observed before
the early 1990s (see, e.g., Korotayev 2005, 2007, 2008, 2012; Korotayev,
Malkov, et al. 2006a). The resultant models provide a rather convincing
explanation of why, throughout most of human history, the world population
followed the hyperbolic pattern with the 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 growth of dN/dt
100 times? The above mentioned models explain this rather convincingly. The
point is that the growth of world population from 10 to 100 million implies
that human subsistence technologies also grew approximately 10 times (given
that it will have proven, after all, to be able to support a population ten times
larger). On the other hand, the tenfold population growth also implies a
tenfold growth in the number of potential inventors, and, hence, a tenfold
increase in the relative technological growth rate. Thus, the absolute
technological growth rate would expand 10 × 10 = 100 times as, in accordance
with equation (4), an order of magnitude higher number of people having at
their disposal an order of magnitude wider technological base would tend to
make two orders of magnitude more inventions. If, as throughout the
7
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, et al.
2006b: 141146).
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Malthusian epoch, the world population (N) tended toward the technologically
determined carrying capacity of the Earth, we have good reason to expect that
dN/dt should also grow just by about 100 times.
In fact, it can be shown (see, e.g., Korotayev, Malkov, et al. 2006a, 2006b;
Korotayev and Khaltourina 2006) that the hyperbolic pattern of the world's
population growth could be accounted for by a nonlinear second-order positive
feedback mechanism that was long ago shown to generate just the hyperbolic
growth, also known 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 faster technological growth faster
growth of the Earth's carrying capacity faster population growth more
people allow for more potential inventors faster technological growth, and so
on (see Fig. 3).
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.
The feedback system described here should be identified with the process of
‘collective learning’ described, principally, by Christian (2005: 146148). The
mathematical models of World System development discussed in this article
can be interpreted as models of the influence that collective learning has on
global social evolution (i.e., the evolution of the World System). Thus, the
rather peculiar hyperbolic shape of accelerated global development prior to the
early 1970s may be regarded as a product of global collective learning. We have
also shown (Korotayev, Malkov, et al. 2006a: 3466) that, for the period prior
to the 1970s, World System economic and demographic macrodynamics,
driven by the above-mentioned positive feedback loops, can simply and
accurately be described with the following model:
Figure 3. Cognitive scheme of the nonlinear second-order positive feedback
between technological development and demographic growth
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201
(5)
aSN
dt
dN
(6)
bNS
dt
dS
The world GDP (G) can be calculated using the following equation:
(7)     
G is the world GDP, N is population, and S is the produced surplus per capita,
over the subsistence amount (m) that is 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); a and b are parameters.
The mathematical analysis of the basic model (not described here) suggests
that up to the 1970s, the amount of S 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, et al. 2006a: 4950). Thus, in the right-hand side of equation (6), S
can be replaced with kN, resulting in the following equation:
2
kaN
dt
dN
Recall that the solution of this type of differential equations is:
)( 0tt C
Nt
which produces a simple hyperbolic curve.
As, according to our model, S can be approximated as kN, its long-term
dynamics can be approximated with the following equation:
(8)
ttkC
S
0
Thus, the long-term dynamics of the most dynamic component of the world
GDP, SN, the ‘world surplus product’, can be approximated as follows:
Grinin et al: Biological and Social Evolution. Cliodynamics 4.2 (2013)
202
(9)
 
2
0
2
tt
kC
SN
This suggests that the long-term world GDP dynamics up to the early 1970s
must be approximated better by a quadratic hyperbola, rather than by a simple
one. As shown in Figure 4, this approximation works very effectively indeed.
Thus, up to the 1970s the hyperbolic growth of the world population was
accompanied by the quadratic-hyperbolic growth of the world GDP, 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 nonlinear second-order positive
feedback loops between the technological development and demographic
growth (see Fig. 5).
Figure 4. The Fit between Predictions of a Quadratic-Hyperbolic Model and
Observed World GDP Dynamics, 11973 CE (in billions of 1990 international
dollars, PPP)a
a R =.9993, R2 =.9986, p <<.0001. The black markers correspond to
Maddison'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:
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203
We have also demonstrated (Korotayev, Malkov, et al. 2006a: 6780) that
the World System population's literacy (l) dynamics are rather
accuratelydescribed by the following differential equation:
(10)
)1( laSl
dt
dl
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.
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. (Additionally, S reflects the
technological level T that implies, among other things, the level of
development of educational technologies.) From a mathematical point of view,
equation (9) can be regarded as logistic where saturation is reached at literacy
level l = 1. 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 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 Fig. 6).
Figure 5. Cognitive Scheme of the World Economic Growth Generated by
Nonlinear Second-Order Positive Feedback between Technological Development
and Demographic Growth
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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)
8
was
growing not just hyperbolically, but rather in a quadratic- hyperbolic way (as
was world GDP). Our empirical test has confirmed this the quadratic-
hyperbolic model describes the growth of the literate population of this planet
with an extremely good fit indeed (see Fig. 7).
Similar processes are observed with respect to world urbanization, the
macrodynamics of which appear to be described by the differential equation:
8
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.
Figure 6. The Fit between Predictions of the Hyperbolic Model and Observed
World Literacy Dynamics, 1 1980 CE (%%)a
a R = 0.997, R2 = 0.994, p << 0.0001. Black dots correspond to World
Bank (2013) estimates for the period since 1970, and to Meliantsev's
(2004) estimates for the earlier period. The grey solid line has been
generated by the following equation:
The best-fit values of parameters (3769.264) and t0 (2040) have been
calculated with the least squares method.
Grinin et al: Biological and Social Evolution. Cliodynamics 4.2 (2013)
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Figure 7. The Fit between Predictions of the Quadratic-Hyperbolic Model and
Observed World Literate Population Dynamics, 1 1980 CE (L, millions)a
aR = 0.9997, R2 = 0.9994, p << 0.0001. The black dots correspond to
World Bank (2013) 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 (2013) data. The grey solid line has been generated by the
following equation:
The best-fit values of parameters (4958551) and t0 (2033) have been
calculated with the least squares method.
(11)
lim
()
du bSu u 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 technological
development, b is a constant, and ulim is the maximum possible proportion of
the population that can be urban. Note that this model implies that during the
Malthusian-Kuznetsian era of the blow-up regime, the hyperbolic growth of
world urbanization must have been accompanied by a quadratic-hyperbolic
growth of the urban population of the world, as supported by our empirical
tests (see Figs. 89).
Within this context it is hardly surprising to find that the general
macrodynamics of largest settlements within the World System are also
quadratic-hyperbolic (see Fig. 10).
Grinin et al: Biological and Social Evolution. Cliodynamics 4.2 (2013)
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As has been demonstrated by socio-cultural anthropologists working across
cultures (see, e.g., Naroll and Divale 1976; Levinson and Malone 1980: 34), for
pre-agrarian, 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 Malthusian-Kuznetsian era, in a
generally quadratic-hyperbolic way.
Turning to a more concrete case study, as suggested at the beginning of this
section, the hyperbolic model is particularly effective for describing the long-
term population dynamics of China, the country with the best-known
demographic history. The Chinese population curve reflects not only a
hyperbolic trend, but also cyclical and stochastic dynamics. These components
of long-term population dynamics in China, as well as in other complex
agrarian societies, have been discussed extensively (see, e.g., Braudel 1973;
Abel 1980; Usher 1989; Goldstone 1991; Chu and Lee 1994; Komlos and
Nefedov 2002; Turchin 2003, 2005a, 2005b; Nefedov 2004; Korotayev 2006;
Figure 8. The Fit between Predictions of the Hyperbolic Model and Empirical
Estimates of World Megaurbanization Dynamics (% of the world population
living in cities with > 250,000 inhabitants), 10,000 BCE 1960 CEa
a R = 0.987, R2 = 0.974, p << 0.0001. The black dots correspond to the
estimates of Chandler (1987), UN Population Division (2013), Modelski
(2003), and Gruebler (2006). The grey solid line has been generated by
the following equation:
The best-fit values of parameters С (403.012) and t0 (1990) have been
calculated with the least squares method. For a comparison, the best fit
(R2) obtained here for the exponential model is 0.492.
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Korotayev and Khaltourina 2006; Korotayev, Malkov, et al. 2006b; Turchin
and Korotayev 2006; Korotayev, Komarova, et al. 2007; Grinin, Korotayev, et
al. 2008; Grinin, Malkov, et al. 2009; Turchin and Nefedov 2009; van Kessel-
Hagesteijn 2009; Korotayev, Khaltourina, Malkov, et al. 2010; Korotayev,
Khaltourina, et al. 2010; Grinin and Korotayev 2012).
As we have observed with respect to world population dynamics, even
before the start of its intensive modernization, the population dynamics of
China were characterized by a pronounced hyperbolic trend (Figs. 11 and 12).
The hyperbolic model describes traditional Chinese population dynamics much
more accurately than either linear or exponential models.
The hyperbolic model describes the population dynamics of China in an
especially accurate way if we take the modern period into account (Fig. 13).
It is curious that, as we noted above, the dynamics of marine biodiversity
are strikingly similar to the population dynamics of China. The similarity
Figure 9. The Fit between Predictions of the Quadratic-Hyperbolic Model
and the Observed Dynamics of World Urban Population Living in Cities with
> 250,000 Inhabitants (mlns.), 10,000 BCE 1960 CEa
a R = 0.998, R2 = 0.996, p << 0.0001. The black markers correspond
to estimates of Chandler (1987), UN Population Division (2013). The
grey solid line has been generated by the following equation:
The best-fit values of parameters (912057.9) and t0 (2008) have
been calculated with the least squares method. For a comparison, the
best fit (R2) obtained here for the exponential model is 0.637.
Grinin et al: Biological and Social Evolution. Cliodynamics 4.2 (2013)
208
probably derives from the fact that both curves are produced by the
interference of the same three components (the general hyperbolic trend , as
well as cyclical and stochastic dynamics). In fact, there is a lot of evidence that
some aspects of biodiversity dynamics are stochastic (Raup et al. 1973;
Sepkoski 1994; Markov 2001; Cornette and Lieberman 2004), while others are
periodic (Raup and Sepkoski 1984; Rohde and Muller 2005). In any event, the
hyperbolic model describes marine biodiversity (measured by number of
genera) through the Phanerozoic much more accurately than an exponential
model (Fig. 14).
When measured by number of species, the fit between the empirically
observed marine biodiversity dynamics and the hyperbolic model becomes
even better (Fig. 15).
Likewise, the hyperbolic model describes continental biodiversity in an
especially accurate way (Fig. 16).
Figure 10. The Fit between Predictions of the Quadratic-Hyperbolic Model
and the Observed Dynamics of Size of the Largest Settlement of the World
(thousands of inhabitants), 10,000 BCE 1950 CEa
a R = 0.992, R2 = 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 best-fit values of parameters (104020618,5) and t0 (2040) have
been calculated with the least squares method. For a comparison, the
best fit (R2) obtained here for the exponential model is 0.747.
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209
YEAR
1900
1800
1700
1600
1500
1400
1300
1200
1100
1000
900
800
700
600
500
400
300
200
100
0
500
400
300
200
100
0
Observed
Linear
Exponential
Figure 11. Population Dynamics of China (million people, following Korotayev,
Malkov, et al. 2006b: 4788), 571851 CE: Fit with Linear and Exponential Modelsa
Linear model: aR2 = 0.469. Exponential model: R2 = 0.600.
1850165014501250105085065045025050
450
400
350
300
250
200
150
100
50
0
observed
predicted
Figure 12 Fit between a Hyperbolic Model and Observed Population Dynamics of
China (million people), 571851 CEa
a R2 = 0.884. The grey solid line has been generated by the following equation:
t
Nt
1915
33431
Grinin et al: Biological and Social Evolution. Cliodynamics 4.2 (2013)
210
t, years
21001800150012009006003000
1400
1200
1000
800
600
400
200
0
predicted
observed
Figure 13. Fit between a Hyperbolic Model and Observed Population Dynamics of
China (million people, following Korotayev, Malkov, et al. 2006b: 4788), 572003
CEa
a R2 = 0.968. The grey solid line has been generated by the following equation:
t
Nt
2050
63150
Figure 14 Global Change in Marine Biodiversity (number of genera, N) through the
Phanerozoic (following Markov and Korotayev 2007) a
a Exponential model: R2 = 0.463. Hyperbolic model: R2 = 0.854. The hyperbolic
line has been generated by the following equation:
t
Nt
37
183320
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211
Figure 15. Global Change in Marine Biodiversity (number of species, N) through the
Phanerozoic (following Markov and Korotayev 2008) a
a Exponential model: R2 = 0.51. Hyperbolic model: R2 = 0.91. The hyperbolic
line has been generated by the following equation:
t
Nt
35
892874
Figure 16. Global Change in Continental Biodiversity (number of genera, N) through
the Phanerozoic (following Markov and Korotayev 2008) a
a Exponential model: R2 = 0.86. Hyperbolic model: R2 = 0.94. The hyperbolic
line has been generated by the following equation:
t
Nt
29
272095
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However, the best fit between the hyperbolic model and the empirical data
is observed when the hyperbolic model is used to describe the dynamics of
total (marine and continental) global biodiversity (Fig. 17).
The hyperbolic dynamics are most prominent when both marine and
continental biotas are considered together. This fact can be interpreted as a
proof of the integrated nature of the biosphere. But why, throughout the
Phanerozoic, did global biodiversity tend to follow a hyperbolic trend similar to
that which we observed for the World System in general and China in
particular?
Figure 17. Global Change in Total Biodiversity (number of genera, N) through the
Phanerozoic (following Markov and Korotayev 2008) a
a Exponential model: R2 = 0.67. Hyperbolic model: R2 = 0.95. The hyperbolic
line has been generated by the following equation:
t
Nt
30
434635
As we have noted above, in sociological models of macrohistorical
dynamics, the hyperbolic pattern of world population growth arises from non-
linear second-order positive feedback (more or less identical with the
mechanism of collective learning) between demographic growth and
technological development. Based on analogy with these sociological models
and diverse paleontological data, we suggest that the hyperbolic character of
biodiversity growth can be similarly accounted for by non-linear second-order
positive feedback between diversity growth and the complexity of community
structure: more genera higher alpha diversity enhanced stability and
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213
‘buffering’ of communities lengthening of average life span of genera,
accompanied by a decrease in the extinction rate faster diversity growth
more genera higher alpha diversity, and so on. Indeed, this begins to appear
as a (rather imperfect) analogue of the collective learning mechanism active in
social macroevolution.
The growth of genus richness throughout the Phanerozoic was mainly due
to an increase in the average longevity of genera and a gradual accumulation of
long-lived (stable) genera in the biota. This pattern reveals itself in a decrease
in the extinction rate. Interestingly, in both biota and humanity, growth was
facilitated by a decrease in mortality rather than by an increase in the birth
rate. The longevity of newly arising genera was growing in a stepwise manner.
The most short-lived genera appeared during the Cambrian; more long-lived
genera appeared in Ordovician to Permian; the next two stages correspond to
the Mesozoic and Cenozoic (Markov 2001, 2002).We suggest that diversity
growth can facilitate the increase in genus longevity via progressive stepwise
changes in the structure of communities.
Most authors agree that three major biotic changes resulted in the
fundamental reorganization of community structure during the Phanerozoic:
Ordovician radiation, end-Permian extinction, and end-Cretaceous extinction
(Bambach 1977; Sepkoski et al. 1981; Sepkoski 1988, 1992; Markov 2001;
Bambach et al. 2002). Generally, after each major crisis, the communities
became more complex, diverse, and stable. The stepwise increase of alpha
diversity (i.e., the average number of species or genera in a community)
through the Phanerozoic was demonstrated by Bambach (1977) and Sepkoski
(1988). Although Powell and Kowalewski (2002) have argued that the
observed increase in alpha diversity might be an artifact caused by several
specific biases that influenced the taxonomic richness of different parts of the
fossil record, there is evidence that these biases largely compensated for one
another so that the observed increase in alpha diversity was probably
underestimated rather than overestimated (Bush and Bambach 2004).
Another important symptom of progressive development of communities is
an increase in the evenness of species (or genus) abundance distribution. In
primitive, pioneer, or suppressed communities, this distribution is strongly
uneven: the community is overwhelmingly dominated by a few very abundant
species). In more advanced, climax, or flourishing communities, this
distribution is more even (Magurran 1988). The former type of community is
generally more vulnerable. The evenness of species richness distribution in
communities increased substantially during the Phanerozoic (Powell and
Kowalewski 2002; Bush and Bambach, 2004). It is most likely there was also
an increase in habitat utilization, total biomass, and the rate of trophic flow in
biota through the Phanerozoic (Powell and Kowalewski 2002).
The more complex the community, the more stable it is due to the
development of effective interspecies interactions and homeostatic
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mechanisms based on the negative feedback principle. In a complex
community, when the abundance of a species decreases, many factors arise
that facilitate its recovery (e.g., food resources rebound while predator
populations decline). Even if the species becomes extinct, its vacant niche may
‘recruit’ another species, most probably a related one that may acquire
morphological similarity with its predecessor and thus will be assigned to the
same genus by taxonomists. So a complex community can facilitate the
stability (and longevity) of its components, such as niches, taxa and
morphotypes. This effect reveals itself in the phenomenon of ‘coordinated
stasis’. The fossil record contains many examples in which particular
communities persist for million years while the rates of extinction and
taxonomic turnover are minimized (Brett et al. 1996, 2007).
Selective extinction leads to the accumulation of ‘extinction-tolerant’ taxa in
the biota (Sepkoski 1991b). Although there is evidence that mass extinctions
can be nonselective in some aspects (Jablonski 2005), they are obviously
highly selective with respect to the ability of taxa to endure unpredictable
environmental changes. This can be seen, for instance, in the selectivity of the
end-Cretaceous mass extinction with respect to the time of the first occurrence
of genera. In younger cohorts, the extinction level was higher than that of the
older cohorts (see Markov and Korotayev 2007: Fig. 2). The same pattern can
be observed during the periods of ‘background’ extinction as well. This means
that genera differ in their ability to survive extinction events, and that
extinction-tolerant genera accumulate in each cohort over the course of time.
Thus, taxa generally become more stable and long-lived through the course of
evolution, apart from the effects of communities. The communities composed
of more stable taxa would be, in turn, more stable themselves, thus creating
positive feedback.
The stepwise change of dominant taxa plays a major role in biotic evolution.
This pattern is maintained not only by the selectivity of extinction (discussed
above), but also by the selectivity of the recovery after crises (Bambach et al.
2002). The taxonomic structure of the Phanerozoic biota was changing in a
stepwise way, as demonstrated by the concept of three sequential ‘evolutionary
faunas’ (Sepkoski 1992). There were also stepwise changes in the proportion of
major groups of animals with different ecological and physiological
parameters. There was stepwise growth in the proportion of motile genera to
non-motile, ‘physiologically buffered’ genera to ‘unbuffered’, and predators to
prey (Bambach et al. 2002). All these trends should have facilitated the
stability of communities. For example, the diversification of predators implies
that they became more specialized. A specialized predator regulates its prey’s
abundance more effectively than a non-specialized predator.
There is also another possible mechanism of second-order positive
feedback between diversity and its growth rate. Recent research has
demonstrated a shift in typical relative-abundance distributions in
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paleocommunities after the Paleozoic (Wagner et al. 2006). One possible
interpretation of this shift is that community structure and the interactions
between species in the communities became more complex. In post-Paleozoic
communities, new species probably increased ecospace more efficiently, either
by facilitating opportunities for additional species or by niche construction
(Wagner et al. 2006; Solé et al. 2002; Laland et al. 1999). This possibility
makes the mechanisms underlying the hyperbolic growth of biodiversity and
human population even more similar, because the total ecospace of the biota is
analogous to the ‘carrying capacity of the Earth’ in demography. As far as new
species can increase ecospace and facilitate opportunities for additional species
entering the community, they are analogous to the ‘inventors’ of the
demographic models whose inventions increase the carrying capacity of the
Earth.
Exponential and logistic models of biodiversity imply several possible ways
in which the rates of origination and extinction may change through time
(Sepkoski 1991a). For instance, exponential growth can be derived from
constant per-taxon extinction and origination rates, the latter being higher
than the former. However, actual paleontological data suggest that origination
and extinction rates did not follow any distinct trend through the Phanerozoic,
and their changes through time look very much like chaotic fluctuations
(Cornette and Lieberman 2004). Therefore, it is more difficult to find a simple
mathematical approximation for the origination and extinction rates than for
the total diversity. In fact, the only critical requirement of the exponential
model is that the difference between the origination and extinction through
time should be proportional to the current diversity level:
(12)   
where No and Ne are the numbers of genera with, respectively, first and last
occurrences within the time interval Δt, and N is the mean diversity level
during the interval. The same is true for the hyperbolic model. It does not
predict the exact way in which origination and extinction should change, but it
does predict that their difference should be roughly proportional to the square
of the current diversity level:
(13)    
In the demographic models discussed above, the hyperbolic growth of the
world population was not decomposed into separate trends of birth and death
rates. The main driving force of this growth was presumably an increase in the
carrying capacity of the Earth. The way in which this capacity was realized
either by decreasing death rate or by increasing birth rate, or both depended
upon many factors and may varied from time to time.
Grinin et al: Biological and Social Evolution. Cliodynamics 4.2 (2013)
216
The same is probably true for biodiversity. The overall shape of the
diversity curve depends mostly on the differences in the mean rates of diversity
growth in the Paleozoic (low), Mesozoic (moderate), and Cenozoic (high). The
Mesozoic increase was mainly due to a lower extinction rate (compared to the
Paleozoic), while the Cenozoic increase was largely due to a higher origination
rate (compared to the Mesozoic) (see Markov and Korotayev 2007: 316, Fig. 3a
and b). This probably means that the acceleration of diversity growth during
the last two eras was driven by different mechanisms of positive feedback
between diversity and its growth rate. Generally, the increment rate
((No Ne)/Δt) was changing in a more regular way than the origination rate
No/Δt and extinction rate Ne/Δt. The large-scale changes in the increment rate
correlate better with N2 than with N (see Markov and Korotayev 2007: 316,
Fig. 3c and d), thus supporting the hyperbolic rather than the exponential
model.
Conclusion
In mathematical models of historical macrodynamics, a hyperbolic pattern of
world population growth arises from non-linear second-order positive
feedback between the demographic growth and technological development.
Based on the analogy with macrosociological models and diverse
paleontological data, we suggest that the hyperbolic character of biodiversity
growth can be similarly accounted for by non-linear second-order positive
feedback between the diversity growth and the complexity of community
structure. This hints at the presence, within the biosphere, of a certain
analogue to the collective learning mechanism. The feedback can work via two
parallel mechanisms: (1) a decreasing extinction rate (more surviving taxa
higher alpha diversity communities become more complex and stable
extinction rate decreases more taxa, and so on), and (2) an increasing
origination rate (new taxa niche construction newly formed niches
occupied by the next ‘generation’ of taxa new taxa, and so on). The latter
possibility makes the mechanisms underlying the hyperbolic growth of
biodiversity and human population even more similar, because the total
ecospace of the biota is analogous to the ‘carrying capacity of the Earth’ in
demography. As far as new species can increase ecospace and facilitate
opportunities for additional species entering the community, they are
analogous to the ‘inventors’ of the demographic models whose inventions
increase the carrying capacity of the Earth.
The hyperbolic growth of Phanerozoic biodiversity suggests that
‘cooperative’ interactions between taxa can play an important role in evolution,
along with generally accepted competitive interactions. Due to this
‘cooperation’ (which may be roughly analogous to ‘collective learning’), the
evolution of biodiversity acquires some features of a self-accelerating process.
The same is naturally true of cooperation/collective learning in global social
Grinin et al: Biological and Social Evolution. Cliodynamics 4.2 (2013)
217
evolution. This analysis suggests that we can trace rather similar
macropatterns within both the biological and social phases of Big History.
These macropatterns can be represented by relatively similar curves and
described accurately with very simple mathematical models.
Addendum
An anonymous referee of this paper suggests that “there is a disconnect
between the verbal description of the model (e.g., as described in the 'cognitive
scheme') and the mathematical expression. The main point is that in the verbal
description you talk about carrying capacity, whereas the mathematical model
does not have carrying capacity (or any equilibrium point). I feel that it would
be better to model the effect of technology on the carrying capacity, rather than
the growth rate. Modifying the model in this way will also get rid of the
pathological blow-up behavior”.
Actually, such models already exist. The smartest model of this sort appears
to have been produced by Sergey Tsirel (2004), who applies two differential
equations. He chooses the classical logistic model of Pierre François Verhulst:
(14)
)1( K
N
rN
dt
dN
where K is the technologically determined carrying capacity of the Earth (as
regards the humans). Tsirel combines this with the TaageperaKremer
equation of technological growth (see above, equation (4)), which results in the
following system of differential equations:
(14)
)1( K
N
rN
dt
dN
(4)
aNK
dt
dK
Tsirel (2004) has demonstrated that this combination describes world
population dynamics up to the 1970s in rather an accurate way.
We feel that Tsirel’s model addresses the referee’s concerns: it traces the
effect of technology on carrying capacity, rather than on growth rate, actually
allowing one to “get rid of the pathological blow-up behavior.” (We would note,
however, that this concern with “blow-up behavior” seems to be a
preoccupation of our American colleagues. The aversion to this phenomenon is
absent among many Russian mathematicians [see, e.g., Kurdyumov 1999]).
Korotayev (2005) has proposed a simplified version of Tsirel’s model:
Grinin et al: Biological and Social Evolution. Cliodynamics 4.2 (2013)
218
(15)
)( KNaN
dt
dN
bNK
dt
dK
Korotayev’s version has also proven capable of accurately describing world
population dynamics up to the 1970s.
Note that it seems generally possible to apply the latter system of equations
to the modeling of biodiversity macrodynamics (we had to discard the Tsirel’s
system immediately due to the impossibility of an analogue of parameter r in
equation (14)). Indeed, Sepkoski (1991a, 1992) and Benton (1995) proposed
the following logistic equation to describe biodiversity macrodynamics (N):
(16)
NNNk
dt
dN )( max
where Nmax is a constant. Equation (16) is basically identical to equation (15).
On the other hand, the materials surveyed above suggest that, like K in
social macrosystems, Nmax in biological systems may be considered not as a
constant, but rather as a variable whose dynamics can be described by the
following equation:
(17)
max
max bNN
dt
dN
Thus, we arrive at another model that seems to be capable of describing both
social and biological macroevolutionary dynamics. However, we had to stop
our work with the model (15)-(16) at a rather early stage as we failed to find
any effective way to estimate values of Nmax empirically.
Acknowledgment
This research has been supported by the Russian Foundation for Basic
Research (Project # 13-06-00501a).
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... This conclusion appears to be of key importance when assessing the stage at which a newly discovered exoplanet is in the process of its chemical evolution towards life. [1][2][3][4][5][6][7]. Another important publication is their mathematical paper [8] relating to the new research field entitled "Big History". ...
... According to Markov and Korotayev, the hyperbolic character of biodiversity growth can be similarly accounted for by a feedback between the diversity and community structure complexity. They suggest that the similarity between the curves of biodiversity and human population probably comes from the fact that both are derived from the interference of the hyperbolic trend with cyclical and stochastic dynamics [1][2][3][4][5][6][7]." This author was inspired by the following Figure 4 (taken from the Wikipedia site http://en. ...
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Full-text available
The discovery of new exoplanets makes us wonder where each new exoplanet stands along its way to develop life as we know it on Earth. Our Evo-SETI Theory is a mathematical way to face this problem. We describe cladistics and evolution by virtue of a few statistical equations based on lognormal probability density functions (pdf) in the time. We call b-lognormal a lognormal pdf starting at instant b (birth). Then, the lifetime of any living being becomes a suitable b-lognormal in the time. Next, our "Peak-Locus Theorem" translates cladistics: each species created by evolution is a b-lognormal whose peak lies on the exponentially growing number of living species. This exponential is the mean value of a stochastic process called "Geometric Brownian Motion" (GBM). Past mass extinctions were all-lows of this GBM. In addition, the Shannon Entropy (with a reversed sign) of each b-lognormal is the measure of how evolved that species is, and we call it EvoEntropy. The "molecular clock" is re-interpreted as the EvoEntropy straight line in the time whenever the mean value is exactly the GBM exponential. We were also able to extend the Peak-Locus Theorem to any mean value other than the exponential. For example, we derive in this paper for the first time the EvoEntropy corresponding to the Markov-Korotayev (2007) "cubic" evolution: a curve of logarithmic increase.
... Following this link between military technologies and socio-cultural development, we might expect to find a positive feedback between technological innovation and population growth at the global scale [2,[28][29][30][31][32] see also [33][34][35][36]. Indeed, a well-known and much discussed theory proposed by economist Michael Kremer and expanded by others suggests exactly this causal link [2]. ...
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... External conflict theories propose that competition between societies, usually taking the form of warfare, imposes a selection regime that weeds out relatively dysfunctional, poorly organized, and internally uncooperative polities, favoring those with larger populations and effective, centralized, and internally-specialized institutions (7,(67)(68)(69)(70)(71). The main proxy for the warfare hypothesis is the Seshat measure of the realized sophistication and variety of military technologies used by polities, MilTech (72). ...
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... Clearly research has been conducted for quite some time but the interdisciplinary nature of the topic meant that it was difficult to find others with the same interest. For example, the group of Russians carried out extensive work in collecting, analyzing, and interpreting data before the Big History movement facilitated the dissemination of their work (see, e.g., Panov 2004Panov , 2005aPanov , 2005bPanov , 2006Panov , 2008Panov , 2011Panov , 2017Nazaretyan , 2015aNazaretyan , 2015bNazaretyan , 2016Nazaretyan , 2017Nazaretyan , 2018Grinchenko 2001Grinchenko , 2004Grinchenko , 2006aGrinchenko , 2006bGrinchenko , 2007Grinchenko , 2011Grinchenko , 2015Grinchenko and Shchapova 2010, 2016, 2017a, 2017bMarkov and Korotayev 2007, 2009Markov et al. 2010;Grinin et al. 2013Grinin et al. , 2014Korotayev 2015Korotayev , 2018. We believe that there are many others that we were not able to identify along with others that were interested but could not participate at this time. ...
... Following this logic, we might expect to find a positive feedback between technological innovation and population growth at the global scale (Taagepera 1976, 1979, Kremer 1993, Komlos and Nefedov 2002, Tsirel 2004, Podlazov 2017) see also (Korotayev 2005, Korotayev et al. 2006, Grinin et al. 2013, Korotayev and Malkov 2016. According to Kremer's much discussed model, "high population spurs technological change because it increases the number of potential inventors… 1 Thus, in a larger population there will be proportionally more people lucky or smart enough to come up with new ideas" (Kremer 1993: 685). ...
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The causes and consequences of technological evolution in world history have been much debated. Of particular importance in many of the theoretical and empirical studies on this topic is innovation in military technologies, details of which are comparatively well preserved in the archaeology and historical record and which are often seen as drivers of broad socio-cultural processes. Here we analyze data on the evolution of key military technologies in a stratified sample of the world’s political systems from the Neolithic to the industrial revolution using Seshat: Global History Databank. Empirically testing a series of previously speculative theories reveals that world population size (as proxy for the potential numbers of innovators), the connectivity between areas of innovation and adoption, and major past innovations such as iron metallurgy and horse riding, all serve as strong predictors of change in military technology. We discuss how the approach showcased here could be extended not only to explain more of the causes and consequences of military innovation but of technological change more generally, with important ramifications for our understanding of the drivers of world history and of the evolution of social complexity.
... Действительно, А. А. Фомин (2018) обращает внимание на то, что на протяжении социальной фазы Большой истории/универсальной эволюции население Земли между каждой парой биосферных революций увеличивалось примерно в одно и то же число раз (где-то порядка 2,8). Отметим, что это совсем не плохо согласуется со многими математическими моделями гиперболического роста численности населения Земли 42 , рассматривающим его как следствие функционирования механизма положительной обратной связи второго порядка между демографическим ростом и технологическим 42 См., например: Подлазов 2000Коротаев, Малков, Халтурина 2005а, 2005б, 2007Taagepera 1976;Kremer 1993;Tsirel 2004;Grinin, Markov, Korotayev 2013 развитием, когда технологическое развитие (наиболее ярко проявлявшееся именно в виде «биосферных революций» типа неолитической или промышленной) значительно ускоряло темпы роста населения, который (в силу действия принципа «чем больше людей, тем больше изобреталей») через механизмы коллективного обучения ускорял наступление каждой следующей «биосферной революции» (как правило, соответствовашей новому технологическому прорыву). При этом А. А. Фомин (2018) достаточно убедительно показывает математически, что «если имеется гиперболический рост количества эволюционных единиц (обобщенное название народонаселения на случай и биологической эволюции), то прирост числа этих единиц в одно и то же количество раз α будет приводить к тому, что промежутки времени между моментами этих приростов будут сокращаться в точно такое же количество раз α» -то есть, если между биосферными революциями население в среднем увеличивается в α раз, то и промежутки времени между каждой последующей парой биосферных революций будет сокращаться в α раз (Отметим, что последняя α это ничто иное, как то, что А. Д. Панов (2005: 128) называет «коэффициентом ускорения исторического времени, показывающим, во сколько раз каждая последующая эпоха 43 короче предыдущей»). ...
... Closely related to our objectives, the new field of historical dynamics and its mathematical formulation, Cliodynamics [Tur11], focuses on finding a scientific modelling of history, including its computer modelling [GMK13]. Collins [Col10] modelled victory or defeat in battle as a set of flow charts for dynamic simulation. ...
Article
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Chapter
The discovery of new exoplanets makes us wonder where each new exoplanet stands along its way to develop life as we know it on Earth. Our Evo-SETI Theory is a mathematical way to face this problem. We describe cladistics and evolution by virtue of a few statistical equations based on lognormal probability density functions (pdf) in the time. We call b-lognormal a lognormal pdf starting at instant b (birth). Then, the lifetime of any living being becomes a suitable b-lognormal in the time. Next, our “Peak-Locus Theorem” translates cladistics: each species created by evolution is a b-lognormal whose peak lies on the exponentially growing number of living species. This exponential is the mean value of a stochastic process called “Geometric Brownian Motion” (GBM). Past mass extinctions were all-lows of this GBM. In addition, the Shannon Entropy (with a reversed sign) of each b-lognormal is the measure of how evolved that species is, and we call it EvoEntropy. The “molecular clock” is re-interpreted as the EvoEntropy straight line in the time whenever the mean value is exactly the GBM exponential. We were also able to extend the Peak-Locus Theorem to any mean value other than the exponential. For example, we derive in this chapter the EvoEntropy corresponding to the Markov-Korotayev (2007) “cubic” evolution: a curve of logarithmic increase.
Chapter
This long Chapter “Evo-SETI Mathematics: Part 1: Entropy of Information. Part 2: Energy of Living Forms. Part 3: The Singularity” is a string of the most important mathematical Theorems making up for our Evo-SETI Theory. If newcomers to Evo-SETI are “scared” by this large amount of mathematics, they may SKIP this Chapter “Evo-SETI Mathematics: Part 1: Entropy of Information. Part 2: Energy of Living Forms. Part 3: The Singularity” in the first instance, and come back to this Chapter “Evo-SETI Mathematics: Part 1: Entropy of Information. Part 2: Energy of Living Forms. Part 3: The Singularity” LATER, when they will have mastered the basic ideas of Evo-SETI theory. Actually, a profound difference exists in between the present Chapter and the rest of this book: the rest of this book is made up by PAPERS that were published by this author during the 2010–2020 decade. Each of these papers covers a rather self-constrained topic (e.g. Evolution of living Species over 3.5 billion years, or Mass Extinctions, or History of Human Civilizations as b-lognormals, or where each planet/exoplanet stands in regard to developing Life, on and so on). Thus, our readers might prefer to read about each topic SEPARATELY by reading the relevant paper, and only at the end they will realize that our statistical equations are so flexible that in reality they make up for a single large mathematical topic: our Evo-SETI Theory.
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Human society is a complex nonequilibrium system that changes and develops constantly. Complexity, multivariability, and contradictions of social evolution lead researchers to a logical conclusion that any simplification, reduction, or neglect of the multiplicity of factors leads inevitably to the multiplication of error and to significant misunderstanding of the processes under study. The view that any simple general laws are not observed at all with respect to social evolution has become totally dominant within the academic community, especially among those who specialize in the Humanities and who confront directly in their research the manifold unpredictability of social processes. A way to approach human society as an extremely complex system is to recognize differences of abstraction and time scale between different levels. If the main task of scientific analysis is to detect the main forces acting on systems so as to discover fundamental laws at a sufficiently coarse scale, then abstracting from details and deviations from general rules may help to identify measurable deviations from these laws in finer detail and shorter time scales. Modern achievements in the field of mathematical modeling suggest that social evolution can be described with rigorous and sufficiently simple macrolaws. The first book of the Introduction (Compact Macromodels of the World System Growth. Moscow: Editorial URSS, 2006) discusses general regularities of the World System long-term development. It is shown that they can be described mathematically in a rather accurate way with rather simple models. In the second book (Secular Cycles and Millennial Trends. Moscow: Editorial URSS, 2006) the authors analyze more complex regularities of its dynamics on shorter scales, as well as dynamics of its constituent parts paying special attention to "secular" cyclical dynamics. It is shown that the structure of millennial trends cannot be adequately understood without secular cycles being taken into consideration. In turn, for an adequate understanding of cyclical dynamics the millennial trend background should be taken into account. In this book the authors analyze the interplay of trend and cyclical dynamics in Egypt and Subsaharan Africa.
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The dynamics of the marine faunal diversity in the Phanerozoic are analyzed on the basis of the total duration of genera (D). The diagramatic representation of D is more regular and thus easier to interpret than the well-known curve of the number of genera (NG). Four periods when D increased at constant rates are identified. These periods of linear growth are separated by crisis periods. The constant rates of D form a rather strict geometric progression. This progression is the first quantitative pattern in the dynamics of the marine faunal diversity of the Phanerozoic that is obeyed so accurately.
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Taxonomic diversity dynamics traditionally interpreted using exponential or logistic models of diversification, both of which are based on the assumption that the rate of origination (and sometimes also the rate of extinction) depends on the level of taxonomic diversity. Paleontological data, however, give inadequate support for this assumption. Therefore, an alternative model is suggested: the generic origination rate is stochastically constant and does not depend on the diversity level; genera differ in their vulnerability; the extinction probability for each genus during each time interval depends on its vulnerability only. Apparently, the most important factor of the increase in diversity in marine biota during the Phanerozoic was a stepwise increase in the mean generic durations. There were four such steps: Cambrian, Ordovician-Permian, Mesozoic, and Cenozoic. This stepwise increase in generic durations was partly due to the successive replacement of dominating groups, but to a larger extent, it was due to the generic durations that increased within each group at each successive step.
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This paper views the standard production function in macroeconomics as a reduced form and derives its properties from microfoundations. The shape of this production function is governed by the distribution of ideas. If that distribution is Pareto, then two results obtain: the global production function is Cobb-Douglas, and technical change in the long run is labor-augmenting. Kortum showed that Pareto distributions are necessary if search-based idea models are to exhibit steady-state growth. Here we show that this same assumption delivers the additional results about the shape of the production function and the direction of technical change. © 2005 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology.