为历史进行数学建模，遇到的困难：
think there are two major reasons explaining this failure. First, these
attempts were inspired directly by successes in physical sciences. Yet physicists
traditionally chose to deal with systems and phenomena that are very different
from those in history. Physicists tend to choose very simple systems with few
interacting components (such as the solar system, the hydrogen atom, etc) or
with systems consisting of a huge number of identical components (as in thermodynamics). As a result, very precise quantitative predictions can be made and
empirically tested. But even in physical applications, such systems are rare, and
in social sciences only very trivial questions can be reduced to such simplicity.
Real societies always consist of many qualitatively and quantitatively different
agents interacting in very complex ways. Furthermore, societies are not closed
systems: they are strongly affected by exogenous forces, such as other human
societies and by the physical world. Thus, it is not surprising that traditional
physical approaches honed on simple systems should fail in historical applica-
tions.
The second reason is that quantitative approaches typically employed by physi-
cists require huge amounts of precisely measured data. Nothing could be further from the
reality encountered by a historical sociologist, who typically lacks data about
many aspects of the historical system he is studying, while possessing fragmen-
tary and approximate information about others. For example, one of the most
important aspects of any society is just how many members it has. But even this
kind of information usually must be reconstructed by historians on the basis of
much guesswork.
一些进展：
First,during the last 20–30 years physicists and biologists have mounted a concerted
attack on complex systems. A number of approaches can be cited here: 【nonlinear dynamics, synergetics, complexity】, and so on. The use of powerful computers has been a key element in making these approaches work.
Second, biologists, and ecologists in particular, have learned 【how to deal with short and noisy datasets】. Again, plentiful computing power was a key enabler, allowing
such 【computer-intensive approaches as nonlinear model fitting, bootstrapping,
and cross-validation】.
Third, there is another hopeful development, this time in social sciences. I am refer-
ring to the rise of quantitative approaches in history, or Cliometrics (Williamson
1991). Currently, there are many investigators that collect quantitative data on
various aspects of historical process, and large amounts of data are already avail-
able in electronic form.
2.1.3
X' = caXY − dX
Y' = −aXY + bY.................................. (2.7)
Consumption is assumed to be proportional to the product of X and Y , for the following reasons. First, more consumers deplete resources faster. Second, when resources are plentiful, each individual consumer depletes resources faster than when resources are scarce. 引自第1页
但是，如果我把方程组写成这样：
X' = mX − nY
Y' = −pX + qY .................................. (my 2.7)
原因1依然可以解释得通；至于原因2，可以理解为当Y不足时，X因自我相互竞争而额外减少数量了。
所以问题是，导致Y减少的因素为什么是kXY,而不是kX?
（今天终于弄明白了，Y' = kXY 即Y'/Y = kX，表示的是Y的自然增长率（而不是Y的增长率）. http://book.douban.com/annotation/36617894/的伏尔特拉鱼群模型提到这个。至于为什么考虑用自然增长率而不用增长率，这个还没想明白 - 2015/11/22 ）
(2016-3-21
为什么考虑用自然增长率?或者说，为什么导致X减少的因素里与X本身的数量有关？
假设X均匀分布在区域S内。因为X与Y是捕食关系，所以不能交织地生活在一起；所以有理由假设S内没有Y，即Y在S外部；所以只有位于S边界上的X才有被捕食的可能。假设S内部X的数量为x，则S边界上X的数量可以近似地表示为sqrt(x)。 这可以解释 “为什么导致X减少的因素里与X的数量有关”。
)
p26
Thus, the imperial collapses occurring in the Hanneman model appear to be entirely due to the assumed delay with which legitimacy follows war success.
This means that if a feedback loop
operates on the scale of years, or even worse, weeks, than it is highly unlikely
that it could cause oscillations, whose average period is measured in centuries.
Centuries-long cycles are typically caused by feedbacks operating on the scale of
human generations (decades or longer).
p29
If the time-scale argument advanced in this chapter is accepted, it does not
mean that geopolitical mechanisms (in the narrow sense) are unimportant in the
explanation of the imperial rise and demise. They are just insufficient, and we
must look to other variables in order to build the required theory.
2.4 SUMMARY
• There are three very general classes of dynamical behaviors. Zero-order
dynamics, boundless growth or decline, arise in systems that are not af-
fected by negative feedbacks. Examples include linear and exponential
growth/decline processes. First-order dynamics characterize systems in
which feedbacks act on a fast scale. First-order dynamics are equilibrial;
examples include asymptotic and logistic growth processes. First-order
dynamics may also be metastable (more than one stable equilibrium is
present). Finally, second-order dynamics arise in systems in which dynam-
ical feedbacks act with a lag. Examples of second-order behaviors include
a single boom/bust dynamic and sustained periodic or chaotic oscillations.
• The geopolitical theory of Randall Collins postulates three main mecha-
nisms explaining territorial dynamics of states: geopolitical resources, lo-
gistical loads, and the marchland position.
• The mathematical model incorporating only the positive feedback between
territory and geopolitical resources exhibits zero-order dynamics. If the
initial state territory is above a certain threshold, than it grows in an ac-
celerating fashion. However, if the initial territory is below the threshold,
then the state shrinks and eventually disappears.
• Adding to the model the negative feedback of the logistical loads leads to
the first-order dynamical behavior, metastability. Again, if the initial terri-
tory is below the threshold, the state loses ground and disappears. However,
starting above the threshold, the territory does not increase without bound,
as in the simpler model, but approaches an upper equilibrium. This equi-
librium is stable with respect to small perturbations.
• In order to examine the positional effects I turn to a spatial simulation
model developed by Artzrouni and Komlos. This model suggests that states
initially enjoying marchland advantage (a higher proportion of boundary
along a coastline) grow to larger sizes than inland states. However, the
model does not exhibit any second-order oscillations: the loser states diss-
appear, while the winners grow to the limits set by logistical factors, where
their size is stabilized.
• Finally, I review the simulation model of conflict–legitimacy dynamics de-
veloped by Hanneman and coworkers. I show that if we translate this
model into differential equations, then we again obtain a first-order system
that is incapable of second-order oscillations.
• An analysis of growth/decline data tabulated by Rein Taagepera suggests
that long periods of imperial decline (more than a century) are frequently
found in the historical record (12 cases out of 31). This finding strongly
suggests that at least in some historical cases imperial dynamics were gov-
erned by second-order mechanisms. However, models based on purely
geopolitical mechanisms do not predict such prolonged declines. Thus, we
must investigate other mechanisms of imperial collapse.
也许N多年后，当这些理论趋于成熟时，会有更简洁的理论来阐述这些问题
5.4 CONCLUSION: THE MAKING OF EUROPE
“At each point in time from 1500 forward, what fea-
tures of a political unit would have permitted us to anticipate whether it would
(1) survive into the following period as a distinct unit; (2) undergo territorial
consolidation, centralization, ... (3) become the nucleus of a national state?”
(1975:40). This is precisely the kind of question that motivates the theory de-
veloped in this book. cannot claim that the metaethnic frontier theory gives a
precise and accurate answer for the territorial dynamics of every European state.
However, I believe that the theory explains the general pattern of European po-
litical development—“the Making of Europe”
Note that the underlying force for state breakdown in both models is the
population growth which leads to a gradual fall in income per head, until even-
tually the surplus over bare subsistence is insufficient to provide for the ruling
class
State bankruptcy and collapse can be prevented only if population
numbers are kept below the critical threshold at which state expenditures match
state revenues.
7.4 SUMMARY
• Population growth in agrarian societies leads to a gradual decrease in per
capita production, due to diminishing returns on labor inputs into agricul-
ture. The surplus available for the state shrinks, and eventually becomes
insufficient to provide for the state’s needs. Thus, unchecked population
growth leads to the fiscal insolvency of the polity.
• State breakdown and ensuing sociopolitical instability have a negative im-
pact on both demographic rates (birth, death, and migration rates) and the
productive capacity of the society. As a result, state breakdown is typically
followed by periods of population decline.
• A simple mathematical model of the interaction between population dynam-
ics and the fiscal health of state predicts recurrent episodes of state building
and population growth followed by state breakdown and population decline.
For the range of plausible parameter values this demographic-fiscal model
predicts that state breakdown should occur at somewhat irregular intervals
2–3 centuries apart.
• A more sophisticated model that keeps separate track of the commoner
and elite segments of population predicts similar dynamics (periodic state
breakdown). Growth of commoner population leads to the expansion of
elite numbers and, initially, strengthening of the state. However, as elite
numbers grow, an increasing proportion of extracted product is appropriated
by the elites for their needs. State revenues decline, while expenditures
increase, and eventually the state becomes bankrupt.
• I develop two models focusing on political dynamics of nomadic elites
interacting with agrarian states. The Ibn Khaldun model investigates the
situation in which nomads periodically conquer the agrarian state and es-
tablish a ruling dynasty there. The parasitic nomad model addresses the
situation in which nomads do not conquer the agrarian state, instead im-
posing a tribute on it. In both cases models predict faster cycles of polity
building and collapse (periods of about a century), compared to cycles gen-
erated by the demographic-fiscal model and its class-structured variants.
• I briefly review two economic models advanced to explain the dynastic
cycle in Chinese history. The basic logic underlying the explanation offered
by these models of the collapse of Chinese empires is also Malthusian-
Ricardian, although they are somewhat more complex than the models I
develop.
• The structural-demographic explanation of early modern state breakdown
developed by Goldstone (1991b) postulates population growth as the pri-
mary cause of state collapse. It is important to emphasize, however, that
in this theory (and in the models I develop) population dynamics do not
directly cause the state breakdown. Rather, population growth promotes po-
litical instability indirectly, by causing the state’s fiscal distress, increased
elite competition and factionalism, and increased mobilization potential of
general populace. I review one of Goldstone’s case studies, the English
Revolution, for which a particularly complete cliometric database exists.
These data are found to be in very good agreement with the quantitative
predictions of the structural-demographic model.
• Finally, I briefly review Fischer’s (1996) theory and data dealing with four
“great waves” of socio-economic dynamics in Western Europe from the
twelfth to twentieth century. Again, my conclusion is that these ideas and
analyses are broadly consistent with the theory developed in this chapter.
• Depending on structural assumptions and parameter values the models, de-
veloped in this chapter, can generate a variety of dynamics. Oscillations
can be either regular, or quite irregular. Average periods are generally long,
but may range from one to three centuries. Changes in commoner popula-
tion may be very drastic, or mild (in the case of “elite cycles”). But one
general conclusion is that agrarian states are characterized by instability
and oscillations on the time scale of centuries. I propose that we call such
dynamics secular cycles or waves.
So it may
be appropriate to think about historical dynamics as a set of cycles of different
periods nested with each other.
because the final collapse of empires
typically occurs during one of the demographic-structural crises
