Douglas R. WhiteDouglas R. WhiteAndrey KorotayevAndrey KorotayevDaria KhaltourinaDaria Khaltourina
Secular Cycles and Millennial Trends
Irvine, 2004Irvine, 2004
NCNr
dtdN )1(
Where:
N (t) - population r - rate of natural growth C - maximum carrying capacity
Fig. 1. The logistics curve and the curve of consumption per capita
population consumption per capita
Nefedov’s Model of Agrarian Demographic CycleQ(t) = F[P(t)]A
F(P) = Q/A = ksps Q = kspsA.
A(P)= kP , if kP< AmA(P)= Am , if kP > Am
AF(Y) = kY , if kY < Am - AT AF(Y) = Am - AT , if kY > Am - AT
M = ksqAFP0 = p0Y(t) W=M+P0
u = (X (t) – M) /Y (t)Half of the surplus is stored for future consumption:
pc = (u+ p0)/2 if u > p0 and (u+ p0)/2 < pm pc = pm if (u+ p0)/2 > pm
If u<p0 , then X(t) < W.P1 = pcY(t)W1=M+P1
Zp = X(t) – W1l = l0+ dl0
Q = ksps AF = ksql AF = lM X(t+1) = lM – H + X(t) – W1
(4))()1(1
)()1(
CtYr
trYtY
R. Pearl's Model
Y – populationr - the rate of natural growth in favorable conditions C - capacity of an ecological niche or the maximal population at available food resources
CtY )(
by
n
CtY
)(
where n is a parameter of compensation suggested by J. Maynard Smith and M. Slatkin
In the model is replaced with
If X(t)<W
X (t) - the quantity of grain after harvest (crop and stocks)W - is the quantity of grain needed to cover minimum consumption and seed
The peasants have grain deficits.
Then the peasants lack sufficient grain in spring sowing even if they consume p0 (the minimal total consumption per capita). Then they sell part of their land in order to compensate for the lack of seed grain.
In some cases the landowners have a limited stock of grain and can not buy all the land sold by the peasants.
Then the peasants reduce their fund of consumption P1 so, that
M+P1 = X(t).
M - the total grain requirements for seed P1 - total consumptionX (t) - the quantity of grain after harvest (crop and stocks)
In this case u < p0
and consumption per capita equals p(u)=P1 / Y(t).
During the famine P1 < p0 Y(t)
and
Y(t)/C = Y(t)/( P1/p0) = p0Y(t)/ P1 > 1 in (4).
Therefore population is reduced.
If the famine threatens destruction of a significant part of the population the authorities distribute
grain to the peasants, increasing consumption up to pu0 (pu0 < p0).
Na = Da/1.5 Ma = ksqAT
Pa0 = p0Ya (t)Pa= pua Ya (t)
Xa (t+1) = (Mal – Ma – Ha)/2 + Xa (t) – PaXr(t+1) = kr (lMa–Ha)/2 – Hr + Xr (t) – Pr
G = (ksql –ksq) Da – Haa )/2 Cr = Pr/p0
Where: 0.75 – minimal cultivated area per person necessary to maintain minimal level of consumption (ha) (for the Late Han period); 1/2 – the rent is assumed to be one half of the total product; A(t) – cultivated area; AF – the area of land belonging to the farmers; Am – maximum size of area under cultivation; AT – land of the tenants; C – capacity of an ecological niche, or the maximum population at available level of agricultural technology; Cr – maximum number of craftsmen; d – random variable; Da – area of land sold by peasants = area of land passing to new tenants; G – income of landowners; H – total amount of taxes in terms of grain; Ha – total amount of taxes paid by tenants in terms of grain; Hr – total amount of taxes paid by craftsmen in terms of grain; kr – % of landowners' income spent for purchase of craft products; ks – multiple-cropping index (sown area divided by cultivated area); l – real productivity; l0 – the output of grain per sowing, harvest/seeds proportion; lM – crop of the next year; M – total grain requirements for seed; Ma – weight of seed grain of the tenants; Na – peasant population decreases; P(t) – the rural population at period t; p(u) – consumption per capita; P0 – minimum total consumption; p0 – minimum total consumption per capita; P1 – total consumption; Pa – total consumption of tenants; Pa0 – minimal total consumption of tenants; pa0 – minimal total consumption per capita of tenants; pc – consumption per capita; pm – maximum consumption; Pr – total consumption of craftsmen; ps – productivity per hectare; pua – consumption per capita of tenents; Q – crop output measured in kilograms; q – quantity of seed needed per hectare; u – the amount of grain available per capita; W – is the quantity of grain needed to cover minimum consumption and seed; W1 – total grain output is used for consumption and seed; X(t) – quantity of grain after harvest (crop + stocks); Xa(t) – Pa – stocks saved in barns of the tenants; Xa(t) – the quantity of grain after harvest (crop and stocks) that tenants have; Xr – stocks of a grain of handicraftsmen; Y(t) – the number of peasants; Ya(t) – number of tenants; Zp – available grain stock by the time of the following crop.
0
10
20
30
40
50
60
57 64 71 78 85 92 99 106
113
120
127
134
141
148
155
162
169
176
183
190
population (documents)
calculated population
farmers
sowing area (calculation)
stocks of the peasants
sowing area (documents)
number of the tenants
craftsmen and servant
Economic dynamics in the period Later HanEconomic dynamics in the period Later Han
Population and consumption in Qing Epoch in ChinaPopulation and consumption in Qing Epoch in China
–▄– – consumption (daily wages, liters of rice)–♦– – population (millions)
Population in Early Tang China (the number of households in Population in Early Tang China (the number of households in millionsmillions))
Population in Late Tang China (the number of households in millions)
Consumption in Babylonia in the 6th – early 5th centuries BC. The numbers indicate the amount of barley in liters that a blue worker
could buy on his daily wage.
Consumption dynamics in Northern India in the late 16th – 17th centuries. The numbers indicate the amount of wheat in liters that a unskilled
worker could buy on his daily wage.
dx/dt = Ax – Bxydy/dt = Cxy – Dy
(where x is population density ["prey"] and y is warfare frequency ["predator"])
(a)
Year
0 200 400 600 800 1000
Pre
y de
nsity
0
2
4
6
8
10
Pre
dato
r den
sity
0
2
4
6
8
(b)
Prey
0 2 4 6 8 10
Pre
dato
r
0
2
4
6
8
DataRegr.
(c)
Prey
0 2 4 6 8 10
Pre
dato
r's ra
te o
f cha
nge
-0.1
0.0
0.1
0.2
DataRegr.
(a) Temporal dynamics of prey (solid curve) and predator (broken curve) predicted by the Lotka-Volterra model with parameters a = 0.02, b = 0.02, c = 0.025, and d = 0.1.
(b) The scatterplot of relationship between P and N; the straight line is regression. (c) The scatterplot of the relationship between the logarithmic rate of change of P (∆p,
defined in the text) and N.
(a)
Year
1950 1955 1960 1965 1970 1975
Pre
y de
nsity
(log
-tran
sfor
med
)
-3
-2
-1
0
1
2
3
Pre
dato
r
0.0
0.2
0.4
0.6
0.8
1.0
(b)
Prey at t
-3 -2 -1 0 1 2 3
Pre
dato
r at t
0.0
0.2
0.4
0.6
0.8
1.0(c)
Prey at t-2
-3 -2 -1 0 1 2 3
Pre
dato
r
0.0
0.2
0.4
0.6
0.8
1.0
Population dynamics of the caterpillar (larch budmoth) and its predators (parasitic wasps). (a) Population oscillations of the caterpillar (solid curve) and predators (broken curve). (b) A scatter plot of the predator against the prey. The solid line is the regression. Broken lines connect consecutive data points, revealing the presence of cycles. (c) A scatter plot of the predator against prey lagged by two years.
(a)
Years
-200 -100 0 100 200 300
log
Pop
ulat
ion
1.2
1.4
1.6
1.8
log
Inte
rnal
War
0.0
0.5
1.0
NW
(b)
log Population
log
War
fare
(c)
log Population
1.2 1.4 1.6 1.8
War
rate
of c
hang
e
-0.5
0.0
0.5
Analysis of the Han Chinese data.
a) Population and warfare trajectories.
b) (The trajectory in the population-warfare phase space.
c) The relationship between the warfare rate of change and population density.
Where:N - number of inhabitants t - timer - population growth rate K - the population size at which surplus equals zero or "carrying capacity" W – internal warfare β - per capita state expenditure rate S - the accumulated state resources (e.g., in kg of grain)ρ - the per capita taxation rate at low population density
2
max
1( )
1( )
( )
dN NrN NWdt K W
dS NN Ndt K WdW aN bW SdtK W k cW
Temporal dynamics of population N (solid curve) and internal warfare I (broken curve)
0
50
100
150
200
250
300
350
400
450
500
200 BCE 0 500 1000 1500 1850
Trend to the Growth of Population (in millions) in China200 BCE – 1850 CE
-0.5
0
0.5
1
1.5
2
2.5
3
-250
0-2
450
-240
0-2
350
-230
0-2
250
-220
0-2
150
-210
0-2
050
-200
0-1
950
-190
0-1
850
-180
0-1
750
-170
0-1
650
-160
0-1
550
-150
0-1
450
-140
0-1
350
-130
0-1
250
-120
0-1
150
-110
0-1
050
-100
0-9
50-9
00-8
50-8
00-7
50-7
00-6
50-6
00-5
50
Trend to the Growth of the Largest State/Empire Territory Size Trend to the Growth of the Largest State/Empire Territory Size (millions of square km) in West Asian/Mesopotamia Centered System(millions of square km) in West Asian/Mesopotamia Centered System
2500 BCE – 550 BCE2500 BCE – 550 BCE
Trend to the Growth of the Largest State/Empire Territory Size (millions of square km) in West Asian/Mesopotamia Centered System
900 BCE – 850 CE
0
2
4
6
8
10
12
-900
-850
-800
-750
-700
-650
-600
-550
-500
-450
-400
-350
-300
-250
-200
-150
-100 -50 0 50 100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
-2
0
2
4
6
8
10
12
-25005-24005-23005-22005-21005-20005-19005-18005-17005-16005-15005-14005-13005-12005-11005-10005-9005-8005-7005-6005-5005-4005-3005-2005-100-5005010052005300540055005600650700750800850
Trend to the Growth of the Largest State/Empire Territory Size Trend to the Growth of the Largest State/Empire Territory Size (millions of square km) in West Asian/Mesopotamia Centered System(millions of square km) in West Asian/Mesopotamia Centered System
2500 BCE – 850 CE2500 BCE – 850 CE
Trend to the Growth of the Largest State/Empire Territory Size (millions of square km) in East Asian/China Centered System
1900 BCE – 1865 CE
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
-1900 1000 0 1000 1865
Population Growth Rate
3.53.02.52.01.51.0.50.0-.5
Rel
ativ
e C
onsu
mpt
ion
Rat
e
3.5
3.0
2.5
2.0
1.5
1.0
.5
0.0
-.5
Population Growth Rate
3.53.02.52.01.51.0.50.0-.5
Terr
itoria
l Exp
ansi
on/A
ggre
ssiv
e E
xter
nal W
arfa
re
4
3
2
1
0
-1
-2
Population Growth Rate X Territorial Expansion/Aggressive External Warfare
Relative Consumption Rate
3.53.02.52.01.51.0.50.0-.5
Terr
itoria
l Exp
ansi
on/A
ggre
ssiv
e E
xter
nal W
arfa
re4
3
2
1
0
-1
-2
Relative Consumption Rate X Territorial Expansion/Aggressive External Warfare
Core Area Population (direct and indirect evidence)
3.53.02.52.01.51.0.5
Ter
ritor
ial E
xpan
sion
/Agg
r. E
xter
nal W
arfa
re4
3
2
1
0
-1
-2
1 – low in comparison with other phases of respective cycle 2 – intermediate in comparison with other phases of respective cycle 3 – high in comparison with other phases of respective cycle
100115
180
350
430
4765
89110
165
0
50
100
150
200
250
300
350
400
450
500
1650 1700 1750 1800 1850
CHINA
PEKING
CHINA = overall population of China (millions)PEKING = population of Peking (tens of thousands)
0
10
20
30
40
50
60
70
80
90
100
1675 1725 1775 1825
Ch.Gr.
Pek.Gr.
Ch.Gr. = growth rate of the overall population of ChinaPek.Gr. = growth rate of Peking population
(r = –.84, p = .078)
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