Douglas R. White
Andrey Korotayev
Daria Khaltourina
Secular Cycles and Millennial Trends
Irvine, 2004
dN
N
 r (1  ) N
dt
C
Where:
N (t) - population
r - rate of natural growth
C - maximum carrying capacity
population
consumption per capita
Fig. 1. The logistics curve and the curve of
consumption per capita
Nefedov’s Model of Agrarian Demographic Cycle
Q(t) = F[P(t)]A
F(P) = Q/A = ksps
Q = kspsA.
A(P)= kP , if kP< Am
A(P)= Am , if kP > Am
AF(Y) = kY ,
if kY < Am - AT
AF(Y) = Am - AT , if kY > Am - AT
M = ksqAF
P0 = 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) – W1
l = l0+ dl0
Q = ksps AF = ksql AF = lM
X(t+1) = lM – H + X(t) – W1
R. Pearl's Model
rY(t)
Y (t  1) 
Y (t)
1  (r 1)
C
(4)
Y – population
r - the rate of natural growth in favorable conditions
C - capacity of by
an ecological niche or the maximal population at available food resources
In the model
Y (t )
C
is replaced with
 Y (t ) 


 C 
n
where n is a parameter of compensation suggested by
J. Maynard Smith and M. Slatkin
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 consumption
X (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
Therefore population is reduced.
in (4).
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) – Pa
Xr(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
Economic dynamics in the period Later Han
60
population (documents)
50
calculated population
farmers
40
sowing area (calculation)
30
stocks of the peasants
20
sowing area (documents)
10
number of the tenants
craftsmen and servant
190
183
176
169
162
155
148
141
134
127
120
113
106
99
92
85
78
71
64
57
0
Population 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
millions)
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 – Bxy
dy/dt = Cxy – Dy
(where x is population density ["prey"] and y is warfare frequency ["predator"])
(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)
8
Prey density
8
6
6
4
4
2
2
0
0
200
400
600
0
1000
800
Year
(b)
(c)
0.2
Data
Regr.
6
Predator
Predator's rate of change
8
4
2
0
Data
Regr.
0.1
0.0
-0.1
0
2
4
6
Prey
8
10
0
2
4
6
Prey
8
10
Predator density
10
(a)
1.0
2
0.8
1
0.6
0
0.4
Predator
Prey density (log-transformed)
3
-1
0.2
-2
-3
1950
0.0
1955
1960
1965
1970
1975
Year
(c)
1.0
1.0
0.8
0.8
Predator
Predator at t
(b)
0.6
0.4
0.6
0.4
0.2
0.2
0.0
0.0
-3
-2
-1
0
Prey at t
1
2
3
-3
-2
-1
0
1
Prey at t-2
2
3
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.
1.8
log Population
1.0
1.6
0.5
1.4
1.2
-200
0.0
-100
0
100
200
300
Years
log Warfare
(b)
log Population
(c)
War rate of change
0.5
0.0
-0.5
1.2
log Internal War
(a)
N
W
1.4
1.6
log Population
1.8
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.


dN
N
 rN  1 
  NW

dt
K (W ) 



dS
N
  N 1 
N

dt
K (W ) 

dW
 aN 2  bW   S
dt
K (W )  k max  cW
Where:
N - number of inhabitants
t - time
r - 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
Temporal dynamics of population N (solid curve) and internal warfare I (broken curve)
Trend to the Growth of Population (in millions) in China
200 BCE – 1850 CE
500
450
400
350
300
250
200
150
100
50
0
200 BCE
0
500
1000
1500
1850
-0.5
-2500
-2450
-2400
-2350
-2300
-2250
-2200
-2150
-2100
-2050
-2000
-1950
-1900
-1850
-1800
-1750
-1700
-1650
-1600
-1550
-1500
-1450
-1400
-1350
-1300
-1250
-1200
-1150
-1100
-1050
-1000
-950
-900
-850
-800
-750
-700
-650
-600
-550
Trend to the Growth of the Largest State/Empire Territory Size
(millions of square km) in West Asian/Mesopotamia Centered System
2500 BCE – 550 BCE
3
2.5
2
1.5
1
0.5
0
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
12
10
8
6
4
2
-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
0
Trend to the Growth of the Largest State/Empire Territory Size
(millions of square km) in West Asian/Mesopotamia Centered System
2500 BCE – 850 CE
12
10
8
6
4
2
0
-2500
-2400
5 -2300
5 -2200
5 -2100
5 -2000
5 -1900
5 -1800
5 -1700
5 -1600
5 -1500
5 -1400
5 -1300
5 -1200
5 -1100
5 -1000
5 5-900
5-800
5-700
-5600
-5500
-54005-300
5-200
5-100
-50050
10052005300540055005600
650
700
750
800
850
-2
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
16.00
14.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00
-1900
1000
0
1000
1865
3.5
3.0
2.5
2.0
1.5
1.0
.5
0.0
-.5
-.5
0.0
.5
1.0
Population Growth Rate
1.5
2.0
2.5
3.0
3.5
Population Growth Rate X Territorial Expansion/Aggressive External Warfare
4
3
2
1
0
-1
-2
-.5
0.0
.5
1.0
Population Growth Rate
1.5
2.0
2.5
3.0
3.5
Relative Consumption Rate X Territorial Expansion/Aggressive External Warfare
4
3
2
1
0
-1
-2
-.5
0.0
.5
1.0
Relative Consumption Rate
1.5
2.0
2.5
3.0
3.5
4
3
2
1
0
-1
-2
.5
1.0
1.5
2.0
2.5
3.0
Core Area Population (direct and indirect evidence)
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
3.5
CHINA = overall population of China (millions)
PEKING = population of Peking (tens of thousands)
500
450
430
400
350
350
300
CHINA
250
PEKING
200
180
165
150
100
50
100
47
0
1650
115
89
110
65
1700
1750
1800
1850
Ch.Gr. = growth rate of the overall population of China
Pek.Gr. = growth rate of Peking population
(r = –.84, p = .078)
100
90
80
70
60
Ch.Gr.
50
Pek.Gr.
40
30
20
10
0
1675
1725
1775
1825