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)