World Population
Phases, Transitions, and
Contemporary Crisis
White, Makov, Korotayev © 2005
What is power-law growth?
1000
900
800
700
600
500
400
300
200
100
0
1000
900
800
700
600
500
400
300
200
100
0
-2
y = 1000τ
R2 = 1
t=10
1
29
38
74
65
56
47
38
29
(a) Time numbering toward singularity at 11
110
y = N0
τ1b=10
2 9 38
(τmax/τb)2
47
56
65
74
83
(b) Time in reverse numbering to singularity at 0
Figure 1. Power-Law Growth in forward time (t) and backward from singularity (τ)
Hypothetical example
r=2
dN/dt = a0N
92
1
10
Premodern example
r=2
dN/dt = a0N
Figure 2. Power-Law Growth Pattern for Hawaiian Population Estimates,
1 – 1832 CE, with 95% confidence intervals (Komori 1992)
Other possibilities for
r=?
dN/dt = a0N
Regime
Growth (a0 > 0)
Zero Growth
Decline (a0 < 0)
Regime
r < 0 Convergent
r = 0 Fixed Change
0 < r < 1 Explosion
r = 1 Malthusian
r > 1 Singularity
Converges up to N*
Decelerating
nd
2 Order Polynomial Rise
Exponential Rise
Power-Law Rise
a0 = 0
a0 = 0
a0 = 0
a0 = 0
a0 = 0
Converges down to N*
Accelerating
nd
2 Order Polynomial Fall
Exponential Decay
Power-Law Decline
r>0
Init. Slow Extinction
Plunge
Rapid Extinction
Slow Death
Table 1: Variations of dN/dt = a0Nr
dN/dt = a0N
r=2
40000000
Nsq 2000
35000000
1995
30000000
1990
1985
1980
1975
1970
25000000
20000000
15000000
10000000
5000000
0
0
20
40
60
80
100
Figure 8: Test of Kapitza’s hypothesis, plotting x = dN/dt, and y = N2
World population over 1M years
100000000
10000000
1000000
100000
10000
1000
100
10
1
0.000001 0.000010 0.000100 0.001000 0.010000 0.100000 1.000000 10.00000 100.0000 1000.000
0
00
000
0.1
Figure 9: Kapitza trend line, plotting x = log (dN/dt) and y = log (N2)
(trend line is for 200 BCE to 1970)
World population over 1M years
China
r=2
dN/dt = a0N
Figure 3. Exponential Growth of Chinese Population in Millions, 1 – 2000 CE and projections to 2050
(Population. History and Projection, International Institute for Applied Systems Analysis)
Time to criticality
3500
3500
3000
3000
2500
2500
2000
2000
1500
1500
1000
1000
500
500
0
-200
y = 89603x-0.8359
R2 = 0.9936
0
0
200
400
600
800 1000 1200 1400 1600 1800 2000
0
500
1000
1500
2000
(a) Double fit least squares K=1.88 x 1011 and k=0.973; R2 =.994 log-predicted/log-actual
(b) Right: Excel Power-law fit, y = (8.96 x 1011)-.8359 R2 =.9936 predicted/actual
4000
3500
3000
2500
2000
1500
1000
500
0
3.6
3.4
y = 1.02x - 13.601
R2 = 0.9977
y = 0.998x + 0.007
R2 = 0.9967
3.2
3.0
2.8
2.6
2.4
2.2
0
1000
2000
3000
4000
(c) Correlation between trend line/actual data
2.2
2.4
2.6
2.8
3.0
3.2
(d) log-log plot for figure (a)
Figure 4. Fit of Actual (Y axis) World Population in Millions
for 200 BCE – 1965 to Power-Law Prediction (X axis), and log-log fit
3.4
3.6
Deviation from trend line (absolute)
150
100
50
0
- 200 - 100
0
100
200
300
400
500
600
700
800
900
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000
-50
-100
-150
-200
-250
-300
Figure 5. Cycles of population deviation from the power-law slope of 200 BCE to 1962
Deviation from trend line (relative)
0.15
0.1
0.05
0
-200
0
200
400
600
800
1000
1200
1400
1600
1800
-0.05
-0.1
-0.15
Figure 6. Cycles of percentage deviation from the power-law slope of 200 BCE to 1962
2000
Deviation from political trend line
Figure 7. Cycles of Political Consolidation (lower values on Y axis) vs. Fragmentation
(Taagepera 1997)
Fit of political - population phases
Taag Taag
1800
Taag
Taag
Taag Taag
1300
800
Taag
Taag
300
-200 1
Taag2
3
4
5
6
7
8
9
10
Figure 8. Congruence in the Phasing of Political (Taag) and Population cycles,
Y axis = Historical dates, X axis = alternate high/low
Population phases and transitions
World
Period
From
1962/63
-200
to1962
-5000 to
-200
1M to
-5000
World
Trend*
Decline
**
Powerlaw
Powerlaw
Exponential
Connecti
vity
High
Generative
Process
**
GeoCentroid
India
Low
World-Systems
China
High
Radial Diffusion
Colonizing Cities
Segmentation
None
Transition
Explanation
n.a.
Mathematical
Singularity
Near East
Intercity
trade network
Africa-Old- Filling of
New World Earth Surface
Transition
Mechanisms
n.a.
e.g., Cost of
Children
e.g., Political
Organization
Nucleation of
Populations
* These trends begin often earlier when examined region by region and power-law distributions do not usually carry over to regions.
** Whether the decline is exponential or power-law is not yet evident, nor can we yet confidently describe the type of period.
Table 2: Summary of Periods and Trends, with Explanations and Mechanisms for World
Demographic Transitions at the end of a world period trend
Power law from 1,000,000 yrs ago
100000000
10000000
1000000
100000
10000
1000
100
10
1
0.000001 0.000010 0.000100 0.001000 0.010000 0.100000 1.000000 10.00000 100.0000 1000.000
0
00
000
0.1
Figure 9: Kapitza trend line, plotting x = log (dN/dt) and y = log (N2)
(trend line is for 200 BCE to 1970)
Discuss implications
Economic
Political
Age Distribution
World-System
Social Organization
Cognition