Computers, Environment and Urban Systems 45 (2014) 89–100
Contents lists available at ScienceDirect
Computers, Environment and Urban Systems
journal homepage: www.elsevier.com/locate/compenvurbsys
Analyzing urbanization data using rural–urban interaction model
and logistic growth model
Shun-Chieh Hsieh ⇑
Department of Land Management and Development, Chang Jung Christian University, Tainan City 71101, Taiwan
a r t i c l e
i n f o
Article history:
Received 22 June 2013
Received in revised form 10 January 2014
Accepted 12 January 2014
Available online 1 February 2014
Keywords:
Urbanization curve
Urbanization dynamics
Urban population
Self-organized criticality
a b s t r a c t
The level of urbanization is a valuable indicator for projections of some global trends. However, urbanization levels may be based on unreliable data. This study proposes a simple method for identifying problems in the time series of urban and rural populations of a country. The time series were fitted to a rural–
urban interaction population model, and improper model coefficients indicated that the time series were
questionable. The upper limit of the urbanization level was calculated to determine whether the trend of
the urbanization level follows the logistic growth model. An analysis of the frequency–spectrum relationship was performed to determine whether the urbanization process is a self-organized criticality and to
consolidate the low possibility for chaos in the urbanization model. Empirical analyses were conducted
using data from the United States, China, and India to verify data reliability and to determine the dynamical mechanism of urbanization. This is critical for demographers, geographers, other scientists, and
policymakers.
Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction
There are multiple indices of the urbanization of a country. The
concentration index, which is related to the distribution and
concentration of urban populations, is the size of the cities relative
to the total population (Casis & Davis, 1946). Ledent (1980)
proposed an alternative measure of urban concentration, an
agglomeration index, which is based on three factors: population
density, population of a large city center, and travel time to the
large city center. The degree or level of urbanization is the percentage of urban population in its total population at any fixed date
(Davis & Hertz, 1951). The rate or speed of urbanization refers to
the change in the degree of urbanization during a period of time
(Durand & Pelaez, 1965). Chen, Ye, and Zhou (2013) differentiated
the urbanization curve to derive the speed of urbanization curve.
To define aforementioned indices of urbanization, in addition to
considering the urban proportion of the population, Arriaga
(1970) also considered the size of the city where an urban
population lives. The tempo of urbanization is defined as the net
difference between the rate of growth in the urban population
and that in the rural population (United Nations, 1974). The scale
of urbanization is defined as RXY, where X is the proportion of
the urban population in units greater than a certain size and Y is
the proportion of the total population in the same units (Gibbs,
⇑ Address: No. 1, Changda Rd., Gueiren District, Tainan City 71101, Taiwan. Tel.:
+886 6 2785 123x2316; fax: +886 6 2785 902.
E-mail address: sch@mail.cjcu.edu.tw
0198-9715/$ - see front matter Ó 2014 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.compenvurbsys.2014.01.002
1966). The level of urbanization is a common demographic definition of urbanization because it is easy to calculate and interpret,
and because of the high availability of data.
In this study, urbanization differs from exogenous urban
growth. Urbanization is an increase in the proportion of a country’s
population that resides in urban areas, in which the city size is not
considered, whereas exogenous urban growth is an increase in the
number of people who live in urban areas. For example, if the
urban population and total population of a country are 4,000,000
and 8,000,000, respectively, then the urban population and total
population will be 8,000,000 and 16,000,000, respectively, fifty
years later. Accordingly, the level of urbanization does not change,
whereas urban growth increases by 4,000,000. The country is
expected to reach a high urbanization level and low urban growth
at the terminal stage of urbanization. Recently, much research has
been conducted on urban size dynamics. Schaffar and Dimou
(2012) studied the dynamics of Chinese and Indian urban
hierarchies from 1981 to 2004, and examined the urban growth
patterns of the rank-size relationship for cities in these countries.
To eliminate problems of urban definitions, Mulligan (2006) projected the urban population above high thresholds and explored
the influence of city-specific initial conditions and national-level
factors on population growth. However, the proportion of urban
dwellers living in large cities exhibits a substantially low correlation with the level of urbanization (Bloom, Canning, & Fink,
2008), which is investigated in this study. Urbanization has a
beginning and an end. By contrast, urban growth is limitless
(Northam, 1975). In current study, no cross-country analysis was
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S.-C. Hsieh / Computers, Environment and Urban Systems 45 (2014) 89–100
China
USA
India
Fig. 1. Percentage of urban population and agglomerations by size class in 1960. Source: United Nations, 2012.
USA
China
India
Fig. 2. Percentage of urban population and agglomerations by size class in 2011. Source: United Nations, 2012.
conducted using a list of cities ranked according to size for each
country, as shown in Figs. 1 and 2, nor were the aggregate urbanization statuses of the regions and the world determined. Stoto
(1979) indicated that the date that the forecast is made is the principal factor determining error. Keyfitz (1981) argued that comparing individual forecasters is essentially futile. This study
investigated the time series reliability of the urbanization level;
however, the urbanization level was not forecasted for the future
and forecasting methods were not compared.
The curves of the change in the level of urbanization over time
are called ‘‘urbanization curves’’ (Knox, 1994; Northam, 1975). The
relationship between the urbanization level and various topics,
namely socioeconomic development (Annez & Buckley, 2009;
Black & Henderson, 1999; Bloom et al., 2008; Chenery & Syrquin,
1975; Fay & Opal, 2000; Henderson, 2003; Jones & Kone, 1996;
Ledent, 1982; Njoh, 2003; Polèse, 2005; Woods, 2003), the
environment and resources (Alig, 2010; Shen, Peng, Zhang, & Wu,
2012; Zhou et al., 2004), and energy consumption and emissions
(Cole & Neumayer, 2004; Krey et al., 2012; Poumanyvong &
Kaneko, 2010; York, 2007), has been explored extensively. Therefore, the level of urbanization has been used as an indicator for projecting various global trends, such as energy use, poverty, and
environment and resource use. (Energy Information Administration, 2012; World Bank, 2011; World Resources Institute, 2003).
Currently, the United Nations (UN) is the only institution that
produces projections of urban and rural population growth on a
global scale. The World Urbanization Prospects (WUP) data set
published biannually by the United Nations Population Division
is the most comprehensive source of estimates and projections of
the urban and rural populations of every country, region, and continent in the world. The published statistics follows the national
census definition of urban population, which differs considerably
among nations (geographical variations) and varies over time
within a single country (historical variations). National definitions
are generally based on demographic, administrative, economic,
sociocultural, and geographic criteria (Frey & Zimmer, 2001). The
UN (1974) detailed discussions on the problems of urban definitions. After discussing numerous definitional problems and the
lack of reliable and current census data, Cohen (2004) concluded
that nearly any statistic on an urban population is merely an
approximation of reality. Bocquier (2005) indicated that the UN
projections were systematically biased, and the problem primarily
originated in the linear regression model used in the projection
method. Montgomery (2008) also indicated that the urbanization
levels were significantly overestimated in the UN projections. This
problem arising from the UN projections raises obvious concerns
regarding data reliability and makes cross-country comparisons
problematic. Because the WUP data set is widely used and referenced, methods for identifying definition and measurement problems in the time series of urban and rural populations are required.
Time-series analyses of empirical population data have indicated that chaos is rare in natural populations (Ellner & Turchin,
1995; Upadhyay & Rai, 1997). Holland (1995) believed that the
interactions that form a city are typically stable. Furthermore, by
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S.-C. Hsieh / Computers, Environment and Urban Systems 45 (2014) 89–100
demonstrating the dynamics of urbanization based on the logistic
growth model (LGM), which has been used to forecast the urbanization level of every country in the world (United Nations, 2002,
2012) to create no possibility for chaos, Chen (2009a) inferred that
the probability of urban chaos is considerably low in the real
world. Reliable statistics for nations’ urban population and
urbanization levels depend on reliable data. Therefore, the primary
objective of this study was to develop a simple method for
identifying problems in the time series of urban and rural populations based on the aforementioned argument. The following
section details a series of systematic approaches to developing a
rural–urban interaction model, the calculation of the upper limit
of the urbanization level, and the development of a frequency–
spectrum relationship (FSR). A schematic framework was applied
to the WUP data set and census data of the United States, China,
and India, and a detailed discussion is subsequently provided. Finally, the paper concludes with a brief summary of this study. A list
of acronyms used in this paper is provided in Table 1.
2. Methods
This section details a series of systematic approaches to analyzing urbanization dynamics. The analysis commenced by fitting the
urban and rural population data to a two-dimensional map to
determine a nation’s urbanization process. The parameter values
in the model were then verified to lie within the bounds of a
reasonable scale. To research the rural–urban interaction model,
logical and empirical analyses were conducted. The upper limit
of the urbanization level was estimated based on the LGM to be
compared with the results derived from the rural–urban interaction model. Finally, fast Fourier transform (FFT) was used to determine an approximate power-law relationship between frequency
and spectral density to obtain a spectral exponent.
2.1. Rural–urban interaction model
The spatial interaction between the rural and urban population
results in urbanization dynamics and can be characterized using
two nonlinear differential equations that are constructed based
on observations and statistical data. The rural–urban interaction
model can be expressed as (Chen, 2009a)
( drðtÞ
dt
duðtÞ
dt
¼ arðtÞ þ buðtÞ /rðtÞuðtÞ
¼ cuðtÞ þ drðtÞ þ urðtÞuðtÞ
ð1Þ
;
where r(t) and u(t) denote the rural and urban population at time t,
respectively; and a, b, c, d, /, and u are parameters. The rural–urban
interaction reduces the rate of rural population growth and raises
the rate of urban growth. Therefore, it produces faster growth in
the urban population than in the rural population (i.e., urbanization). If / and u are constants, then the aforementioned model is
analogous to the Lotka-Volterra model (LVM; Dendrinos & Mullally,
1985; Volterra, 1938), and the urbanization curve is a J-shaped
curve. An analogy can be drawn between the rural–urban interaction in urban systems and the predator–prey interaction in ecosysTable 1
Acronyms used in the text.
Acronym
Full description
FFT
FSR
LGM
LVM
SOC
UNM
WUP
Fast Fourier transform
Frequency–spectrum relationship
Logistic growth model
Lotka-Volterra model
Self-organized criticality
United Nations model
World Urbanization Prospects
tems (Chen, 2009b). The sizes of the urban and rural populations
affect each other. If / = //[r(t) + u(t)] and u = u/[r(t) + u(t)], then
the rural–urban interaction model corresponds with the UN model
(UNM; Karmeshu, 1988; Ledent, 1980; United Nations, 1980), and
the urbanization curve is an attenuated S-shaped curve. The former
hits its carrying capacity and continues causing a population
increase, whereas the latter reaches its carrying capacity and stabilizes. Discretizing Eq. (1) yields a two-dimensional map, such as
( DrðtÞ
Dt
DuðtÞ
Dt
¼ arðtÞ þ buðtÞ /rðtÞuðtÞ
¼ cuðtÞ þ drðtÞ þ urðtÞuðtÞ
:
ð2Þ
For simplicity, the notation of parameters is not changed
despite the error caused by the continuous-discrete conversion.
For dimensional uniformization, the rural and urban populations
are divided by the initial value of the rural population. Thus, we obtain r(0) = 1. The model exhibits no periodic oscillation or chaotic
behaviors when the parameter values used in the model lie within
the following reasonable ranges (Chen, 2009a): 0 < a; /; u < 1
and 0 6 b; c; d < 1.
The level of urbanization can be expressed as
LðtÞ ¼
uðtÞ
c:
uðtÞ þ rðtÞ
ð3Þ
where c indicates the upper limit of urbanization (usually set at
100%). Taking the derivative of Eq. (3) yields
dLðtÞ
LðtÞ
;
¼ ðu aÞLðtÞ 1 dt
c
ð4Þ
for UNM with b = c = d = 0 and / = u (closed system). Eq. (4) is
analogous to the logistic equation first created by Verhulst (1838).
In a closed system such as the entire world, the decrease in rural
population is equal to the increase in urban population caused by
the rural–urban interaction. The difference between parameters
u and a dominates the behavioral features of the closed urbanization dynamics. Thus, the rural region and rural–urban interaction
determine the progress of urbanization.
2.2. Logistic growth model
Because the urbanization level of a nation exhibits clear upper
and lower limits and its growth is not of uniform speed, the
increase in the urbanization level over time exhibits an S-shaped
curve, as demonstrated by Northam (1975). The curve can be
formulated as a sigmoid function. The LGM is the most common
representation of the urbanization process because it can be estimated in a straightforward manner by using ordinary least-squares
regression (Keyfitz & Caswell, 2005), and because ‘‘most other
models of S-shaped curves are much more complicated to estimate’’ (Mulligan, 2013). Although the LGM has often been criticized for being applied to population forecasts (Keyfitz & Caswell,
2005), it has been proved useful in summarizing historical changes
in population size and for short-term projections (Berry, 1973;
Keyfitz, 1980; Leach, 1981; Marchetti, Meyer, & Ausubel, 1996;
Mulligan, 2013; Rogers, 1995a). For several years, the UN assumed
that the urban–rural growth difference of countries follows a logistic path and estimated it based on the experience of numerous
countries (United Nations, 2002, 2012). The standard three-parameter LGM can be expressed as
LðtÞ ¼
c
1 þ aebt
;
ð5Þ
where a is the location parameter (it shifts the model in time but
does not affect the model’s shape; Oliver, 1966), b is the relative
growth rate at substantially low urbanization levels, and c refers
to the upper limit of the urbanization level. By letting t = 0, we
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S.-C. Hsieh / Computers, Environment and Urban Systems 45 (2014) 89–100
obtain a = c/L0 1, where L0 represents the initial value of L(t). A
lower (high) value for a indicates that the growth process begins
earlier (later) and a low (high) value for b indicates that the growth
process of reaching the upper limit of the urbanization level occurs
slowly (rapidly). Each country follows its own urban transition,
which leads to various urban saturation levels. Based on cross-sectional time-series data from 11 countries, Rao, Kanneshu, and Jain
(1989) determined that the value of b is below 0.05 in practice. In
other words, the value of b is generally low. This finding supports
the assumption that urbanization dynamics based on the LGM create no possibility for chaos. Therefore, the urbanization level is not
sensitive to initial conditions and the upper limit of the urbanization level is asymptotically reached independently of the initial value of L(t).
The upper limit of the urbanization level can be estimated using
natural logarithms on both sides of Eq. (5) to obtain the following
linear regression equation:
ln
c
LðtÞ
1 ¼ ln a b t;
ð6Þ
where the intercept indicates the estimated starting time of the
urbanization process and the slope indicates the rate of change during the process. When the process is half complete (L(t) = c/2) and
the logistic curve reaches its inflection point (where the growth rate
is maximal), the left side of Eq. (6) can be set to zero. This indicates
that the acceleration stage changes to the deceleration stage at time
t = ln a/b. Therefore, the urbanization process without counterurbanization can be divided into four stages: the initial stage, acceleration stage, deceleration stage, and terminal stage (Chen, 2012). The
J-shaped curve of urbanization at the initial and acceleration stages
is an exponential growth curve, which indicates that the urbanization process is not sustainable. When a J-shaped curve reaches the
deceleration stage, it converts to an S-shaped curve with inflection
near the upper limit of the urbanization level.
The derivative of Eq. (5) is
dLðtÞ
LðtÞ
:
¼ bLðtÞ 1 dt
c
ð7Þ
This equation is analogous to Eq. (4) and indicates how the instantaneous change dL(t)/dt in the urbanization level is related to the
intrinsic growth bL(t), which in turn is constrained by the continually diminishing factor 1 L(t)/c. The right side of Eq. (7) equals
zero when L(t) = c, which must be the point at which growth stops
on the urbanization curve. Thus, the urbanization level grows exponentially under the constraints of an upper limit, producing a typical S-shaped curve. Discretizing Eq. (7) yields a one-dimensional
map in the form
b
Ltþ1 ¼ ð1 þ bÞLt L2t :
c
ð8Þ
Let xt = bLt/[(1 + b)c], then Eq. (8) can be normalized and we obtain
xtþ1 ¼ ð1 þ bÞxt ð1 xt Þ:
ð9Þ
The quadratic map approaches a fixed state b/(1 + b) when 0 < b < 2
(May, 1976).
narrow regime near the boundary between chaos and order, called
the ‘‘edge of chaos’’ (Packard, 1988). Kauffman (1993) indicated
that the rate of evolution of evolving systems is maximized near
the edge of chaos. By conducting a parameter analysis of urbanization dynamics, Chen (2009a) concluded that the spatial complexity
of a self-organized urban system occurs on the edge of chaos rather
than in a chaotic state. The urbanization process can be regarded as
a phase transition from a rural to an urban settlement (Andersson,
Rasmussen, & White, 2002) and an SOC (Allen, 1997; Portugali,
2000). This phase transition may explain why an observed urbanization process often displays no characteristic time or length
scale. SOC is observed in several simple cellular automation models
and its chain reaction is a fractal process (Batty & Xie, 1994, 1999;
Portugali, 2000). An analysis of a power-spectrum relationship can
be conducted using the urbanization data of a country. The power
spectra of such urbanization processes obey a power-law relationship as follows
Pðf Þ / f g ;
where f refers to the frequency and g is the spectral exponent. The
spectral exponent is associated with the profile dimension Ds
according to the following formula (Peitgen & Saupe, 1988):
g ¼ 5 2Ds ¼ 2H þ 1;
Self-organized criticality (SOC) refers to the tendency of large
complex systems with numerous degrees of freedom naturally to
drive systems to a critical state in which minor events can cause
chain reactions of various sizes, and complements the concept of
chaos, in which simple systems with few degrees of freedom can
display complex behavior (Bak & Chen, 1991; Bak, Tang, & Wiesenfeld, 1987). Complex systems tend to naturally evolve toward a
ð11Þ
where H is the Hurst exponent (Feder, 1988). The FSR is a typical
mathematical indication of the SOC of urban systems (Chen & Zhou,
2008). The 1/f fluctuation and fractal growth are regarded as the
‘‘fingerprint’’ and the ‘‘signature’’ of SOC in time and space, respectively (Bak, 1996). SOC is characterized by uncontrolled fractal
growth independent of scale (Batty, 2005; Batty & Xie, 1999).
According to Bak (1996), if only 0 < g < 2, then Eq. (10) can be
considered to indicate a 1/f fluctuation in practice.
3. Empirical analysis
The United States, China, and India are the top three most populated countries accounting for 75% of the urban population of the
world in 1950, 2000 and 2030 (United Nations, 2002). The reliability of the rural and urban population data from these countries’
population censuses and the WUP 2011 data set (United Nations,
2012) were verified in this study.
3.1. United States
Table 2 shows the rural and urban populations of the United
States reported in the population censuses and the WUP 2011 data
set. The definition of U.S. cities was changed in 1950 and adopted
in 1970. We used only the census data from 1790 to 1960. Let r(t),
u(t), r(t)u(t) and r(t)u(t)/[r(t) + u(t)] be independent variables and
entry statistics, and Dr(t)/Dt, and Du(t)/Dt be dependent variables
and exit statistics. A multivariate stepwise regression analysis
based on least squares computation yielded the following models
8
< DrðtÞ ¼ 0:02584rðtÞ 0:03615
Dt
: DuðtÞ ¼ 0:05044
Dt
2.3. Frequency–spectrum relationship
ð10Þ
rðtÞuðtÞ
rðtÞþuðtÞ
rðtÞuðtÞ
rðtÞþuðtÞ
:
ð12Þ
This model is a UNM in which all types of statistic can pass the
tests at a .01 significance level. Stepwise regression involves multiple regressions and the weakest correlated variable is removed in
each regression. The regression yields a UNM or an LVM, which
is a combination of independent variables that most accurately
explains the dependent variables. The estimates depend on time
series, which must be a sufficient length, and change if the data
set conforming to the same urban definition is divided into two
periods (e.g., 1790–1870 vs. 1880–1960 in the United States), thus
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S.-C. Hsieh / Computers, Environment and Urban Systems 45 (2014) 89–100
Table 2
The United States’ rural and urban populations and urbanization levels. Source: http://www.census.gov/population.
Census data
WUP data (unit: thousands)
t
Dt
r(t)
u(t)
L(t)
t
Dt
r(t)
u(t)
L(t)
1790
1800
1810
1820
1830
1840
1850
1860
1870
1880
1890
1900
1910
1920
1930
1940
1950
10
10
10
9.8125
10
10
10
10
10
10
10
9.7917
9.7917
10.25
10
10
10
1960
10
1970
1980
1990
2000
10
10
10
10
3727559
4986112
6714422
8945198
11733455
15218298
19617380
25226803
28656010
36059474
40873501
45997336
50164495
51768255
54042025
57459231
61197604
54478981
66259582
54045425
53565309
59494813
61656386
59061367
201655
322371
525459
693255
1127247
1845055
3574496
6216518
9902361
14129735
22106265
30214832
42064001
54253282
69160599
74705338
90128194
96846817
113063593
125268750
149646617
167050992
187053487
222360539
.0513
.0607
.0726
.0719
.0877
.1081
.1541
.1977
.2568
.2815
.3510
.3965
.4561
.5117
.5614
.5652
.5956
.6400
.6305
.6986
.7364
.7374
.7521
.7901
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
2015
2020
2025
2030
2035
2040
2045
2050
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
56570.772
56200.599
55906.195
56088.792
55293.233
57729.547
60356.639
61498.878
62574.509
60589.249
59073.418
57192.943
55424.913
53932.505
52691.099
51656.798
50539.187
49273.034
47892.552
46425.864
44916.864
101242.268
114951.692
130420.02
143363.716
154170.632
161378.811
169468.365
179620.874
190764.588
205734.468
223422.892
239627.353
254959.035
269952.635
284410.734
298101.47
311140.702
323615.877
335567.923
347027.900
358183.662
.6415
.6716
.7000
.7188
.7360
.7365
.7374
.7449
.7530
.7725
.7909
.8073
.8214
.8335
.8437
.8523
.8603
.8679
.8751
.8820
.8886
Note: Census data based on new urban definition are indicated in Italic.
indicating a bias. The growth rate of the rural population depends
on rural population size and the rural–urban interaction, whereas
that of the urban population is proportional only to the rural–
urban interaction but not directly associated with rural and urban
population sizes. Therefore, the United States’ urbanization process
was primarily migration-lead, and population migration between
rural and urban sectors depended only on rural–urban interaction.
Consequently, the original UNM was simplified to the form shown
in Eq. (12). According to Eq. (12), the rural population cannot spontaneously flow into the urban sector, and vice versa. The phase portrait of the United States’ urbanization, as shown in Fig. 3,
indicated that the urban population increased slowly as the rural
population increased until the urbanization level was 71.66%,
and then increased rapidly as the rural population decreased. In
other words, rural-to-urban migration was initially the primary
contributor to urbanization, but urban natural increase subsequently became the chief cause of urbanization. By using a similar
approach, we also obtained a UNM based on the WUP data from
1950 to 2050. However, the model with unreasonable parameters
was abnormal, as shown in Table 5.
The fit of the LGM based on the census data from 1790 to 1960
and the original urban definition yielded the upper limit of the
urbanization level c = 0.7073. The acceleration stage changed to
the deceleration stage in 1891. When we used the census data
based on the urban definition from 1950 to 2000, the fit of the
LGM yielded the upper limit of the urbanization level c = 0.7945.
The urbanization trend was better fit when using the 1950 urban
definition than when using the original definition. When we used
the WUP data from 1950 to 2050, the fit of the LGM yielded an
abnormal upper limit of the urbanization level, as shown in Table
5. The changing trend of the United States’ urbanization level, as
shown Fig. 4, indicated that the UNM can clearly describe the United States’ rural–urban interaction process of the recent 200 years.
By using FFT and least squares computation, we obtained the
FSR based on the census data as
Pðf Þ ¼ 0:0009f 1:8348
ðR2 ¼ 0:9547Þ:
ð13Þ
1
Census 1790-1960
Census 1970-2000
150
Urbanization level
0.8
u (t )
100
WUP 2011
UNM-Census 1790-1960
0.6
LGM-Census 1790-1960
LGM-Census 1950-2000
0.4
50
0.2
L = 71.66%
0
0
1750
0
5
10
15
1800
1850
1900
1950
2000
2050
2100
Year
r (t)
Fig. 3. Phase portrait of the United States’ rural–urban interaction model.
Fig. 4. The changing trend of the United States’ urbanization levels based on the
UNM and the LGM.
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S.-C. Hsieh / Computers, Environment and Urban Systems 45 (2014) 89–100
Table 3
China’s rural and urban populations and urbanization levels. Source: http://www.stats.gov.cn/tjsj/ndsj.
Census data(unit: ten thousands)
WUP data (unit: thousands)
t
Dt
r(t)
u(t)
L(t)
t
Dt
r(t)
u(t)
L(t)
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
48404
49031
49668
50320
50970
52016
53180
53643
54703
55270
54834
53131
53155
55633
57523
57548
59496
61229
62820
64696
66554
68568
70518
72244
73867
75268
76394
77373
78306
79009
79048
79566
79897
80175
80738
80344
80754
81146
81625
82370
83164
83898
84574
83566
83400
83212
83041
82943
82483
82043
81472
80786
79563
78241
76851
75705
74544
73160
71496
70399
68938
67113
5763
6165
6632
7162
7826
8250
8285
9185
9950
10724
12373
13076
12704
11662
11649
12951
13042
13313
13548
13838
14117
14424
14711
14933
15344
15591
16026
16344
16668
17250
18494
19139
20175
21479
22270
24013
25097
26361
27675
28656
29540
30435
31249
33605
35117
36638
38080
39446
41143
42718
44314
45957
48064
50212
52376
54283
56212
58288
60633
62403
64512
66978
.1064
.1117
.1178
.1246
.1331
.1369
.1348
.1462
.1539
.1625
.1841
.1975
.1929
.1733
.1684
.1837
.1798
.1786
.1774
.1762
.1750
.1738
.1726
.1713
.1720
.1716
.1734
.1744
.1755
.1792
.1896
.1939
.2016
.2113
.2162
.2301
.2371
.2452
.2532
.2581
.2621
.2662
.2698
.2868
.2963
.3057
.3144
.3223
.3328
.3424
.3523
.3626
.3766
.3909
.4053
.4176
.4299
.4434
.4589
.4699
.4834
.4995
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
2015
2020
2025
2030
2035
2040
2045
2050
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
485765.396
524062.767
551613.736
581825.231
672878.467
755823.828
792851.113
814900.298
842378.617
838120.410
813792.013
751576.031
681049.007
608163.067
541428.390
483452.501
435427.009
397142.173
362325.039
327678.511
293992.031
65006.037
84296.918
106656.358
128465.068
141744.374
159217.126
190319.525
241678.921
302816.612
375866.200
455324.724
556017.458
660286.145
761579.451
846363.122
911803.946
957649.059
984445.798
998581.458
1004089.577
1001611.732
.1180
.1386
.1620
.1809
.1740
.1740
.1936
.2287
.2644
.3096
.3588
.4252
.4923
.5560
.6099
.6535
.6874
.7125
.7338
.7540
.7731
Another FSR based on the WUP data was obtained as follows:
Pðf Þ ¼ 0:0002f
1:4745
2
ðR ¼ 0:9872Þ:
3.2. China
ð14Þ
Eqs. (13) and (14) indicate that the mathematical criteria of SOC fit
well with the United States’ urbanization for both the census data
and WUP data. The profile dimension of the United States’ urbanization level was 1.5826, based on the census data, and 1.7628,
based on WUP data.
Table 3 shows the rural and urban populations of China reported in the population census and the WUP data. China is experiencing a process of rapid urbanization. The urbanization level
increased from 26.62% to 49.95% between 1990 and 2010. A multivariate stepwise regression analysis based on the census data
from 1949 to 2010 yielded the following model:
95
S.-C. Hsieh / Computers, Environment and Urban Systems 45 (2014) 89–100
( DrðtÞ
Dt
DuðtÞ
Dt
¼ 0:02515rðtÞ 0:03656rðtÞuðtÞ
¼ 0:02446rðtÞuðtÞ
ð15Þ
:
This model is an LVM in which all types of statistic can pass
the tests at a .01 significance level. Shen (2005) also argued that
the urban–rural difference in the population growth rate was not
stable in the period of 1982–2000. Therefore, to achieve a precise
estimate of the urbanization level by using the UNM is impossible.
The phase portrait of China’s urbanization, as shown in Fig. 5,
indicates that the urban population increased slowly as the rural
population increased until the urbanization level was 26.11%, which
is considerably lower than that of the United States, and then
increased rapidly as the rural population decreased. In other words,
migration ceased to dominate the urban increase at a point at which
the urban population was still much lower than the rural population. By using a similar approach, we also obtained an LVM based
on the WUP data from 1950 to 2050. However, the model with
unreasonable parameters was abnormal, as shown in Table 5.
From 1961 to 1977, the antiurbanization period, rural to urban
migration was tightly restricted in China and urbanization was in a
rapidly declining stage (Chan & Zhang, 1999). In late 1978,
economic reforms were initiated in China and the urbanization
level ascended. From 1978 to 1999, rural–urban migration
3.5
3
Pðf Þ ¼ 2:5462f 1:6149
Pðf Þ ¼ 0:0012f 1:6786
u (t)
2
1
L = 26.11%
0.5
ð17Þ
2
1
Fig. 5. Phase portrait of China’s rural–urban interaction model.
Table 4 shows the rural and urban populations of India reported
in the population census and WUP data. India exhibits low urbanization levels. A multivariate stepwise regression analysis based on
the census data from 1901 to 2011 yielded the following model:
( DrðtÞ
1
Dt
DuðtÞ
Dt
Census 1949-2010
WUP 2011
LVM-Census
Urbanization level
ðR2 ¼ 0:9742Þ:
3.3. India
0
r (t)
LGM-WUP
0.6
0.4
0.2
0
1940
ð16Þ
Eqs. (16) and (17) indicate that the mathematical criteria of SOC fit
well with China’s urbanization for both the census data and the
WUP data. The profile dimension of China’s urbanization level
was 1.6926, based on the census data, and 1.6607, based on the
WUP data.
1.5
0.8
ðR2 ¼ 0:9824Þ:
Another FSR based on the WUP data was obtained as follows:
2.5
0
predominantly contributed to urban population growth in China
(Zhang & Song, 2003). Since the onset of the economic reform era
in 1979, China’s urbanization has developed rapidly and accelerated after 1995, as shown in Fig. 6. Generally, China’s urbanization
process can be divided into three parts: the random process, periodic process, and trend process (Chen, 2007). The estimated regression relationship between the urbanization level and years from
1978 to 2010 based on census data follows exponential growth.
Furthermore, the definition of urban population has changed substantially over time (Chan & Hu, 2003; Shen, 2005; Zhang & Zhao,
1998). Therefore, the fit of the LGM based on census data from
1949 to 2010 yielded unreasonable parameters, as shown in Table
5. It is difficult for the world’s most populous country to exceed its
urbanization level of 80% (Chen & Luo, 2006). When we used the
WUP data from 1950 to 2050, the fit of the LGM also yielded an
unreasonable upper limit of the urbanization level, as shown in Table 5. The changing trend of China’s urbanization level, as shown in
Fig. 6, indicates that China’s rural–urban interaction process in the
recent 60 years cannot be adequately described using the LVM.
Chen (2007) also observed that the common model cannot describe China’s urbanization process because of autocorrelation
and random disturbance. China’s urbanization process is a composition of the first-order autoregressive model and the high-order
(even infinite-order) moving-average model.
By using FFT and least squares computation, we obtained the
FSR based on the census data as
1960
1980
2000
2020
2040
2060
Year
Fig. 6. The changing trend of China’s urbanization levels based on the LVM and the
LGM.
¼ 0:01852rðtÞ þ 0:25567uðtÞ 0:05053rðtÞuðtÞ
¼ 0:03333rðtÞuðtÞ
:
ð18Þ
This model is an LVM and it is abnormal because a few parameter
values in the model lay outside reasonable ranges, as shown in
Table 5. By using a similar approach, we also obtained an LVM based
on the WUP data from 1950 to 2050. However, the model with
unreasonable parameters was also abnormal, as shown in Table 5.
During the four decades from 1961 to 2001, natural increase
(the difference of births and deaths) accounted for greater than
50% of urban population growth in India (Kundu, 2011). India’s
urbanization process is not primarily migration lead, but is a
product of demographic explosion caused by natural increase.
Because India’s urbanization curve is a J-shaped curve rather than
an S-shaped curve, as shown in Fig. 7, the fit of the LGM based on
the census data from 1901 to 2011 yielded unreasonable parameters, as shown in Table 5. When we used the WUP data from 1950
to 2050, the fit of the LGM also yielded an unreasonable upper limit
of the urbanization level, as shown in Table 5.
By using FFT and least squares computation, the FSR based on
the census data was obtained as
96
S.-C. Hsieh / Computers, Environment and Urban Systems 45 (2014) 89–100
Table 4
India’s rural and urban populations and urbanization levels. Source: http://censusindia.gov.in/Census_Data_2011/.
Census data
WUP data (unit: thousands)
t
Dt
r(t)
u(t)
L(t)
t
Dt
r(t)
u(t)
L(t)
1901
1911
1921
1931
1941
1951
1961
1971
1981
1991
2001
2011
10
10
10
10
10
10
10
10
10
10
10
10
212544454
226151757
223235046
245521249
274507283
298644156
360298168
439045675
523866550
628836076
741660293
833087662
25851873
25941633
28086167
33455989
44153297
62443934
78936603
109113977
159462547
217551812
285354954
377105760
.1084
.1029
.1118
.1199
.1386
.1729
.1797
.1991
.2334
.2570
.2778
.3116
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
2015
2020
2025
2030
2035
2040
2045
2050
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
308483.929
334924.536
367571.756
403151.116
444426.573
489393.294
538360.177
593481.426
650555.737
707862.144
762313.429
806755.007
845838.901
879711.939
903865.56
916766.695
917669.536
906218.493
884362.011
854129.722
816624.748
63372.571
71449.484
80272.403
93249.265
109447.317
132703.388
161698.412
191009.416
223229.712
256624.011
291584.678
333287.856
378775.426
428508.756
483043.512
542190.888
605812.799
673583.693
742667.375
810389.351
875382.883
.1704
.1758
.1792
.1879
.1976
.2133
.2310
.2435
.2555
.2661
.2767
.2923
.3093
.3276
.3483
.3716
.3977
.4264
.4565
.4869
.5174
Table 5
The parameter values in the rural–urban interaction model and the LGM.
b
c
d
/ or /
u or u
a
0.02584
0
0
0.00067
0
0
0
0
0.03615
0
0.05044
0.05962
21.2505
0.5740
0.0303
0.0126
0.7073
1.0371
LVM
LVM
0.02515
0
0
0.01079
0
0
0
0
0.03656
0.00960
0.02446
0.00645
0.5166
10.7014
0.0092
0.0394
0.0655
0.9674
LVM
LVM
0.01852
0.02502
0.25567
0
0.03333
0
0
0.00307
0.05053
0
0
0.00813
92.0234
0.9926
0.0112
0
8.7289
0.0014
Country
Data
Model
US
Census
WUP
UNM
UNM
China
Census
WUP
India
Census
WUP
a
c
b
Note: Abnormal values are indicated in bold.
Pðf Þ ¼ 0:0003f 1:7779
ðR2 ¼ 0:8176Þ:
ð19Þ
Another FSR based on the WUP data was obtained as
Pðf Þ ¼ 0:0002f 1:6021
ðR2 ¼ 0:9729Þ:
ð20Þ
Eqs. (19) and (20) indicate that the mathematical criteria of SOC fit
better with China’s urbanization for the WUP data because the time
series of the census data was not sufficiently long. The profile
dimension of India’s urbanization level was 1.6111, based on the
census data, and 1.6990, based on the WUP data.
4. Discussion
In general, the mathematical criteria of SOC fit well with the
urbanization data. This study demonstrated that the values of the
parameters in the urbanization model could be changed widely
without affecting the emergence of SOC. The FSR also consolidated
the rare possibility for chaos that is characterized by a power spectrum 1/f0 in the urbanization model.
The phase plots indicate that the lower the value of the rural
population is, the sooner natural increase exceeds migration. In
the United States, when the country was mostly urbanized and little rural population was left to migrate to cities, natural increase
began to exceed migration during urbanization, which occurs
similarly in most developed countries. However, in China, natural
increase began to exceed migration even when the country was
still primarily rural, which occurs in most developing countries
(Bocquier, 2005).
Based on the census and the WUP data set, the urban population and urbanization level in the United States, China, and India
are shown in Figs. 8 and 9. The coefficients of variation of the
root-mean-square error were 2.41% (United States) and 3.05%
(China) for urban population, and 0.14% (United States) and
4.30% (China) for urbanization level. A comparison of the projected
urbanization levels in the United States, China, and India is shown
in Table 6. Clearly, the explanatory powers of the UNM and LVM
decreased when the estimation period was extended from 40 to
90 years. The projection model for the urbanization level in developing countries is enhanced by including exogenous variables
(Mulligan, 2013) and uncertainty assessment (Alkema, Gerland,
et al., 2011). The upper limits of the urbanization level estimated
by Alkema, Gerland, et al. (2011) are more reasonable than those
estimated in this study, which were 84.4% and 64.4% for China
and India, respectively.
Although the LGM generates a curve that tends toward an exponential form at low values, its maximal slope, or point of inflection
(where the growth rate is maximal), is always at half of the value of
the upper limit. This is unsatisfactory, because the factors that
determine the urbanization at which each country grows fastest
are complex; therefore, it is unlikely that all urbanization occurs
fastest when countries are at half of their urban saturation level.
Mulligan (2006) asserted that the logistic model should be used
with caution when examining urbanization growth in less developed countries. Birch (1999) proposed a generalized LGM that
could potentially generate its point of inflection at any point.
However, the low asymptote is not equal to zero, and obtaining
97
S.-C. Hsieh / Computers, Environment and Urban Systems 45 (2014) 89–100
0.8
0.6
Census 1901-2011
0.7
WUP 2011
0.5
L (t ) = 4E-11e
0.0113t
2
R = 0.9952
0.3
WUP-China
0.6
Expon. (WUP 2011)
0.4
Urbanization level
Urbanization level
Expon. (Census 1901-2011)
0.2
Census-China
0.5
WUP-US
Census-US
0.4
WUP-India
Census-India
0.3
0.1
L (t ) = 1E-10e
0.2
0.0108t
2
R = 0.9776
0
1900
1950
2000
2050
0.1
1950
750000000
650000000
WUP-China
Urban population
Census-China
550000000
450000000
350000000
WUP-US
Census-US
WUP-India
Census-India
250000000
150000000
50000000
1950 1960 1970 1980 1990 2000 2010 2020
Year
Fig. 8. Urban populations based on the census and WUP data for the United States,
China, and India.
separate estimates for all the parameters in this sigmoid equation
is difficult. In addition, Meade and Islam (1995) observed that the
simple LGM often outperformed generalized LGMs, which have the
disadvantage of losing clear physical interpretations for their
parameters.
O’Neill, Balk, Brickman, and Ezra (2001) systematically compared the approaches and results of various projections. Some scenarios and probabilistic projection models can account for
uncertainty in projected trends of fertility, mortality, and migration but require additional detailed historical information (Alkema,
Gerland, et al., 2011; Goldstein, 2004; Lee, 1998; Lutz, Sanderson, &
Scherbov, 1998; Rogers, 1995b). However, no generally accepted
approach to characterizing this uncertainty is available. Furthermore, detailed data on fertility, mortality, and migration may not
be available, adequate, or reliable in some countries.
Because the UN projections are based on the extrapolation of
historical trends, Bocquier (2005) proved that this can be attributed mainly to an inappropriate projection model that systematically biases the urban estimates upward, and also to the quality
of the available data. In countries such as China, the problem
may be caused by the data used in the projection than by the model. The understanding of the level of urbanization and its scale in
developing countries is challenged by differences in the definition
of the term urban and, consequently, the lack of reliable data. The
definition of urban plays a crucial role.
1970
1980
1990
2000
2010
2020
Year
Year
Fig. 7. The exponential growth trend of India’s urbanization level.
1960
Fig. 9. Urbanization levels based on the census and WUP data for the United States,
China, and India.
The types of urbanization in countries differ across time and
space. Urbanization occurs in cities of various sizes and types.
Some projections have indicated that urbanization is concentrated
in the large cities of developing countries (Henderson, 2002). Other
projections have indicated that most urbanization is expected to
occur in small- and medium-sized cities of one million or fewer
people (Montgomery, 2008). Urbanization can occur through rural–urban migration, natural population increase, and annexation
(Cohen, 2004). Urbanization during and after the Industrial Revolution can be attributed solely to rural–urban migration; however,
today, in many parts of the world, urbanization is fueled by both
rural–urban migration and high urban fertility (Mulligan, 2013).
Urbanization in East Asia and the Pacific has been caused mostly
by industrialization and job opportunities in urban areas, whereas
several countries in Africa have experienced urbanization with little or no economic growth (Cohen, 2004; Fay & Opal, 2000; Weeks,
1994). Urbanization caused by migration may be a result of urban
pull, which was the chief cause of urbanization in Europe and the
USA, whereas migration is attributed to rural push in Africa and
other developing countries, resulting from agricultural stress,
political instability, and natural disasters (Dutt & Parai, 1994).
Almost all projections have indicated that urbanization is occurring rapidly and at large scales in developing countries, which undergo rapid demographic changes (Angel et al., 2005). Therefore,
the projection model for urbanization levels may not be enhanced
by including exogenous variables, such as socioeconomic and technological factors (Kelley & Williamson, 1984). The urbanization
that occurs in developing countries is primarily the outcome of
demographic transition. Economic and other considerations are
secondary influences (Dyson, 2011).
Including exogenous variables in the projection model for
urbanization levels would complicate the calculation and interpretation. The proposed projection model is endogenous and based
only on available data. The objective of this study was not to offer
a projection model with explanatory power, using several exogenous socioeconomic variables that could explain the urbanization
level and its trend, but rather to identify problems in the projections using only known quantities without knowing the characteristics of the initial population (size, age structure, and vital rates).
5. Conclusion
This study analyzed the trends in urbanization levels in the
United States, China, and India. Based on the census and WUP data
98
S.-C. Hsieh / Computers, Environment and Urban Systems 45 (2014) 89–100
Table 6
The comparison of projected urbanization levels in the United States, China, and India.
Country
USA
China
India
Urbanization level in 2030 (%)
Urbanization level in 2050 (%)
WUP 2011
Alkema, Gerland, et al. (2011) and Alkema,Raftery, et al. (2011)
Bocquier (2005)
This study
WUP 2011
Mulligan (2013)
This study
86.0
68.7
39.8
–
51.1
35.6
75.2
39.8
30.9
81.3
76.0
–
88.9
77.3
51.7
90.6
77.3
51.7
84.9
94.7
–
sets, urbanization dynamics were examined by using the rural–urban interaction model and the LGM. Insight into the patterns and
processes of the urbanization trends in the United States, China,
and India were obtained by systematically examining of the census
and WUP data. However, the cases in which counterurbanization
patterns exist, such as in Denmark, France, the Netherlands, Sweden, and Switzerland, (Champion, 2001; Kontuly, 1998) were not
considered in this study.
Based on the WUP data for the United States, China, and India,
the derived rural–urban interaction model and the upper limit of
the urbanization level were all abnormal or unreasonable. The
available WUP data set was not satisfactory which may be caused
by the various definitions of urban areas, the lack of reliable and
current demographic data, or the projection method. For projecting
the urban population, the urban–rural growth difference method
was adopted by the UN. Disregarding certain country-specific aspects, this method is based on the assumption that the level of
urbanization follows a logistic growth pattern (United Nations,
1980). The actual urban–rural growth difference for a country
may not be as uniform as the UN method assumes. The manner
in which the UN projection manages the issue of uncertainty is
unsatisfactory. Alkema, Gerland, et al. (2011) determined that
China‘s growth differential was projected to decrease linearly until
it reached the global norm, whereas India‘s growth differential was
projected to increase linearly until it reached the global norm. The
overestimation of the urban population based on the UNM was
particularly more pronounced for developing countries (Bocquier,
2005). Therefore, the WUP data for China and India were not reliable. Most projections were based on the WUP data with five-year
intervals, and are, thus, partly based on interpolations; therefore,
using the original estimation from the census data is preferred.
The LGM performs well for most animal populations because
the niches that encase their populations are of constant size
(Marchetti et al., 1996). Thus, the LGM is more suitable for closed
urbanization dynamics in which the rural region and rural–urban
interaction determine the progress of urbanization. The growth
of human population exhibits the elasticity of the human niche;
thus, the LGM may exhibit fleeting limits. Therefore, a fixed carrying capacity should not be considered in long-term projections of
urbanization levels (Cohen, 1998). However, an advantage of the
aggregate methods, such as the LGM, is that longer time series
are available for the total population, compared with the length
of series for variables such as age-specific fertility, mortality, and
migration. Additionally, if ecological and economic factors limit a
growth trend more directly on the total population than on the
vital rates, then projecting total population directly may be sensible (O’Neill et al., 2001). In other words, aggregate methods are
more suitable in such a situation.
The management of uncertainty in population forecasting has
recently received increased interest among researchers (Keilman,
Pham, & Hetland, 2002; Lutz & Goldstein, 2004; Raftery, Li,
Ševčíková, Gerland, & Heilig, 2012; Wilson & Rees, 2005). For
example, Alkema, Raftery, et al. (2011) addressed uncertainties in
the factors that cause drops in the fertility level. Lutz and
Samir (2010) advocated probabilistic projections to address
uncertainty and indicated that a universally accepted approach
to quantitatively describing the uncertainty of population projections has not yet been developed. New techniques for the probabilistic projection of urbanization levels must be developed.
The urbanization process in rapidly urbanizing countries can be
described using the J-shaped curve rather than the S-shaped curve
(Cadwallader, 1996; Haggett, 2001; Mulligan, 2013). Additional
studies on the S-shaped curve have been conducted. Another research challenge that was also considered by Chen (2012) was to
explore the general principle of the J-shaped curve of urbanization
and its underlying rationale. This is critical for demographers,
geographers, other scientists, and policymakers because an insight
into the urbanization process is the basis of social, economic, cultural, and environmental planning and policy making.
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