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A Leslie-Type Urban-Rural Migration Model, and the Situation of Germany and Turkey

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Movements in the age structure of a population are often accompanied by substantial rural-urban migration. It is therefore compelling to analyze the implications of fertility, mortality, and migration patterns together. We use a joint Leslie-type population model of urban and rural populations which projects the current population structure into the future, allowing for migration in both directions. This model permits an analysis of the long-run (stable) population properties, such as urbanization, under the assumption that current conditions persist, as well as an analysis of rural and urban populations in isolation, when migration is computationally eliminated. Applying the model to the female populations below age 50 of Germany and Turkey, we find that the actual urbanization is lower (higher) than long-run urbanization in Germany (Turkey, respectively). Among our findings is also that the slight long-run growth of the Turkish population is due to rural-urban migration, while Turkish urban areas have a below-replacement fertility.
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(will be inserted by the editor)
A Leslie-type urban-rural migration model, and the
situation of Germany and Turkey
Harald Schmidbauer ·Angi R¨osch ·
Narod Erkol
(last compiled: May 24, 2012)
Abstract Movements in the age structure of a population are often accom-
panied by substantial rural-urban migration. It is therefore compelling to an-
alyze the implications of fertility, mortality, and migration patterns together.
We use a joint Leslie-type population model of urban and rural populations
which projects the current population structure into the future, allowing for
migration in both directions. This model permits an analysis of the long-run
(stable) population properties, such as urbanization, under the assumption
that current conditions persist, as well as an analysis of rural and urban pop-
ulations in isolation, when migration is computationally eliminated. Applying
the model to the female populations below age 50 of Germany and Turkey, we
find that the actual urbanization is lower (higher) than long-run urbanization
in Germany (Turkey, respectively). Among our findings is also that the slight
long-run growth of the Turkish population is due to rural-urban migration,
while Turkish urban areas have a below-replacement fertility.
Keywords Leslie-type model ·Population projection ·Urban-rural mi-
gration ·Stable population theory ·Population growth ·Urbanization ·
Germany ·Turkey
Harald Schmidbauer
Istanbul Bilgi University, Istanbul, Turkey
E-mail: harald@hs-stat.com
Angi R¨osch
FOM University of Applied Sciences, Munich, Germany
E-mail: angi@angi-stat.com
Narod Erkol
Barcelona Graduate School of Economics
E-mail: naroderkol@gmail.com
2 Schmidbauer, R¨osch, Erkol
1 Introduction
“The world is undergoing the largest wave of urban growth in history”, the
United Nations Population Fund warns in an 2007 online release.1Meanwhile,
by the year 2010, world urbanization has arrived at the 50% level, with a
five-year urban population growth of 1.9% (compared to an overall population
growth of 1.2%), which is an aggregate of values spanning from 0.7 for more
developed regions to 4.0 for least-developed countries, according to the UNFPA
State of World Population report 2010 [23].
Urbanization and economic growth are often closely linked, but urban-
ization as well concentrates poverty. Urbanization results from internal mi-
gration. Therefore, on the other hand, the increasing urbanization is insofar
problematic as it contributes to the aggravation of the structural weakness of
rural regions which may just be a major push factor of rural-urban migration.
Urbanization may also be linked to decreasing fertility and shrinking, hence
ageing populations, as there is a gap in fertility levels between urban and rural
regions throughout the world.2The present study compares Germany (“devel-
oped” with respect to urbanization as well as growth, according to the UNFPA
report [23]) and Turkey (also “developed” with respect to urbanization, but
“less developed” with respect to growth, according to the UNFPA report [23]).
Germany, among the more developed countries, is attributed a 74% level
of urbanization in the UNFPA report [23], along with an indiscernible ur-
ban growth (the reported growth rate is 0.0%) within five years. Accelerated
urbanization in Germany became apparent from the mid-nineteenth century
onward; it had its origin in industrialization and its impulse to the formation
of new urban settlements (which, according to K¨ollmann [10], became possi-
ble only with the abolition of the older municipal constitutions of “guild and
trade restrictions designed to discourage migration”). Tracing the trends of
internal migration in German history, Mai et al. [15] distinguish a phase of
long-distance migration (from rural areas in the East to urban areas in the
West), progressively replacing migration from urban hinterland, so that by
the early twentieth century about 50% of the German population had become
internal migrants, as K¨ollmann [11] calculates. Between the wars, for reasons
of supply and ideology, beginnings of a trend back to the countryside could
be observed. When the World War II aftermath with its reallocation of pop-
ulation had abated in the late 1950s, a phase of suburbanization solidified in
Western Germany, leading to the emergence of urban sprawls and a persist-
ing large-scale de-concentration of population. However, rural-urban migration
remained relevant particularly for the young. — The situation in East Ger-
many before 1990 was different, where regional concentration of population
had prevailed. A sharp drop in younger age groups could be witnessed in the
1990s, which was triggered in particular by east-west migration after German
1http://www.unfpa.org/pds/urbanization.htm, accessed: 2011-05-14
2l.c.
A Leslie-type urban-rural migration model 3
reunification and amplified by low fertility rates; cf. [5], [15]. Thus, suburban-
ization trends in East Germany have attenuated by now, with urban sprawls
gradually coming into being.
The ongoing phenomenon of rural depopulation used to hit east-German
regions in particular, but not exclusively. It is selective with respect to the
younger age of migrants, and is accompanied by low total fertility levels (1.33
in 2010; cf. [23]) having persisted for more than three decades. Mai et al. [15]
assert that total growth of the German population today rests solely on an
increase in life expectancy, the ageing of the population being notably pro-
nounced in rural areas.
With a 70% proportion of population living in urban areas today, Turkey is
on a level with more developed regions like Germany, while its five-year urban
growth rate of 1.9% is characteristic for less-developed countries according to
the 2010 UNFPA report [23]. With delay from the trend in Germany, urban-
ization in Turkey started on a large scale in the early 1950s from a level of 20%
only. For the Turkish population between 1955 and 2000, Gedik [6] investigated
the Alonso theory of differential urbanization which postulates cycles of three
evolutionary phases, urbanization, polarization reversal, and counter urbaniza-
tion, where growth rates are highest for large, medium, and small settlement
sizes, respectively. Gedik found evidence for a phase of pre-concentration in
small cities in the 1950s, large-city urbanization that followed, and polariza-
tion “dispersal” starting in 1980 with highest growth rates in medium-sized
cities dispersed throughout the country.
Simultaneously, Turkey experienced a pronounced change in fertility. Within
three decades, the total fertility rate halved down to barely 2 today (2.09 in
2010, c.f. [23]). The Turkey Demographic and Health Survey 2008 [8] inves-
tigates fertility preferences and behavior of Turkish women by residence, and
provides information on maternal and child health. Among the findings is
that, though total urban fertility is below replacement and the urban-rural
differential appears to be contracting over time, clear above-replacement lev-
els in South and East Anatolia persist. At each age class, rural women tend to
bear more children than women in urban centers where a trend that fertility
decreases with a higher educational level can be observed. As a result, the
urban-rural gap with respect to the median age at first birth is broadening. A
significantly higher proportion in urban centers of women in the working ages
certainly adds to this gap, and thus the effects of rural-to-urban migration of
economically active women. Furthermore, the findings of the survey indicate
a significant urban-rural differential in child mortality which appears to be
correlated with the mother’s young age and educational level.
From a formal point of view, the phenomenon of urbanization (respectively
rural depopulation) can be analyzed as a result of migration probabilities and
their interaction with urban- and rural-specific fertility rates and survival prob-
abilities. The approach may vary in several basic respects: The focus may be
either on forecasting the future population using forecasts of mortality, fer-
tility and migration, or on population projection in order to find answers to
4 Schmidbauer, R¨osch, Erkol
what the population would be like in the long run if mortality, fertility and mi-
gration would evolve (or persist) in a certain way. The effects of demographic
and environmental conditions on the dynamics of populations may be stud-
ied in discrete or continuous time, in a deterministic framework, or using a
probabilistic model with random variation in births, deaths, and migration.
In her overview of probabilistic approaches to demographic and popula-
tion forecasting, Booth [2] identifies three widely-used frameworks: methods
on the basis of sample data on individual expectations about future devel-
opments or expert opinions, structural modeling methods based on theories
on relations between demographic variables and processes, and extrapolative
methods using time series models to detect patterns and trends in the past
and extrapolate them into the future. A time series approach recently adopted
by Hyndman, e.g. [9], involves functional data models to forecast mortality,
fertility, and migration.
The Leslie population model (in recognition of Leslie’s work, cf. [14]) falls
under the category of projection. In its classical formulation, it is a discrete-
time and age-structured (respectively stage-structured, cf. [13]) transition ma-
trix model for the evolution of a closed population in time, but can be extended
to a Leslie-type model which allows for immigration; e.g. [4], [20]. The spec-
tral properties of the matrix provide insight into asymptotic population growth
rates and stable stage structures. A detailed review of matrix population mod-
els, including Leslie models for populations in time-varying, deterministic or
stochastic environments, is given by Caswell [3]. Many applications and de-
velopments in spatial demography have their root in the (also matrix-based)
multiregional cohort-component model introduced by Rogers (cf. [18], [19]),
which, accessing a multispatial life table, describes the dynamics of a popula-
tion which is dispersed over different spatial patches and allows for migration
in between. An example is the multiregional multinational cohort-component
projection model by Kupiszewska and Kupiszewski, cf. [12] for its revised form
to capture international migration. For an ecological system in a multi-patch
stochastic environment with different time scales for migration and vital rates,
recently Alonso and Sanz [1] showed the application of an aggregation method
in order to obtain a reduced stochastic Leslie model.
Our contribution is a mathematical model conceived for urban and rural
populations, which is an extension of the classical Leslie model and allows for
migration from rural to urban areas and in the opposite direction. It can be un-
derstood as a particular formulation of the multiregional model by Rogers [19]
installing two regional types, urban and rural, but at the same time two kinds
of inhabitants, natives and migrants. It is able to project the current urban-
ization structure into the future and permits sensitivity analyses of the impact
of different vital patterns and migration scenarios on population growth and
urbanization, as well as insight into the trade-off between fertility, mortality
and migration with respect to stability.
This model is introduced in Section 2. First, a hypothetical population is
considered in Section 3, then an application to the populations of Germany
and Turkey is presented in Section 4. Finally, Section 5 gives a summary and
A Leslie-type urban-rural migration model 5
some outline for further research. A further challenge in our studies was how
we may obtain the information needed as input to our model. The data used
in our study, including data sources and data processing, are specified in the
Appendix. All computations are carried out in R [17].
2 The model
Our model deals with four populations of females: city natives, village natives,
and two classes of migrants which are distinguished by destination into city
migrants and village migrants. Time proceeds in discrete steps; for illustration
purposes we choose 15 years in this and the following section (but 5-year steps
when applied to Germany and Turkey in Section 4). Accordingly, the four
populations are structured by three 15-year intervals of age covering repro-
ductive ages. The age-specific fertility, mortality and migration patterns are
assumed to be constant through time. They may be village- or city-specific.
In particular, the migrants’ vital rates may differ from those of the natives
as well, while second generation migrants are assumed to behave like natives
with this respect; they are counted as natives actually.
Our model is defined by the relationship and the matrix displayed in Ta-
ble 1; all symbols are defined in Table 2. Schematically, the matrix MIcan
also be written as
MI
[city native]
[city native] MI
[city migrant]
[city native] MI
[village native]
[city native] MI
[village migrant]
[city native]
MI
[city native]
[city migrant] MI
[city migrant]
[city migrant] MI
[village native]
[city migrant] MI
[village migrant]
[city migrant]
MI
[city native]
[village native] MI
[city migrant]
[village native] MI
[village native]
[village native] MI
[village migrant]
[village native]
MI
[city native]
[village migrant] MI
[city migrant]
[village migrant] MI
[village native]
[village migrant] MI
[village migrant]
[village migrant]
(1)
with sub-matrices MI
[from]
[to] indicating possible transitions within one time step.
Such a sub-matrix will be a zero matrix whenever the corresponding transition
is impossible (as in the case MI
[city native]
[city migrant]). The element-by-element sum of
submatrices along a column of the partitioned MIwill result in a usual Leslie
matrix, which in turn defines population streams channeled to several possible
destinations by means of the migration pattern.
Theorem: Consider an age-structured population evolving according to Nt=
MI·Nt1.
Let the following conditions be satisfied: (i) migration probabilities from city
to village and from village to city are positive for any (not necessarily the
same) age class, (ii) survival probabilities are all positive, and (iii) fertility
rates are positive for any two adjacent age classes. Then:
6 Schmidbauer, R¨osch, Erkol
a) There exists a stable population structure ˜
Nand a λRsuch that
λ·˜
N=MI·˜
N. (2)
Here, λis the maximum eigenvalue of MI.
b) The future long-run growth rate of an initial population is given by the
maximum eigenvalue of MI.
c) All four population segments (city native, city migrant, village native, vil-
lage migrant) will ultimately grow with the same rate.
Proof: The projection matrix MIis a non-negative square matrix, irreducible
and primitive. Therefore, classical Perron-Frobenius theory can be applied, see
e.g. Seneta [21] and Caswell [3].
Irreducibility and primitivity can be evaluated from the transition diagrams
in Figures 1 and 2: The “life cycle graph” is strongly connected, i.e. each pair
of nodes is connected in the sense that one node can be reached from the
other within a finite number of transitions. The greatest common divisor of
loop lengths is 1. This holds if only the conditions of the theorem are satisfied.
Then, according to the Perron-Frobenius theorem, there exists a real and
positive eigenvalue λwhich dominates any other eigenvalue of MI. The eigen-
vectors associated to λare strictly positive and unique to constant multiples.
In particular, there exists a right eigenvector ˜
Nsuch that equation (2) holds.
It follows what is known as the strong ergodic theorem, that λcompletely
determines the long-term dynamics of the population:
limt→∞
Nt
λt=c·˜
N(3)
In the long run, the population will grow at a rate given by λ, with a stable
population structure proportional to ˜
N. In particular, this ultimate growth
rate carries over to all four population segments.
A Leslie-type urban-rural migration model 7
city native
city migrant
village native
village migrant
nc1t
nc2t
nc3t
n
c1t
n
c2t
n
c3t
nv1t
nv2t
nv3t
n
v1t
n
v2t
n
v3t
=
fc1¯mc1fc2¯mc2fc3¯mc3f
c1¯m
c1f
c2¯m
c2f
c3¯m
c30 0 0 0 0 0
pc1¯mc10 0 0 0 0 0 0 0 0 0 0
0pc2¯mc20 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 fv1mv1fv2mv2fv3mv3f
v1m
v1f
v2m
v2f
v3m
v3
0 0 0 p
c1¯m
c10 0 pv1mv10 0 p
v1m
v10 0
0 0 0 0 p
c2¯m
c20 0 pv2mv20 0 p
v2m
v20
0 0 0 0 0 0 fv1¯mv1fv2¯mv2fv3¯mv3f
v1¯m
v1f
v2¯m
v2f
v3¯m
v3
0 0 0 0 0 0 pv1¯mv10 0 0 0 0
0 0 0 0 0 0 0 pv2¯mv20 0 0 0
fc1mc1fc2mc2fc3mc3f
c1m
c1f
c2m
c2f
c3m
c30 0 0 0 0 0
pc1mc10 0 p
c1m
c10 0 0 0 0 p
v1¯m
v10 0
0pc2mc20 0 p
c2m
c20 0 0 0 0 p
v2¯m
v20
·
nc1,t1
nc2,t1
nc3,t1
n
c1,t1
n
c2,t1
n
c3,t1
nv1,t1
nv2,t1
nv3,t1
n
v1,t1
n
v2,t1
n
v3,t1
Table 1 The relation Nt=MI·Nt1
8 Schmidbauer, R¨osch, Erkol
Symbol Definition
fci the average number of girls born to a native of city in age
class i, and surviving to the next age class
fvi the average number of girls born to a native of village in age
class i, and surviving to the next age class
f
ci the average number of girls born to a migrant (from village to
city) in age class i, and surviving to the next age class
f
vi the average number of girls born to a migrant (from city to
village) in age class i, and surviving to the next age class
pci the probability that a native of city now in age class i, survives
to be in i+ 1
pvi the probability that a native of village now in age class i,
survives to be in i+ 1
p
ci the probability that a migrant (from village to city) now in
age class isurvives to be in i+ 1
p
vi the probability that a migrant (from city to village) now in
age class isurvives to be in i+ 1
mci probability that a city native in age group imigrates to village
¯mci probability that a city native in age group idoes not migrate
to village; ¯mci = 1 mci
mvi probability that a village native in age group imigrates to city
¯mvi probability that a village native in age group idoes not mi-
grate to city; ¯mvi = 1 mvi
m
ci probability that a migrant (from village to city) in age group i
migrates again (back to village)
¯m
ci probability that a migrant (from village to city) in age group i
does not migrate again (back to village); ¯m
ci = 1 m
ci
m
vi probability that a migrant (from city to village) in age group i
migrates again (back to city)
¯m
vi probability that a migrant (from city to village) in age group i
does not migrate again (back to city); ¯m
vi = 1 m
vi
nct age structured city population of natives in period t
nvt age structured village population of natives in period t
n
ct age structured population of migrants from village to city in
period t
n
vt age structured population of migrants from city to village in
period t
Table 2 Explanation of symbols in the model
3 A hypothetical example
The following hypothetical example is meant to illustrate the dynamics of the
model outlined in Section 2, and to demonstrate how the model can contribute
to analyzing the successive development of a population. The example is based
on a population broken down into three 15-year age classes: 0–15, 15–30 and
A Leslie-type urban-rural migration model 9
3 2 1 1 2 3
3
2
1
1
2
3
village native citynative
village migrant
citymigrant
Fig. 1 Transitions between states (except migrant to migrant transitions)
1 2 3
3 2 1
village migrant
citymigrant
Fig. 2 Transitions between migrant states
30–45. The first step in constructing the population dynamics is to define two
Leslie matrices describing two populations, “city” and “village”, in isolation
(with no migration between them). The two populations are then linked to-
gether by specifying fertility rates and survival probabilities for migrants and,
most crucially for our purposes, a migration pattern between them. In what
follows, we shall investigate two among many possible migration patterns with
respect to the stable development of the resulting population. When analyzing
actual populations, the migration pattern has to be inferred from population
statistics, and in this case the procedure of linking populations together can
be reversed to study city and village populations in isolation, as we shall see
in Section 4.
10 Schmidbauer, R¨osch, Erkol
3.1 Two populations in isolation
Let two Leslie matrices be given as
Mcity =
0.10 0.20 0.20
0.95 0 0
0 0.90 0
, Mvillage =
0.30 0.90 0.70
0.90 0 0
0 0.85 0
.(4)
Their respective maximum eigenvalues are: λcity = 0.7086, λvillage = 1.2699.
The resulting growth of the stable populations in a 15-year interval is there-
fore 29.14% (city) and +26.99% (village), corresponding to annual rates
of 2.27% and +1.61%, respectively. When considered in isolation, the city
population is thus shrinking, while the village population is growing (when
stability is reached). The next step is to link the two populations together via
specification of a migration pattern between them. The two cases we consider
are: (i) there is no further migration after migrating once, (ii) migration is
location-specific.
3.2 Linkage I: no migration back to the origin
The assumptions connecting city and village populations are:
Assumption 1: Migration probabilities may depend on the origin (city or
village), but are equal across age classes:
mc1=mc2=mc3=mc, mv1=mv2=mv3=mv.(5)
Assumption 2: A migrant will stay at her destination and won’t migrate
back:
m
c1=m
c2=m
c3= 0, m
v1=m
v2=m
v3= 0.(6)
– Assumption 3: Migrants’ survival probabilities are obtained as arith-
metic means of city native and village native survival probabilities in their
respective age class:
p
ci =p
vi = 0.5·(pci +pvi ), i = 1,2,3.(7)
Assumption 4: As far as fertility is concerned, migrants will “split the
difference” between city and village fertility, that is, the age-specific fertility
rate of migrants is obtained as the arithmetic mean of the respective age-
specific fertility rates of city and village population. In symbols:
f
ci =f
vi = 0.5·(fci +fvi ), i = 1,2,3.(8)
This leads to the matrix MIdisplayed in Table 3. A comparison of the matrices
MIin Tables 1 and 3 reveals that MI
[city migrant]
[village migrant] and MI
[village migrant]
[city migrant] are
now zero matrices, reflecting our assumption that migration back is impossible
(m
c=m
v= 0). Considering the growth properties of city (shrinking) and
village (growing) populations, the theorem in Section 2 implies that there is
A Leslie-type urban-rural migration model 11
0.10 ¯mc0.20 ¯mc0.20 ¯mc0.20 0.55 0.45 0 0 0 0 0 0
0.95 ¯mc0 0 0 0 0 0 0 0 0 0 0
0 0.90 ¯mc0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0.30mv0.90mv0.70mv0 0 0
0 0 0 0.90 0 0 0.90mv0 0 0 0 0
0 0 0 0 0.85 0 0 0.85mv0 0 0 0
0 0 0 0 0 0 0.30 ¯mv0.90 ¯mv0.70 ¯mv0.20 0.55 0.45
0 0 0 0 0 0 0.90 ¯mv0 0 0 0 0
0 0 0 0 0 0 0 0.85 ¯mv0 0 0 0
0.10mc0.20mc0.20mc0 0 0 0 0 0 0 0 0
0.95mc0 0 0 0 0 0 0 0 0.95 0 0
0 0.90mc0 0 0 0 0 0 0 0 0.90 0
Table 3 Matrix MI, resulting from the assumptions in the example
a migration pattern, expressed by mcand mv, that will ultimately lead to
astationary total population. The transient behavior of the population is
illustrated in Figure 3, with plots of the 12 series (four population groups:
native city, migrant city, native village, migrant village; each broken down
into three age groups) in Nt=MI·Nt1for t= 1,...,25, where the initial
population is given by
N0
0= (1000,1000,1000
| {z }
city native
,0,0,0
| {z }
city migrant
,1000,1000,1000
| {z }
village native
,0,0,0
| {z }
village migrant
),(9)
and with migration probabilities mc= 0.1 (from city to village) and mv= 0.3
(from village to city) The maximum eigenvalue of MI, as displayed in Table 3,
equals λ= 0.9920, so that the population shrinks in the long run: The low
fertility of the city population prevails due to the high village-to-city migration
(mv= 0.3). Lowering mvsomewhat (to mv= 0.28, say) would make the
population grow.
In which way do growth and urbanization of the population in the long run
depend on fertility? This question can be discussed by letting stable growth
and urbanization depend on two factors, designated as fcfactor and fvfactor
in Figure 4, with which fertility rates fci,f
ci and fvi,f
vi, respectively, are
multiplied. This method of investigating the impact of fertility is suitable
because fertility rates have no theoretical upper bound. Figure 4 shows the
contour lines of growth (that is, the maximum eigenvalue of MI, hence the
growth in a 15-year interval) and urbanization when city-specific and village-
specific fertility rates are varied in a range from 50% to +100% of their
original levels (Table 3).
Urbanization is computed as the share of population in the first six compo-
nents of the population vector. (Technically, urbanization is the sum of the first
six components of the right eigenvector belonging to the maximum eigenvalue,
divided by the total sum.) The higher slope of the growth surface along the
fvfactor axis points to the more important role of village fertility for growth
under the given mortality and migration regime. A higher village fertility can
offset a higher city fertility with respect to urbanization, at the same time
leading to higher growth.
12 Schmidbauer, R¨osch, Erkol
(a) city native (b) village native
0 5 10 15 20 25
0 200 400 600 800 1200
age group 0 − 15
age group 15 − 30
age group 30 − 45
0 5 10 15 20 25
0 200 400 600 800 1200
age group 0 − 15
age group 15 − 30
age group 30 − 45
(c) city migrant (d) village migrant
0 5 10 15 20 25
0 200 400 600 800 1200
age group 0 − 15
age group 15 − 30
age group 30 − 45
0 5 10 15 20 25
0 200 400 600 800 1200
age group 0 − 15
age group 15 − 30
age group 30 − 45
Fig. 3 Evolution of initial population
fv factor
fc factor
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
fv factor
fc factor
0.59
0.6
0.61
0.62
0.63
0.64
0.65
0.66
0.67
0.68
0.69
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
Fig. 4 Impact of fertility on growth (left) and urbanization (right), example
3.3 Linkage II: location-specific migration probabilities
Only a single migration is possible for each person under Assumption 2 above.
The two Leslie matrices in (4) can also be linked together such that the maxi-
mum number of two migrations in a model with three age groups (nine migra-
A Leslie-type urban-rural migration model 13
mv
mc
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
mv
mc
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Fig. 5 Impact of migration on growth (left) and urbanization (right)
tions in a model with ten age groups, see Section 4 below) become possible.
One way to achieve this is by replacing Assumption 2 with
Assumption 2’: A migrant adopts the migration pattern of her current
location:
m
ci =mci, m
vi =mvi , i = 1,2,3.(10)
This amounts to making migration probabilities location-specific in the sense
that the status of a potential migrant is irrelevant.
The long-run impact of migration probabilities on growth and urbanization
can be studied via the matrix MIin Table 3, whose structure will now be
modified by letting long-run growth and long-run urbanization be a function
of mvmvi and mvmvi . In contrast to the procedure when investigating
the impact of fertility, we substitute a new set of migration probabilities, mc
and mv, rather than multiplying by a factor, because a probability is bounded
by one. Plots of resulting contour lines are displayed in Figure 5. In particular,
a higher city-to-village migration can offset a higher village-to-city migration
in terms of equal long-run growth, where the ratio depends on the location
of (mv, mc). For example, high village-to-city migration will lead to below-
replacement overall fertility if city-to-village migration is low (the lower right
corner of the left-hand plot in Figure 5). At the same time, urbanization will
approach a high level (the lower right corner of the right-hand plot).
4 Analyzing the populations of Germany and Turkey
The goal of this section is to analyze the populations of Germany and Turkey
on the basis of the model defined in Section 2. Data concerning age structure,
fertility, survival, and migration are presented in the Appendix. The first step
will be to give an account of the assumptions made to obtain the projection
matrix MI. This will enable us to compare the actual populations of Germany
14 Schmidbauer, R¨osch, Erkol
and Turkey with their respective stable counterparts, and to discuss the impact
of fertility and migration levels on long-run growth and urbanization, similar
to the procedure undertaken in Section 3.
4.1 Definition of MI
The subsequent analysis is based on ten five-year age groups, covering ages 0
to 50. All data used in the definition of MIare reported in the Appendix. On
this basis, we obtain the entries of the projection matrix MI(see Table 1) as
follows:
Natives’ fertility rates fci,fvi, survival probabilities pci,pv i, and migration
probabilities are taken from Table 7 (Germany) and Table 8 (Turkey).
Migrants’ probabilities to migrate again are obtained as indicated in As-
sumption 2’ in Section 3, that is: migration probabilities m
ci and m
vi are
location-specific.
Migrants’ survival probabilities p
ci and p
vi are obtained as arithmetic
means of city native and village native survival probabilities in their re-
spective age class; this is Assumption 3 in Section 3.
Migrants’ fertility rates f
ci and f
vi are obtained as arithmetic means of
city native and village native fertility rates in their respective age class;
this is Assumption 4 in Section 3.
4.2 Actual and stable populations
Given the initial population N0and the projection matrix MI, the model
permits a comparison of actual and stable populations in terms of numerical
characteristics (Table 4) and in terms of histograms of age distributions in
Figures 6 and 7.
Germany Turkey
actual stable actual stable
urbanization 85.3% 87.5% 65.0% 58.7%
growth, city, annual 1.0806% 1.2748% 0.894% 0.302%
growth, village, annual 1.8154% 1.2748% 1.239% 0.302%
total growth, annual 1.1874% 1.2748% 1.015% 0.302%
median age, city 29.53 28.85 21.81 24.42
median age, village 29.40 28.64 19.75 22.66
Table 4 Characteristics of (female) populations, age 0–50: actual and stable
City and village populations at time tcan be obtained by adding native and
migrant parts of the vector Nt; urbanization is the share of total population
belonging to the city part. Growth measures referring to the actual population
result from relating the initial population N0to N1=MI·N0; for the stable
A Leslie-type urban-rural migration model 15
population, the maximum eigenvalue of MIequals the growth factor, and
stable growth must be equal for all population parts. The median age is found
using linear interpolation of cumulative age group frequencies. Median age
here refers to the female population aged 0–50.
In the case of Germany, actual village population shrinks much faster than
it would in the case of stability. Median age is very similar in city and village; it
is lower in for the stable population. This is confirmed by the age distributions
shown in Figure 6: the actual population of Germany is older than its stable
counterpart. Frequencies of the stable population in the histograms are essen-
tially increasing, which is a consequence of the population being shrinking,
but there is an exception: the slight trough in village population, age group
25–30, is due to migration.
0 10 20 30 40 50
0.00 0.02 0.04 0.06 0.08 0.10 0.12
age
0 10 20 30 40 50
0.00 0.02 0.04 0.06 0.08 0.10 0.12
age
Fig. 6 Actual (left) and stable (right) female population of Germany
The actual Turkish growth figures in Table 4 reflect the young age struc-
ture of the population when compared to its stable counterpart, which is also
obvious from Figure 7. The relation between city and village with respect to
urbanization and median age is the opposite of that in Germany, actual me-
dian age being lower than stable median age. The hump (age groups 25–35)
in the otherwise monotonically decreasing histogram frequencies in the case of
the city age distribution is again due to migration.
4.3 City and village populations in isolation
Partitioning the projection matrix MIaccording to (1) reveals the charac-
teristics of city and village population if there were no migration. Projection
16 Schmidbauer, R¨osch, Erkol
0 10 20 30 40 50
0.00 0.02 0.04 0.06 0.08 0.10 0.12
age
0 10 20 30 40 50
0.00 0.02 0.04 0.06 0.08 0.10 0.12
age
Fig. 7 Actual (left) and stable (right) female population of Turkey
Germany Turkey
maximum eigenvalue, MI ,city 0.9375 0.9954
maximum eigenvalue, MI ,village 0.9412 1.0385
growth, city in isolation, annual 1.2829% 0.0922%
growth, village in isolation, annual 1.2057% 0.7587%
median age, city in isolation 28.85 24.90
median age, village in isolation 28.61 22.30
Table 5 Populations characteristics, age 0–50, without migration
matrices for the populations evolving in isolation are obtained as
MI,city =MI
[city native]
[city native] +MI
[city native]
[village migrant],
MI,village =MI
[village native]
[village native] +MI
[village native]
[city migrant] .
(11)
(Computationally removing migration in this way creates a situation which is
similar to the starting point in Section 3.) The projection matrices MI ,city and
MI,village can be analyzed like a classical Leslie matrix. A comparison of the
models characteristics in isolation, as given in Table 5, with the characteristics
of the full model (Table 4) will then reveal the balancing effect of migration.
For example, long-run growth in Turkish cities arises only through migration,
while internal fertility is below replacement level. The difference in city and
village fertility is much smaller in the case of Germany. The diminished miti-
gating effect of migration in Germany is also reflected in the smaller difference
between city and village median age.
A further interpretation of growth rates of city and village in isolation
is that any migration pattern, with which MI,city and MI,village are linked
together (see Section 3), will lead to an overall stable growth between the
values of city and village growth.
A Leslie-type urban-rural migration model 17
4.4 Impact of fertility levels on growth and urbanization
As in Section 3.2, stable growth and urbanization can be plotted as functions
of factors with which city fertility fc(fc,1, . . . , fc,10) and village fertility
fv(fv,1, . . . , fv,10 ) are multiplied. The result is displayed in Figures 8 and 9
where these factors, designated as fcfactor and fvfactor respectively, range
from 0.5 to 2.0, corresponding to lowering fertility levels down to 50% and
increasing them up to 200% of actually observed levels. We proceed with mi-
grants’ fertility according to Assumption 4 above. “Growth” in Figures 8 and 9
refers to five-year intervals.
fv factor
fc factor
0.875
0.9
0.925
0.95
0.975
1
1.025
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
fv factor
fc factor
0.835
0.84
0.845
0.85
0.855
0.86
0.865
0.87
0.875
0.88
0.885
0.89
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
Fig. 8 Impact of fertility levels on growth (left) and urbanization (right), Germany
fv factor
fc factor
0.925
0.95
0.975
1
1.025
1.05
1.075
1.1
1.125
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
fv factor
fc factor
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
Fig. 9 Impact of fertility levels on growth (left) and urbanization (right), Turkey
18 Schmidbauer, R¨osch, Erkol
There is a substitution effect between city and village fertility, which is
more or less constant across the factor range considered for the population in
Germany, but not Turkey, where the importance of village fertility increases
in a non-linear fashion as city fertility falls below village fertility. This sub-
stitution has a big impact on urbanization, with urbanization contour lines
standing almost perpendicular to growth contour lines. The shape of urban-
ization contour lines is very similar for Germany and Turkey.
Fertility levels in Germany are well below replacement level. For example,
with village fertility unchanged (fvfactor = 1), it can be shown that city
fertility would have to be raised by about 61% in order to reach replacement
level.
4.5 Impact of migration on growth and urbanization
The impact of migration probabilities mcand mvon growth and fertility is
shown in Figures 10 and 11. The procedure used to modify the projection
matrix MIis as explained in Section 3.3. As stated earlier, five-year growth
will be between that indicated by the maximum eigenvalues of city and village
population in isolation (given in Table 5). The impact of the balance between
mvand mcon urbanization can be grave when both probabilities are small,
according to the right-hand plots in Figures 10 and 11. Contour line shapes
are similar for the populations of Germany and Turkey, but with growth again
at different levels.
mv
mc
0.9375
0.938
0.9385
0.939
0.9395
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
mv
mc
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Fig. 10 Impact of migration on growth (left) and urbanization (right), Germany
A Leslie-type urban-rural migration model 19
mv
mc
1
1.005
1.01
1.015
1.02
1.025
1.03
1.035
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
mv
mc
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Fig. 11 Impact of migration on growth (left) and urbanization (right), Turkey
5 Summary and conclusions
We use a projection-matrix based population model which takes rural-urban
and urban-rural migration explicitly into consideration. Perron-Frobenius the-
ory provides insight into the stable behavior of this model. This approach con-
stitutes a platform for the analysis of the impact of fertility rates, survival
probabilities, as well as migration patterns on long-run characteristics of a
population. The interplay between fertility, mortality and migration, in par-
ticular: how village and city populations are connected together via migration,
is illustrated using hypothetical examples.
Applying this model to the female populations of Germany and Turkey,
as represented by their respective age structures, fertility rates, and survival
and migration probabilities in 5-year age intervals up to age 50, insight could
be obtained in two ways: (i) comparing actual population characteristics with
their stable counterparts; (ii) analyzing the stable behavior of city and village
populations in isolation, when migration is computationally removed from the
populations.
In Germany, both actual and stable, city as well as village actual popula-
tions are shrinking. The strongest actual negative growth (1.8% annually)
is observed for the village population, which is substantially greater than in
the case of the stable growth (1.3%); the rapid depopulation of rural areas
(relative to urban areas) in Germany thus appears as a transient phenomenon.
Median age of actual populations is slightly higher than in stable populations:
the actual population of Germany is actually older than is would be in the
long run, if current demographic parameters persist. Long-run urbanization
levels were found to be very similar to its current level, if slightly higher.
Turkish population characteristics differ from those of the German popula-
tion in almost every respect. Current population growth is positive and higher
than in stability (+0.3% annually), and highest (+1.2% annually) in the case
20 Schmidbauer, R¨osch, Erkol
of the actual village population. Median age of the city population is markedly
higher than of the village population, and stable median age is higher than
the actual median age (which explains the currently higher growth rate): the
current young population structure of Turkey is, in this sense, a transient phe-
nomenon, and so is the high level of urbanization, which will recede in in the
long run if current demographic parameters persist.
When considered in isolation, we find little difference between stable char-
acteristics of city and village populations of Germany, while the stable city
population of Turkey was found to shrink, to the effect that the slight over-
all population increase of Turkey is due to migration. We further find that a
slight change in migration patterns can have a profound impact on long-run
urbanization in Germany as well as in Turkey.
The framework on which this study is based provides ample opportunity
for further research. Two directions are: (i) to elaborate further characteristics
of actual populations, and assess the impact of public policies on the future
population structure and its economic consequences, such as the age depen-
dency ratio; (ii) to gain further theoretical insight into the model, for example,
the transformation of the model into a Markov chain (this was found useful
for other Leslie-type models; see, for example, Schmidbauer and R¨osch [20]),
which would create a platform for the introduction of reproductive values in
order to study migration and fertility from a novel perspective.
Appendix: Data for Germany and Turkey
Germany
All data used in our study of Germany are retrieved from the “Regionaldaten-
bank Deutschland”3and refer to years 2008 and 2009.
Definition of city/village population. The population living in ur-
ban areas (the “city population”) is the population living outside the rural
areas. Our definition of rural areas adopted the OECD urban-rural typol-
ogy (see [16]). In a first step, local administrative units (“Gemeinden, Samt-
/Verbandsgemeinden”, LAU2) with a population density below 150 inhabi-
tants per square kilometer were classified as rural. On the next higher terri-
torial level (“Kreise, kreisfreie St¨adte”, NUTS 3 regions), as it provided the
data input to our model in the first place, we defined rural areas according to
the share of regional population living in rural LAU2 (more than 50%) and
the absence of urban centers with more than 200 000 inhabitants accounting
for at least 25% of the population. The basic year of this definition was 2008.
The population living in such defined rural (respectively predominantly rural)
3http://www.regionalstatistik.de
A Leslie-type urban-rural migration model 21
areas constituted our “village population” in Germany. All further data were
aggregated on the basis of this territorial typology.
Table 6 shows the initial (by end of year 2009) female city and village
populations of Germany and Turkey (for which an explanation will be given
below) used in this study. (The two groups covering ages below 10 were defined
by splitting and combining the provided age groups proportionally to lengths.)
The share of female village population amounted to 14.5% at that time, with
respect to the total female population (i.e. covering all ages).
Germany Turkey
age group city village city village
0–4 1 434 569 233 617 2 040 448 1 267 489
5–9 1 505 624 264 152 2 105 014 1 276 474
10–14 1 632 282 297 973 2 146 147 1 272 645
15–19 1 779 434 323 606 2 334 348 1 339 118
20–24 2 101 192 316 612 2 312 556 1 156 954
25–29 2 154 056 301 496 2 076 129 998 454
30–34 2 035 725 300 608 1 734 423 817 621
35–39 2 219 078 364 288 1 683 480 793 328
40–44 2 869 289 484 606 1 377 598 663 653
45–49 2 924 688 522 709 1 112 760 590 090
Table 6 The initial (female) age-structured populations of Germany and Turkey
Fertility. Female births during 2009 by age groups of the mother were
related to the midyear (average) female population to estimate 1-year fertility
rates, and 5-year rates assuming linearity. (Age groups “below 20 years” and
“40 years and older” were treated as 5-year intervals.)
Mortality. Probabilities of surviving 5 years from the beginning of each
age group are estimated from the age-specific cases of death during 2009 to-
gether with a linear approximation of the population at risk at the beginning
of each age group (one fifth of the corresponding midyear female population
plus half of deaths).
Migration. Migration probabilities from city to village and vice versa,
covering 5 years from the beginning of each age group, were obtained in sev-
eral steps. In the first step, age-specific probabilities of departure from urban
(rural) areas are estimated from the cases of departure during 2009. Compu-
tations are analogous to the estimation of probabilities of dying within 5 years
from the beginning of each age group. As departing from an urban area is
not synonymous with arriving to a rural, the share of arrivals to a rural area
among all arrivals during 2009 is used to obtain directional migration proba-
bilities. These were further adjusted for internal migration by a factor 0.8321
of non-foreign departures.
The resulting rates and probabilities are given in Table 7.
22 Schmidbauer, R¨osch, Erkol
fertility survival migration
rate probability probability
age group city village city village to city to village
0–4 0.0000 0.0000 0.9964 0.9966 0.1514 0.0247
5–9 0.0000 0.0000 0.9996 0.9996 0.1496 0.0263
10–14 0.0000 0.0000 0.9995 0.9994 0.1488 0.0271
15–19 0.0223 0.0231 0.9991 0.9986 0.2983 0.0443
20–24 0.0951 0.1086 0.9989 0.9984 0.4632 0.0531
25–29 0.1897 0.2299 0.9988 0.9983 0.4154 0.0562
30–34 0.2184 0.2200 0.9982 0.9981 0.1543 0.0235
35–39 0.1084 0.0870 0.9973 0.9971 0.1522 0.0257
40–44 0.0211 0.0154 0.9953 0.9956 0.1517 0.0263
45–49 0.0000 0.0000 0.9917 0.9915 0.1504 0.0277
Table 7 German (female) population data used for the pro jection matrix MI
Turkey
Age-specific fertility rates of the urban and rural population in Turkey for the
year 2003 are published by the Hacettepe University Institute of Population
Studies [7]. — Survival probabilities (year 2000) are taken from WHO [24],
abridging the first two intervals. — Substitutes for migration probabilities were
obtained as the number of female migrants (village to city or city to village),
divided by the total female population in that age group (again village or city,
respectively). Both series are published by the Turkish Statistical Institute [22].
Since city- and village-specific survival probabilities were not available, we
assume that urban and rural populations have the same survival probabilities.
Furthermore, we assume that migrants immediately adopt the destination’s
(and the natives’) migration probabilities, that is, m
ci =mci,m
vi =mvi .
As to fertility rates, we repeat the assumption made in Section ??: Migrants’
age-specific fertility rates are obtained as the arithmetic means of natives of
both locations.
Finally, the initial (female) population N1of Turkey (for the year 2000;
see [22]) is given as shown in Table 6.
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Finite Non-Negative Matrices.- Fundamental Concepts and Results in the Theory of Non-negative Matrices.- Some Secondary Theory with Emphasis on Irreducible Matrices, and Applications.- Inhomogeneous Products of Non-negative Matrices.- Markov Chains and Finite Stochastic Matrices.- Countable Non-Negative Matrices.- Countable Stochastic Matrices.- Countable Non-negative Matrices.- Truncations of Infinite Stochastic Matrices.
Article
Age–sex-specific population forecasts are derived through stochastic population renewal using forecasts of mortality, fertility and net migration. Functional data models with time series coefficients are used to model age-specific mortality and fertility rates. As detailed migration data are lacking, net migration by age and sex is estimated as the difference between historic annual population data and successive populations one year ahead derived from a projection using fertility and mortality data. This estimate, which includes error, is also modeled using a functional data model. The three models involve different strengths of the general Box–Cox transformation chosen to minimise out-of-sample forecast error. Uncertainty is estimated from the model, with an adjustment to ensure that the one-step-forecast variances are equal to those obtained with historical data. The three models are then used in a Monte Carlo simulation of future fertility, mortality and net migration, which are combined using the cohort-component method to obtain age-specific forecasts of the population by sex. The distribution of the forecasts provides probabilistic prediction intervals. The method is demonstrated by making 20-year forecasts using Australian data for the period 1921–2004. The advantages of our method are: (1) it is a coherent stochastic model of the three demographic components; (2) it is estimated entirely from historical data with no subjective inputs required; and (3) it provides probabilistic prediction intervals for any demographic variable that is derived from population numbers and vital events, including life expectancies, total fertility rates and dependency ratios.
Article
Current population-forecasting efforts generally adopt minor variants of the cohort-survival projection method. This technique focuses on a population disaggregated into cohorts, a group of people having one or more common characteristics at a point in time and, by subjecting each cohort to class-specific rates of fertility, mortality, and net migration, generates a distribution of survivors and descendants of the original population, at successive intervals of time. Although cohort-survival methods take on a large number of variations, they all are essentially trend-based, dynamic, aspatial models of growth. The temporal element is introduced by a recursive structure which operates over a sequence of unit time intervals. The spatial dimension, when it is included at all, typically is accommodated by replicating the analysis over as many areal units as comprise the study area. Realistically, however, time and space need to be considered jointly in population-forecasting models. The need for interregional models which systematically introduce place-to-place movements and simultaneously consider the spatial as well as the temporal character of interrelated population processes is becoming increasingly apparent. Recently several demographers have taken advantage of the conceptual elegance and computational simplicity of matrix methods of population analysis. Their models, however, assume a “closed” population which is subject only to the processes of fertility and mortality. These, therefore, are not directly applicable to interregional “open” systems in which migration is frequently a much more variable and important contributor to population change than births or deaths. However, a natural extension of the demographer's matrix model allows one to incorporate place-to-place migration and provides an integrated interregional population-forecasting model which easily may be programmed for any of the current generation of digital computers. Such a model is outlined in this paper.
Article
Leslie’s work, rather than that of his predecessors Bernardelli and Lewis, is most commonly cited in the widespread literature using matrices, largely for the reason that Leslie worked out the mathematics and the application with great thoroughness. Some of his elaboration was designed to save arithmetic—for example his transformation of the projection matrix into an equivalent form with unity in the subdiagonal positions. Such devices, like a considerable part of classical numerical analysis, are unnecessary in a computer era.
Article
"We consider a Leslie-type model of a one-sex (female) population of natives with constant immigration. The fertility and mortality schedule of the natives may be below or above replacement level. Immigrants retain their fertility and mortality, their children adopt the fertility and mortality of the natives. It is shown how this model may be written in a homogeneous form (without additive term) with a Leslie-type matrix. Reproductive values of individuals in each age group are discussed in terms of a left eigenvector of this matrix. The homogeneous form of our projection model permits the transformation into a Markov chain with transient and recurrent states. The Markov chain is the basis for the definition of genealogies, which incorporate immigration. It is shown that genealogies describe the life histories of individuals in a population with immigration. We calculate absorption times of the Markov chain and relate them to genealogies. This extends the theory originally designed for closed populations to populations with immigration." (SUMMARY IN FRE)
Immigration into a below replacement population. Reproduction by immigration -the case of Germany
  • G Feichtinger
  • G Steinmann
Feichtinger, G., & Steinmann, G. (1990): Immigration into a below replacement population. Reproduction by immigration -the case of Germany. Forschungsbericht Nr. 127, TU Wien.
  • A Gedik
Gedik, A. (2003): Differential Urbanization in Turkey, 1955-90. Tijdschrift voor Economische en Sociale Geographie 94, 100-111.