Urban is still an interesting topic to discuss whether in government or public studies. As economy grows in cities, many people are attracted to come to cities from villages to try their luck. In this paper, we investigated urban area development using von Thunen’s economic location theory. By using stochastic Cellular Automata which views land location as agents that will change their states in agriculture, industry, and service series, here we show how the urban areas dominated by economy, industry, and service activities develop to their surroundings and form areas with typical activities with urban area. Even for the farther agriculture area from urban center, von Thunen’s theory is still valid.von Thunen’s Theory, urban, Stochastic Cellular Automata, Markov matrices

Ivan Mulianta

Yun Hariadi

English

Research Papers in Economics

  1
Urban Area Development in Stochastic Cellular Automata 
 
Ivan Mulianta1, Yun Hariadi2 
 
1.  Research Assistant in Bandung Fe Institute, email: ivanmulianta@yahoo.de  
2.  Department of Dynamical System Modeling, email: faust_doktor@yahoo.com     
 
Abstract 
Urban is still an interesting topic to discuss whether in government or public 
studies.  As economy grows in cities, many people are attracted to come 
to cities from villages to try their luck.  In this paper, we investigated urban 
area development  using von Thunen’s economic location theory.  By 
using stochastic Cellular Automata which views land location as agents 
that will change their states in agriculture, industry, and service series, here 
we show how the urban areas dominated by economy, industry, and 
service activities develop to their surroundings and form areas with typical 
activities with urban area.  Even for the farther agriculture area from 
urban center, von Thunen’s theory is still valid. 
 
Keywords: von Thunen’s Theory, urban, Stochastic Cellular Automata, 
Markov matrices 
 
1.1 Introduction 
 
Urban is still an interesting topic to discuss whether in government or 
public studies.  As economy grows in cities, more people are coming 
there to try to find their luck.  According to Pitirim Sorikin (1959), many 
people wanted to improve their social class hoping to vertical mobility, so 
they came to cities.  Instead of vertical mobility, most of them only receive 
horizontal mobility (e.g., farmer becomes labor).  Of course, the rapid 
growth of people coming to cities becomes a problem especially to 
government deciding policies.     
Government tries to make some policies to regulate the number of 
urbanization.  Some make the city as a closed city which means no one 
can enter the city, like in Jakarta when Ali Sadikin was a governor.   
Besides making city as a closed city, there are also some policies to make 
satellite cities (suburb) such as Ciputat, Bintaro, Bekasi.  Although the 
making of satellite cities mobilize people from city for housing but there 
are other problems rising, i.e.: transportation and traffic jam.  We see 
nowadays there are a lot of new flyovers being built but still cannot 
answer the traffic jam problem.  The city government also has to opt with 
the problem of land use whether to build new settlement for urban poor,   2
build new shopping malls, build new houses, or else.  These are the 
problems that the city government has to concern with.   
 
1.2 Several Definitions 
 
To give limitation on our paper, we want to propose some 
definitions, they are: 
a) Urbanization is redistribution from the countryside to the city.   
According to that, urbanization is measured as numbers of 
population who reside in urban area.    
b) Urban Structure is an arrangement of urban area (Wikipedia).  In this 
paper we use von Thunen’s. 
c)  Suburbs are inhabited districts located either on the outer rim of a 
city or outside the official limits of a city (the term varies from 
country to country), or the outer elements of a conurbaniton. 
 
1.3 Previous Works  
 
The urban problem has played great attention for centuries; so here 
we would like to consider some previous works. 
 
1.  John Heinrich von Thunen’s works gave a predictive model of rural 
development around an idealized isolated urban center, imposing 
several simplifications in an attempt to focus on some of the 
fundamental processes at work in settlement patterns and rural 
economic activity. It is presently a regular component of 
introductory economic geography courses. Despite criticism for 
oversimplifications, it has persisted in geographic, economic, and 
rural development curricula. 
 
His founding was meant as generalization towards discovering laws 
which govern agricultural prices and translate them into land use 
patterns.  (Yuya Sasaka & Paul Box, 2003). 
 
He also considered four generalized types of economic and 
agricultural activities.  
 
2.  Yuya Sasaki and Paul Box (2003) 
 
Yuya Sasaki and Paul Box developed a model to demonstrate 
whether a collection of autonomous individuals can contribute to 
the formation of this optimal pattern, without any system-level 
optimization capabilities. They also analyzed the mechanism that 
leads to an emergent spatial optimization by applying theories of   3
positive feedbacks and lock-in.  They used agent-based modeling 
with cellular automata, and the software called SWARM. 
 
2. Model 
 
We develop von Thunen’s location theory using stochastic cellular 
automata approach to model the emergence of urbanization and the 
change of the urbanization.  
 
Von Thunen’s location theory divided economic location structure 
into four types of agricultural activities that surrounded a city.  Each of 
them will form a ring that scopes an area through another, visualized in 
figure 1.  
    
 
Figure 1 Von Thünen's isolated state (source: JASSS)   4
 
In this paper, we made four other agricultural activities types in one type, 
which is non-urban; hence von Thunen’s location model becomes a city 
surrounded by non-urban area.  Generally, economic activities in a city 
which are high and dynamic will attract people from non-urban area; 
hence it makes a city as an urban area directly. Furthermore, besides 
using that assumption, we take a closer look at economic area divided 
from von Thunen’s location area, especially in city area and non-urban 
area (four agricultural activities types).  The result of that, von Thunen’s 
location model becomes: 
Agriculture  Industry             Service 
 
 
We divide into these categories based on assumption that a city area will 
become a center of economic service; suburban becomes economic 
industrial area; and village become economic agricultural area.  We tried 
to simulate how the emergence appears in the suburban area and its 
change.  As time goes by, numbers of people in city area will increase 
until reaching city space’s limit to push to shift some of economic activities 
to the outer city area, including settlement problem. 
 
While in village area in which economic agricultural activities has 
s o m e  k i n d  o f  s u p e r i o r i t y  c o m p a r e d  t o  c i t y  i n  t h i s  c a s e  m o r e  s p a c e  
availability, this difference will become an important factor in relationship 
of city and village.  The city needs more space to fulfill economic activities 
and the other sides; village needs high economic activities to support its 
economic activities, instead of the increasing numbers of population in 
each side which demand more spaces and appropriate economic 
activities (availability of job fields). 
 
We limit factor which pushes the change of land-use function in this 
model only to economic problem.  Land-use function in location is 
s y m b o l i z e d  a s  a  c e l l  w h i c h  h a s  l o c a t i o n    N j i ,..., 1 , =  and in an area 
M l k ,..., 1 , =  in this paper M = 3 (agriculture, industry, service).  At each 
specific time there is impossible for two different land-use functions.  
1 ) ( , ,..., 1 , , 0 ) ( , 1 ) ( = = ≠ = = ∑
l
k
l
l
i
k
i t N M l l k t N t N  
 
Aggregately, all cells being in area I aredefined as: 
 
) ( ) ( t N t N
i
k
i
k ∑ =    5
 
Meanwhile in total all cells being in all areas are 
 
∑ =
k
k t N t N ) ( ) ( 
 
The change of state from area k to l in time from t to t+1 defined as: 
 
1 ) 1 ( , 1 ) ( , 1 = + = = ∆ t N t N N
l
i
k
i
kl
i  
 
And sum of total cells which migrate from area k to l is: 
 
∑∆ = ∆
i
kl
i
kl N N  
 
Hence total cells which choose area l in time t+1 are: 
 
∑∆ = +
k
kl l N t N ) 1 ( 
 
Can be written as probability change: 
 
∑ = +
k
k kl l t N P t N ) ( ) 1 ( 
 
With: 
 
) (t N
N
P
k
kl
kl ∆
=  
 
Equation above is explained as a change probability area from k to 
l globally, but how does probability change area from k to l at cell i?  
Claudia Maria de Almaida et al [Almaida, C. M. (2002)] proposed general 
equation, i.e.: 
 
)] , ), ( ( ), ,..., ( [ ) (
1 M l N t N g X X f t p i
l
j
k
i
E
i i
kl
i ∈ Ω ∈ Θ =  
 
First element from functionΘ , is f(.) explain factors influenced cells in 
some different area state meanwhile function g(.) describes neighbors 
influenced from cell i, especially the equation can be defined as: 
 
∑ ⎥
⎦
⎤
⎢
⎣
⎡
+ −
+
Ψ = +
e
ie kl
kl
kl
i W
t P
t P
t p
~
exp
) 1 ( 1
) 1 (
) 1 (    6
 
With ie W
~
, describes weight to measure whether relevant or not a 
factor to change from k to l in cell i meanwhile Ψ  is a normalized constant 
to probability function.  If neighbor element is considered as a factor 
influenced to cell i in taking decision, thus: 
 
) 1 (
4
) (
) 1 ( ˆ +
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
= + ∑ Ω ∈ t p
t N
t p
kl
i
j
i
j kl
i
i γ  
 
Equation above is trying to connect between cells behavior to the 
global condition which in this model assumed neighbors from each cell is 
4. 
 
3.  Simulation and Numerical Analysis  
 
In the next simulation, we involved three conditions in an area, i.e.: 
agriculture, industry, and service.  Each area location in one specific time 
is only representative one from conditions above.  But change dynamic 
between conditions, which explain change probability area state 
becomes another area approach with Markov matrices, which is: 
 
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
1 0 0
2 . 80 . 0
5 . 15 . 85 .
P  
 
  From left to right (column) or from top to bottom (row), each of 
them explain change probability from state of agricultural area, industrial 
area, and service.  In practice, Markov matrices above can be resulted 
through comparing change area state in previous years. Also, we may 
see that Markov matrices above are government policy to arrange its 
area.  In this simulation, matrices above are used as a constant unit, 
although this can be extended as a function of time. 
 
  Next are simple steps for simulation in this paper (World X size x 
World Y Size: 100 x 100): 
1.  In initial stage, area is divided into agriculture, industry, and service.  
The difference between urban and non-urban area is laid in 
percentage from that each state, also each land location has 
economic weight factor, ie W
~
  in this simulation it is a function of 
distance from urban center.  This condition is an initial condition in 
simulation stage.   7
2.  Each location from land will have state change or don’t have push 
from Markov matrices.  This change caused by Markov matrices is 
global push which will be accepted by an agent as external pusher 
while to change itself influenced by environment around, in this 
term its neighbors’ decision. 
3.  Each land location will receive influence from its neighbors based 
on the number of probability neighbors’ value to stay in its state. 
4.  The result from 2 and 3 will decide land state whether change or 
not. 
5.  This process is iterated from procedure 2. and so on 
 
 
Figure 2 
Initial condition (t=0), in center is urban area dominated economic activities in industries 
(yellow) and services (red).  Meanwhile in non-urban area is dominated by agricultures 
(blue) 
 
 
 
Figure 3 
This is 5th iteration from initial condition in Figure 2, shows that it is start to change of land 
state, in urban area is increasing in economic services and this increasing is still higher 
than in land for industry.   While it is also happens in outer urban area.   8
 
 
      
Figure 4 
This is 10th iteration, shows that existed land state in urban area begins to spread to its 
surrounding.  Further that spread is still can be observed to its initial condition and also 
the emergence of new land state group, what is called suburb. 
 
 
 
 
Figure 5 
This is 15th iteration, shows that urban area structure has spread, so thus not only urban 
center becomes bigger but there are now new colonies that have almost the same 
characteristic like in urban center   
 
 
   9
4.  Conclusion   
 
W e  l o o k  f u r t h e r  t o  v o n  T h u n e n ’ s  a g r i c u l t u r a l  e c o n o m i c  l o c a t i o n  
theory, simply in simulation shown that agricultural area which is closer to 
urban area will decrease step by step and is replaced by economy 
activities which is closer to urban area.  
 
This simulation result also shows the emergence of new group 
economy activities.  This is because of neighborhood factor which 
influences to change condition an area.  In other word, the change of an 
area is more caused by neighbor’s influence. 
 
Simulation in this paper is a very simple simulation type, needed to 
develop furthermore, such as using in-constant Markov matrices and 
irregular area or map based. 
 
 
 
 
Acknowledgement   
 
The research is done under financial support from Surya Research Inc. 
Both authors thank to all BFI colleagues for the critics, discussion, and 
suggestion.  All faults remains to the authors.  
 
 
References 
 
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the Evidence”, 30th March 2004, URL: 
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Life Chances and Social Mobility: An Overview of the Evidence”,

Urban Area Development in Stochastic Cellular Automata

http://128.118.178.162/eps/urb/papers/0412/0412001.pdf

Urban Area Development in Stochastic Cellular Automata

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