890
ARE FRENCH METROPOLITAN AREAS
EVOLVING AS SELF-ORGANIZING SYSTEMS ?
Denise PUMAIN, Bertrand ROEHNER,
‘Lena SANDERS, SAININJULIEN
Centre de Gographie Théorique et Quantitative
13 rue du Four, 75006 PARIS, France
. However, it needs to be tested in real-world situations. ques-
tions are : Until which extent the same set of equations is able to simila-
te various observed urban evolution 7 and how many in
necessary to
we tried to apply the model to a sample of french metropolitan areas.
‘The INTRA URBAN model was drawn up by researchers of the "Brussel's
school” in systems dynamic [1] [2]. It is one of the first models of hifur-
cation-type designed to describe the dynamics of spatial systems. That kind
of model use equations which were first established to similate auto-orga-
nization phencmena in open physical systems situated far from equilibrium.
‘These models allow two kinds of changes in the system : progressive evolu-
‘tion along deteministic trajectories defined by non-linear equations ; and
sudden bifurcations, or change in trajectories, which may be produced by
xandam fluctuations. These bifurcations alter the structure of the system,
Such a conception of change is rather attractive for social sciences [3]
and seems to open new ways to the study of urban dynamics.
Inded, two series of models already designed and presented by P, Allen
have proved their ability to reproduce theoretical and plausible long-range
or medium-range evolutions of fictitious spatial systems : on the one hand
by simulating the development of a central places network [4] ; and on the
other hand by generating various internal configurations of econamic acti-
vities and residences in a large metropolitan area [2].
Before recording particular problems encountered and preliminary re-
sults in our tentative applications, we shall first recall the main specifi-
cities and mathematical structure of the INTRA URBAN model.
I - A DYNAMIC MODEL FOR INTRA-URBAN SPATIAL STRUCTURES
ne characteristic of INTRA-URBAN model, of particular interest for
geographers, is that it is basically a dynamic and spatial model,which is a
great advantage comparing for example to FORRESTER's urban dynamics model.
In ALLEN's model, urban space is considered as an open system, made up of
about twenty parts (urban districts or commmes). Each town-part is compe-
ting with each other for attracting jobs and residents. Each towm~part is
characterized by :
~ a number of jobs in different ecmomic activities (industry, exporting
tertiary, short-range and medium-range tertiary activities) ;
- a number of resident population having different occupational status
(blve collars and white collars) ;
~ a location respective to other tom~parts and accessibility indices to
transportation and information networks.
Starting from a given initial configuration, the model calculates
variations in the number of jobs and residents of each kind in each town-
part, according to spatial and non-spatial interactions between them.
891
Contrary to its apparent over-simplification of urban structure, the
model recaptures the high complexity of urban dynamics by defining these
interactions. They are designed as mathematical relationships between va~
riables or parameters standing for empirical regularities, theoretical hy-
pothesis or conceptual entities often encountered in the litterature about:
spatial dynamics, but not having yet been put all together in a single mo-
del. INTRA URBAN model thus incorporates :
~ a logistic type model of growth for jobs and population of each kind in
every town-part 7
- the principles of economic base theory, dividing jobs between exporting
activities (the growth of which depends upon an external demand) and non-
exporting activities (the growth of which depends upon local demand, by
means of induction rates) ;
~ a distance-decreasing-interaction model, of the exponential type with
intervening opportunities, for people working in a town~part and choosing
a residence location ;
= same principles of location theory for econamic activities, including an
“internal cooperativity" parameter standing for externalities and simila-
rities in locational preferencies of each activity sector, and a “sensi-
tivity to crowding" parameter figuring unequal needs in floorspace accor
ding to the nature of econanic activity, competition for space, and sat
ration effects ;
~ The model. also incorporates hypothesis about actor behaviour in an urban
context, for example grouping and segreyative tendencies inside and bet-
ween social groups } information theory is requested to introduce some
delay in reaction of entrepreneurs or individuals to a potential demand
and to define a parameter quoting for unequal information diffusion about
quality of places among urban actors.
TI ~ DESCRIPTION OF THE EQUATIONS
Each equation describes the variation, per unit of time, of the num
ber of jobs or residents in a town~part j. Each equation takes the follo~
wing form :
agg mays &-y3)
where yj figures the number of jobs or residents already located in amej,
a (settled for € or 7) is a speed of reaction to offer or to demand, and
XK is a potential demand which determines the magnitude of the variation.
(See tables 1 and 2 for the significance of parameters). This variation is
then first a function of preexisting quantities of jobs or residents in
considered town-part, which are growing according to a logistic curve.
1 ~ Bor activities depending upon an external demand (equation (1), indus~
try and exporting tertiary), this variation is also a function :
~ of external demand D ;
~ of relative attractivity of each zone j for this kind of activity ; it is
a ratio between intrinsec attractivity of zone j and the’ sum of all at~
tractivities of zones j. ‘The attractivity of a zone j is proportional to:
~ the volume of jobs of the same kind already present in that zone 7
~ the internal cooperativity of the activity being considered ( p para-
meter) 7
~ the sensitivity to crowding (7) ;
and inversely proportional to :
- production cost
Table 1 : Equations of INTRA-URBAN models.
with |
Ss] = nuber of employments of activity E in zone j
E = 1 = industry
= 2 = exporting tertiary
with vase
s" = nutber of employments of function u in zone j
= short-range function (local tertiary)
u = 4 = nediumrange function (regional tertiary)
= white collars
= distance between j and j'
for the significance of parameters, see table 2.
Table 2 : Significance of parameters
values in
‘Symbol Significance plese caeiavce
@ (17,2| For each zone, accessibility index to alto 1
transportation and information network
b (2) | Distance-decay parameter for each resident 58
population category
B(2,2)| Induction rate of short range and medium 5.3 53
range tertiary function by residents of
each category
Ci (2) | Elasticity of demand for the cost of each 2. 2.
service
Co (4) | Information available to the actors lo- 10- 10- 10
€ (4) | Speed of reaction of entrepreneurs to the | .001 .001 .Ol .O1
demand
171 (2) | Speed of adjustment of resident population 1005 005
to employment offer
@ (4) | Transportation cost for each activity +02 .0001 .025 .005
# (4) | Production cost for each activity rdraid.
p (2) | Sensitivity to crowding for residents 1000 500
p (4) | Internal cooperativity for each activity .Ol .O1 .0011 .OOL1
sector
@ (2) | Grouping tendency for a category of 2 22
residents
7 (4) | Sensitivity to crowding for each activity | 1000- 3000- 1000- 2000
2(2,4)
D (2)
Share of each category of residents working
in each activity sector
External demand
1. 63.7.5 .5 5 5
500 - «350 =
~ transportation cost 9 weighted by accessibility index of zone j to
transportation network @j 7
~ saturation of space in tom-part being considered, as measured by total
number of jobs and residents in that zone.
Each attractivity is exponentiated by the mean of a CO parameter
which figures the level of unanimity in territorial preferences among ur-
ban actors.
2 - For short-range and medium-range tertiary activities (equation (2),
local and regional services), employment variation is a function of inter-
nal demand emanating from each zone,
Internal demand for a type of services (function u) in a zone j' is
proportional to the total number of resident population, weighted by the
mean level of demand per inhabitant for this kind of activity (induction
rate $), and inversely proportional to the cost of this service. Cost is
affected with an exponent Ci which figures degree of elasticity of services
demand according to their cost. Cost incorporates production cost # and
transportation cost %, weighted by the distance §3j' between place of pro-
duction j and place j' where consumers are living. This potential demand for
services from zone j' to zone j is weighted by relative attractivity of zo-
ne j for zone j' (it is the ratio between attractivity of j for j' and sum
of attractivities for j' from all towm-parts).
Attractivity of a zone j for a zone j' is proportional to +
- the volume of preexisting jobs of type u in j;
~ the internal cooperativity of function u (p) ;
- the intensity of the crowding supported by the function u ;
and inversely proportional to
= production cost fl ;
= transportation cost % weighted by the distance §jj' between zone j and
zone j' 7
- crowding in zone j as measured by total number of employments and resi-
dents already located at j.
Each attractivity is exponentiated by the mean of a Co parameter
standing for degree of uniformity in territorial preferences among the
actors.
- For the two categories of residents (equation (3), blue collars and
white collars), variation in a zone j is function of the number of resi-
dents of each category employed by the four econamic activities in every
other tom-part j', weighted by residential attractivity Rjj' of zone j
for people working in zones j'. Residential attractivity is a ratio between
intrinsic attractivity of zone j upon labor force working in j' and the
sum of residential attractivity of all tom-parts.
Residential attractivity of a zone j for a category of people working
in a zone j' is proportional to :
~ number of residents of this category already present in j ;
~ affinity between menbers of a population of the same type a) +
~ intensity of crowding supported by this category of residents pv.
Residential attractivity decreases :
~ proportionally with spatial crowding as measured by total number of em-
ployment and residents in j +
894
~ exponentially with distance between residential place j and working place
4, according to a gradient b which measures sensitivity of each catego-
xy to distant commuting.
‘he program realizes integration of each variable by an iterative
procedure. It is possible to choose the time-lag for calculation and dura~
tion of the simulation. Finally, the program allows for introduction, at
various stages of the simlation, of random fluctuations in the variable
values.
III - APPLICATION
For the first test of the model, we choosed the urban agglomeration
of Rouen. It offered sufficient similarities with the fictitious metropoli-
tan area used by P, Allen for theoretical similations : a large urban area
(400.000 inhabitants), divided between 17 communes, the commmne of Rouen
being in a central position and being more than ten times larger by resi-
dents and employments number than other communes of the agglomeration. Hea-
vy industry is mainly located, as in the theoretical case, along a river,
the Seine.
1 - The data
For each of the 17 cammmes, we collected comparable data for 1954,
1962, 1968 and 1975 (dates of censuses). Labor force is registred at work-
place and had to be divided between four types of economic activity. Exis-
ting nomenclature allows only direct calculation for number of people em-
ployed in industry. lo distinguish three levels (local, regional and expor-
ting) among tertiary activities, we applied the minimm requirement tech-
nique. The two categories of resident population also stemmed from census
10
data (socio-professional nomenclature), by grouping all workers as “blue
collars" and all others as "white collars".
Relative location of the 17 commnes are figured by geographic coor-
dinates of their centre of gravity. Their accessibility index for comni-
cation lines needed by industry was quantified’ by separating commmes si-
tuated along the Seine river ( @ (j,1) = .5) and others ( @ (j,1 = 1:).
As for information network, only the commune of Rouen was given a more fa~
vorable situation ( @ (j,2) = .1) than the others ( @ (3,2) = 1.).
2 ~ First simulation : american-type evolution
In order to specify how the Rouen's evolution differed from some ame-
rican-like theoretical cases stuied by Allen et al. [2], we made a simla~
‘tion using the same set of parameters (values in table 2). Results are
plotted on figure 1 for the commune of Rouen (Center), all other commnes
being aggregated (Periphery), with dashed Lines representing observed evo-
lutions, for each of the six variables. These graphics show the following
tendencies :
~ a very rapid decline of industrial employment in the agglomeration center
and a stagnation in periphery (in fact, industry stagnated between 1954
and 1975 in Rouen and growed slowly at the periphery, a little more in
a few cammmes situated along the river) ;
- a slight move of concentration of exporting tertiary in the center (on
the contrary, Rouen lost employments mainly to the benefit of two.or
three commmnes situated on the right bank) ;
~ a very large growth of the number of local tertiary employments, which
are developing quicker in the periphery than in the center (this is al-
most the observed evolution) +
895
ut
calierage & ['américaine
TERTIAIRE
LOCAL
TERTIAIRE
FONDAMENTAL,
INDUSTRIE
contre
calibrage & faméricaine
ROUEN
Figure 1 (suite) :
OLS BLANCS
OUVRIERS
896
- after a quick departure from the center, regional tertiary employment
concentrates again on that commune, while stagnating at the periphery
(the observed evolution carries on the first tendency) ;
- blue collars are concentrating heavily in the center of the agglomeration
(the observed evolution is almost the opposite) ;
~ white collars leave completely the center and disperse in the whole peri-
phery, more and more farthest (actually, the number of white collars was
growing faster in the periphery but not leaving the centre of the agglo-
meration) .
This first trial shows strong differencies between similated and ob-
served evolution. In order to fit results to observed data, we had to chan~
ge the values of the parameters. It was done according to three principles:
~ same of the parameters could be estimated,directly from the data being
considered : induction rates (number of local and regional tertiary
employment per inhabitant), external demand D (total number of employment
of this activity sector at the end of the period), growing rates € and
1) for every employment and resident category.
~ other parameters could be induced from other data : the proportion of
blue collars and white collars working in each of the four economic ac~
tivities was determined by applying estimations established for french
urban population’; sensitivity to crowling T and p were estimated from
data about space needed by economic activities on the one hand and mean
density by housing-type in Rouen agglameration on the other hand.
~ the remaining parameters had to be estimated, following the relationships
between their value and their effects on urban configuration, as esta~
blished by Allen et al. [2].
14
3 - Calibration
We started calibration in a very empirical way, changing one value of
parameter after another and comparing results obtained by simlation with
observed values for each of the six variables in every commme at 1975.
‘These "empirical" similations did not yet lead to a good adjustment.
We learned however much through them about sensitivity of the model to dif-
ferent values of parameters. the main issues are the following :
~ different sets of values for parameters can generate the sane type of spa-
tial structure, for example an almost camplete decentralization of indus-
‘try reported upon one or more peripheral commmes, or a highly concentra-
ted distribution of industry or exporting tertiary. These kinds of bifur-
cations are among the most frequently encountered issues of the model.
They are mostly determined by respective values of the p parameters (ta~
ble 3).
= Resulting evolution obviously depends on choosen tine-lag for similation.
A long period of time furthers apparition of important bifurcations, whe-
reas a shorter period allows a better similation of smoth increase or
decrease as observed in the Rouen comme for industry and exporting ter-
tiary. Sensitivity of the model to parameters values also changes then +
for example, variation in the p parameters is less determinant upon spa-
tial repartition when time period is short. The model is then more sensi~
tive to € and 7 parameters.
~ fram one simulation to another, evolution of spatial repartitions or va~
riation of total quantities of employments and residents are not always
directed to the outcome that one would have intuitively expected, by trus-
ting in general significance given to parameters. These counter-intuitive
Table 3 : Values of parameters and resulting configuration for exporting
activities.
b 5 43 B .1 005 .25 .05
B .24 .24 .32 .32 #keolow e
ca 200 2 pv 1000. 800,
© 10 10 8 8 o 2 22
+4412 416.10 T 2000. 3000. 1000. 2000.
13 5 2.75 2 1 9 3 7 3 7
D 503 204 p (3Y = p (4) = .0015
ralues for : Resulting configuration for :
pa p (2) industry exporting tertiary
02 202 concentrated in the centre concentrated
.015 015 concentrated dispersed
014 014 dispersed (located on 3 concentrated
peripheral zones)
Ro sol Jdispersed (located on 1 concentrated
peripheral zone)
results explain themselves on the whole by non-linearities of the system,
butmake rather difficult a progressive "manual" calibration, because of
the number of iterative operations that they imply in choosing values for
parameters by trial and error.
‘That is why, after more than a hundred of these tentative similations,
we decided to use an automatic technique for calibration.
To fit the model's parameter to observed data, we used a non-linear
least square method. the actual computation was performed with the help of
a library subroutine (called MINUIT) designed by nuclear physicists. It is
a flexible minimization program incorporating three different minimization
procedures which are used successively and in conbination, Tt may in prin-
ciple handle as many as forty parameters. Tt appears however preferable for
an efficient convergence to subdivide the parameter set into smaller sub-
sets (of four or five parateters each) to be treated successively.
‘The adjustments that we obtained until now are nit good enough to be
considered as a "best fit" and to allow calculation of residuals. But we
have a serie of interesting results with various configurations according
to different values of parameters. In a number of cases, we obtained a good
estimation of the total number of employments and residents for the whole
agglomeration but a bad repartition between commmes of the center and of
the periphery ; in some other cases we obtained an almost perfect adjuste-
ment for the center but all variables for the other commes were underes-
timated. As the model is very sensitive to parameter values, the minimiza-
tion program can give very different results, depending upon the initial
set of values choosen and of the variations which are allowed to them. It
898
ud
is therefore necessary to examine in detail each result and to modify the
initial configuration of parameters values according to their significance
and effect on variables evolution. As these effects are not always the sa~
me, depending upon the total configuration being considered, the adjustment
procedure can take rather a long time. However, it seems to be consistent
from one urban case to another, as preliminary results with data for urban
agglomerations of Nantes and Bordeaux already showned.
For each of those agglomerations, main points of disagreement between
predicted and actual urban evolution concern rates of variation of employ~
ment in exporting activities and their spatial repartition. ‘lwo further di-
rections for experiment can then be suggested :
~ by replacing final potential demand D by a progressive allocation of new-
ly offered employments, less brutal variations could be generated +
~ by modifying accessibility indices of cammmes, final observed repartition
could be better estimated and explanation for these differencies resear~
ched afterwards.
We think very plausible to improve the global quality of adjustment
between observed and calculated evolution by choosing firstia better set of
parameters and using the calibration procedure. That is why we do not pro-
pose for the manent a final judgment about the assumed too high complexity
of the model ( as in Wilson[S1), or about its ability to reproduce observed
evolutions. Qne problem will nevertheless arise when we will find - if we
é so ~ a "satisfying" fit : is the corresponding set of parameters actual-
ly the better one ? and is it unique ? Answers to these questions are, of
course, critical if we want to use the INTRA URBAN model as a tool for com
paring evolutions of various urban agglomerations - as we hoped by starting
this study. It is possible that more insights in theoretical issues of the
model are necessary before trying applications ~ but on the reverse, only
applications can reveal practical utility of the model. What is more uryent-
ly needed is a serious improvement in the general methodology required for
applications of such complex dynamic models to real-world situations.
‘REFERENCES
(1] ALLEN, P.M., SANGLIER, M., 1978 : Dynamic models of urban growth. Jour-
nal of Social and Biological Structures, n° 1, pp. 265-280,
and 1979, n° 2, pp. 269-278.
[2] ALLEN, P.M., SANGLIER, M., BOON, F., DENEUBOURG, J.L., DE PAIMA, A.,
1981 : Models of urban settlement and structure as dynamic
self-organizing systems. Washington (D.C.), US Department of
Transportation, contract T.S.C. 1640.
[3] PRIGOGINE, I., STENGERS, I., 1979 : La nouvelle alliance. Paris, Gal-
‘Limard.
[4] ALLEN, P.M., SANGLIER, M., 1979 : A dynamic model of growth in a cen-
tral place system. Geographical Analysis, vol. 11, n° 3,
pp. 256-272.
[5] WILSON, A.G., 1981 : Catastrophe theory and bifurcation : Applications
to Urban and Regional Systems. London, Croom Helm.
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