Approach Dynamics

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Modeling Population Dynamics in the City:

from a Regional to a Multi-Agent Approach

ITZHAK BENENSON*

Departmentof^Geography,Tel-Aviv University,Ramat-Aviv,Israel, 69978 (Received^16November1998)

Thepaperreviews differentapproaches tourbanpopulationfrom thepoint ofview of the theory of complex systems. Regional models deal with large numbers of urban regions involved inexchanging populationand resourcesamong themselves.In contrast,ecological modelsdeal with severalqualitativelydifferenttypesof relationships betweenasmall number ofcomponents, aimed at understanding the most general laws ofurban dynamic. Two relatively new approaches, namely CellularAutomataand multi-agentones describethe macro-processes resulting fromuniform collectiveprocesses atthe micro-leveloflandparcels and migrating city individuals. Recent results ofthe multi-agent simulations regarding abstractand real-worldsystemsarepresentedin moredetails.

Keywords." Urban modeling,Regional modeling,Ecological modeling, CellularAutomata, Multi-agent simulations, GIS-basedmodeling

URBANMODELING:FROM INTEGRAL DESCRIPTION TOTHE DESCRIPTION OF CITY COMPONENTS

Approachestourban population dynamics model- ing evolvedinparalleltotheevolutionofthetheory of complex systems. Historically, the regional approach dated from the sixties and rooted in cybernetics of the fifties, was the first developed.

Regional modelingisbasedonthe presentationof a geographical system (city, metropolitan, adminis- trativeregion)bymeansof

"zones",

whichexchange population, goods, capital,etc. among themselves.

Most regional models are economically oriented,

witheachzone characterizedby avector ofsocio- economicindicators.Componentsofthis vector are numbers/proportions of population groups in a zone according to their age, culture, education or employment,aswellasthe numbers/proportionsof jobs in different industries, dwellings of^different

kinds,servicesof differenttypes,etc.

The initial goal of regional approach was the simulationofreal systems.This operational aspect turned out to be problematic after its initial applications. On the one hand, even a minimal division of a city’s territory into zones and its population into socio-economic groups requires dozens of equations andthe estimation ofhundreds

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149

ofparameters.Ontheother,althoughthe behavior of solutions of the regional models is insensitive to some ofparameters and sensitive toothers, the sets of the most influential parameters cannot be identified until constructionofthemodel hasbeen completed. The inconsistencies wereresolved dur- ing the 1970s and 1980s with the development of the general theory of complex systems. Seminal works by Prigogine (Prigogine, 1967; Nicolis and Prigogine,

1977)

^on "dissipative

systems"

and by Haken

(1978, 1993)

on"synergetics" made clear that opencomplex systems (including urban

ones)

reveal commonlaws ofbehavior. Stated briefly, according to Haken, for example, the evolution of external

"control" parameters entails the bifurcation of city dynamics. In the vicinity of the point of bifurcation, the trajectory of thesystem belongsto a low-dimensional subspace, which is defined by slowly varying characteristics of the system and can be described by few "order

parameters"

that

"enslave" the rest of

system’s

rapidly changing

"fast" characteristics. The new factors, which re- present slow order and fast characteristics of the system are functions ofthe original variables, but donotcoincide withthem.

The results of complex system theory, which became the "paradigm"forregional analysis,essen- tiallyinfluenced thefutureof urban modeling. The paradigm basic idea is as follows:

A

qualitative understandingof urbandynamics doesnotdemand adetaileddescriptionofall theinterrelationsestab- lished among the components.We arerequired to understand only themostimportantand,perhaps, latent orders of urban dynamics, to classify their possible interactions, andto interpret qualitatively different outcomes.Asfor quantitative description of urban dynamics, the problem remains open if the order parameters uncovered by the mathe- maticalanalysis differfromthose variables we are interested in.

Consequently, we can understand urban dyna- micsqualitativelybyreducing the numberof com- ponents andsocio-economic indicatorsof the model from hundredsto afewunits.Incontrast, inorder tostudy those dynamics quantitatively,wehaveto

include the additional information necessary to provide clues about the internal orders of a certainurban system.

The achievements made by the qualitative approach are based on the application of mathe- matical ecology models to urban systems (Day, 1981; Dendrinos and Mullally, 1982; 1985;

Zhang, 1989;Dendrinosand Sonis,

1990).

Regard- ing population dynamics, ^we can cite studies of competition between two social groups for space (Zhang,

1989),

or studies of "predator-prey"

relationships between density and economic status ofcity population(Dendrinos^{and Mullally,} 1982;

1985).

Thenovelsuccessor tomulti-component regional modelingisthe Cellular

Automata (CA)

approachto modeling cityinfrastructure(Tobler, 1979; Phipps, 1989; 1992; Phipps and Langlois, 1997; Whiteand Engelen, 1993; Batty and Xie, 1994; Durrett and Levin, 1994;

Wu,

1996; White etal., 1997; Rapini and Rabino,

1997).

Formally, a CA is a regular latticeof cells appearinginalternating states,which changes accordingtothestateof the cellitselfandits neighbors.In urban applications, cellstatesreflect propertiesofthe city’sparcels, namely, thetypeof land usageoreconomicpotential.The definition of cellstates as landuses entails essential constraints onCAapplications. First,itimplieslimitations on the cellsize thelattershould be sufficiently small to retain internal homogeneity; second, it places lowerlimits on thedurationofthemodel’siteration land usage does not change in days or even months. At the same time, for properly chosen characteristic cell sizeandtimestep, theestimation ofCA parameterscan be done straightforwardly, according to the city maps of consecutive years (Whiteand Engelen,

1993).

An

important brand of CA models simulates the fractalstructureof the city. Fractal andCAap- proaches differ in the mechanisms they use for generating urban patterns. From the broad spec- trum of local interaction mechanisms, fractal models utilize the Diffusion Limited Aggregation scheme,whichallowstogenerate spatial patterns of given fractaldimension.

CA and fractal approaches ^are very promising regarding the description ofthe physicalstructure ofa city.

However,

for obvious reasons, they do notaccountfor"soft" component,namely, forthe structureand distributionofthe city’s population.

Spatial population dynamics inthe city is an out- come of the individuals’ decisions and residential behavior. Thelatter are consideredexplicitlyin the framework ofthemulti-agent

(MA)

models

(Maes,

1995; Openshow, 1995; Portugali and

Benenson,

1995; 1997;Conteetal.,1997;Troitzschetal.,

1996).

Thegeneralaimof

MA

modelsis toinvestigate the consequences of social interactions in the urban context. Individuals arerepresentedin

MA

models asfree agents whoexhibittheeconomicandcultural properties of human beings and whochange their location inthe city. Inthe

MA

framework, spatial and temporal resolutions are constrained from above: first, the residential decision is based on information on separate apartments or houses;

second, relatively high numbers of individuals change theirresidence during any month or even shortertime interval.Asaresult,

MA

models study rapidly transpiring urbanphenomena, particularly the emergence of residential segregation in the urban space.

To conclude, regional models deal with large numbers ofurban regions involved in exchanging population and resources among themselves. In contrast, ecological models, which deal with sev- eral qualitatively different types of relationships between asmall number of components, aimed at understanding the most general laws of urban dynamic.

A

newsetofmodels, CAand

MA

models describethe macro-processes resulting from collec- tive behavior atthemicro-levelof land parcels and migrating city individuals. More specifically, CA modelsdealwiththe relatively slow changes under- gone by the urban infrastructure and provide the backgroundfor

MA

simulationsof relatively rapid changesinthe population’s spatialstructure.

In this paper different approaches to urban modelingwith respectto population dynamics are compared.

A

hierarchical combinationofCAand

MA

micro-approachesissuggestedasatool forthe

simultaneousmodelingofurbaninfrastructureand population dynamics.

THE MAINSTREAMS OFURBAN MODELING

2.1 Black-boxMacro-Modeling 2.1.1 EcologicalApproach

Much effort has beeninvested in studies of urban population dynamics suggestingdifferent types of interactions between population groups and the otherurban components. Mostof theseresultscan beconsidered asapplications of ecological models to urban situations. Mathematical ecology

(see

Murray, 1993, among numerous

texts)

has an almost hundred-year history ofdevelopment, and dynamic models ofsimple ecological systems con- sisting of one or two interacting populations are among the most elaborated examples of this approach. Qualitatively, the behavior of amodel’s solutionsdependson thefeedback betweenpopula- tion density and growth rate for a one-species system or on the type ofinter-species interaction for a two-species system. Several qualitatively differenttypes offeedback relations(linear, expo- nential,

etc.)

and interactions (competition, pre- dator-prey, parasite-host,

etc.)

have beenstudied and their consequences classified. Many of the subsequent results regarding plant and animal species have direct urban interpretations. For example, Dendrinos and Mullally

(1982, 1985)

considerurban dynamics asinterplay between the city’s populationsizeandanindividual’s economic status:

dX/(X, dt)

^c.

(Y- Ym) -/3.

X

dY/(r.dt)--

-y

(Xm X), (1)

where X denotes relative population size, Y is deflated per capita income,X" isthe city’s carrying capacityin terms ofpopulationsize,

Ym

^isprevailing

deflated per capitaincome.

The solutions of

(1)

depend on the type of socio-economic interactions, the latter determined by the parameters c,

/3

^and 5. The estimates of the parameters for 32 US metropolitan areas (Dendrinos and Mullally,

1982)

^{best fit} those interactions between population size and income thatcorrespondformallytothe predator-prey type ofrelationshipsandprovide oscillating convergence to an equilibrium. According to the parameter estimates,the periodofconvergingoscillationsfor citiesinthesample ranges between 20 and 150 years.

Theabove model doesnot accountdirectlyfor the critical process of spatial diffusion. The simplest way to do so is to assume a radial symmetry of population distribution and to add the diffusion term to the equation for non-spatial population dynamics. For alogistic description of population growth, the resulting model looksasfollows:

dN/dt

^c.

(1 N) N+

D

(1/r dN/dr

+ d2N/dr2), (2)

whereristhe distancefrom thecenterofthearea.

O’Neil

(1981),

who investigated the abilityof this simplemodel toapproximatetheprocessofexpan- sion of the black ghetto in Chicago during 1968- 1972,obtained agood agreementof the solution of the above equation with the observed data. The equation that best fits the experimental data is as following:

dN/dt

^0.191

(1 N).

^N

+

^0.128

(1/r. dN/dr + d2N/dr2).

The coefficient of determination equals

r2=0.79,

andissignificantat p

<

^0.001.

Zhang

(1989)

has been developing atheoretical modeloftheurban dynamicsin whichthe popula- tionconsistsoftwogroups competingforspace:

dX/dt

^c.

(a-

b.X- c.

Y)

^X-

all.

^{X. Y}

+ Dx" d2X/dr ^2,

d

Y/

^dt

^/3. (a-

^{b X-c.}

Y)

^Y-

d2

^{X. Y}

(4)

+ Dr.

^d2

Y/dr2,

where groupdensities

X(r, t)

^and

Y(r, t)

dependon distancertoCBD.Possible interactionsbetween the membersofthe groupsareexpressed by positiveor negative values of

dl

^and

d2.

^Zhang

⁽¹⁹⁸⁹⁾

^interprets

apositive

dl

and negatived2,for instance,asadesire ofless-educated populationXto live withthemore educated population Y. Expressions

Dx.d2X/dr

²

and D

y.dZY/dr

² describe the diffusion of the individuals.

Whensolvingfor

(4)

^weeventuallyobtain eithera homogeneous city where onlyX- or Y-individuals remain,or acitywhereboth groups persist. Zhang

(1989)

has proved that the number of qualita- tivelydifferent outcomes of

(4)

is lessthanforthe corresponding non-spatial model. Namely, the oscillating solutionsdo not exist, and we cancon- clude, therefore, that migration stabilizes urban system dynamics.

Tosummarize,withinthe frameworkofecologi- calapproach urban dynamicsis definedbyinterac- tionsbetweenlimitednumber of socialgroups.The

system’s

dynamic patterns are qualitativelyclassi- fiedaccordingto thetypes ofinteraction.Forthose specific situations,wherelow numberof variables is sufficient, acorrespondence between model results and experimental datacanbeachieved.

2.2.2 Regional Models

Ecological models intentionally simplify urban population structures and urban systems. The de- scriptionofurban population dynamicsthat canbe attunedtorealworldsituationsdemandsplenty of components and relationships are accounted for, and regional models try to account explicitly for urban complexity.Intheregionalframework,each zone is characterized usually by two sets ofvari- ables,the first onerepresenting the propertiesof the city’s physicalenvironmentandthesecond those of the city population.Toillustrate,PeterAllenand his colleagues

(Allen

and Sanglier, 1979; 1981; Allen, 1982;Allenetal., 1986; Engelen,1988),by adopting thisapproach, attemptedtosimulatethedynamics of the city ofBrussels, the economicdevelopment of North Holland, the economics of Belgium as

a whole, etc. We consider here the model of an artificialcity, "Brusseville",whose structurereflects that of Brussels (Engelen,

1988).

According to the logic of regional modeling, Brusseville is divided into N:36 spatial zones.

Eachzone is characterized by avector vi, which consistsof"population" and"socio-economic"sub- vectors.The populationsub-vector,inturn,consists oftwo setsofcomponents,representing the proper- tiesof white- andblue-collar residents. Each set is characterized bythe number ofactive residentsxi,

the number of non-active residents ni, and the variablesrepresenting migration flows. The socio- economicsub-vectoraccountsforthequalityof the neighborhoods, the housing stockHi,andthelevel ofemployment of different types (finance, ^heavy industry, etc.). Thecrucial element of themodelis thedefinition ofthezones’interactions, the number ofwhichforasystemdivided intoNzonesisofan order of N

2.

Bearingin mind thepotential applica- tions, the authors intentionally keep the number of independent parameters describing these inter- actions to an orderofN. Todo so, theyintroduce the potential attractivity

Ri

^{of zone anddescribe}

residential movements asatwo-stage process.Atthe firststage emigrants leave thezoneswherethey live;

atthe secondthey choosea newresidenceaccording tothezone’sattractivity, irrespective of the param- etersofthe zonetheyleft.Inorderto account forthe distances between the zones oforigin anddestina- tion,theattractivityof zone for the migrant from

zone./is

given by

Ri"

^exp(--,

do. ^),

^{where d0.is the}

distancebetweenzones andj.

The dynamicsofpopulation groupsis described inthe modelbymeansofthe logistic relationships.

Forinstance,thedynamics ofthe activepopulation ofzone is described as

dx/dt-

^eg

ngi (Ji

^g

xig), (5)

where g denotes the population group (white- orblue-collar), andc^gis a net rateofemployment of non-active residents _ni as white- or blue- collar employers, and

Ji

^{is an} overall potential employment.

The dynamics ofthespatial patternsgeneratedby the Brusseville model are studiedaccordingto scen- arios. In one scenario, for instance, thecanal that crossesthe cityisreplaced byalineofhills(Engelen,

1988).

As a result ofthis dramatic change of the city’sinfrastructure, the industry thatwas concen- trated along the canal spreads out along the peri- meterofthecity, wherecrowdingisless and landis cheaper. Followingthesechangeswhite-collar resi- dentsmovefrom the city’soutskirtstowardsthe new hillarea, wheretheyhavegoodaccess tothe CBD andarefar from thenuisancesmarredby industry.

VanWissenandRima

(1988)

constructedacom- prehensive regional model of Amsterdam popu- lation dynamics. They represent Amsterdam and surrounding area by means of 20 zones. In each zone, 11 dwelling types and24typesofhouseholds of four different sizes are distinguished. The intensity ofmigrationand theresidential choiceof each familyisdependent upon the ageofthe headof household (accordingto 5-year age categories) and onthenumber of family members

(seven

groups). In addition, immigration, emigration and birth and death processesareincluded. Theparameters of the modelwereestimatedbasedondetailedhousehold and migration data for each zone for the period 1971-1984.Thequality andresolutionof empirical dataweresufficient toprovide^averygood approx- imation of population and household dynamics.

For thirteen zones, the R² statistics ofcorrespon- dence betweenactual data and model resultswere higherthan0.9; for the remaining zones, excluding one, it was notless than 0.5. Based on this corre- spondence,twoscenariosofAmsterdam population and household dynamics for 1985-2000 were compared. The first reflects central government plans to build newdwellingsinAmsterdam, while the second reflects local government measures to decrease construction quotas in the expanding suburbs. The short period of prognosis implies rather similarpredictionsforboth scenarios. They divergeatthe level of 10 percentorless formostof the parameters, including total population and number of households as well as population and household numbers per^zone.

Batty

andLongley

(1994)

^{consider a}similar, but lessdetailedmodel ofresidential choice inGreater London. Intheir study the city is divided into 32 zones, each one described by the percentages of dwellings of four types, namely ofpurpose-built flats, convertedflats,rowhouses andsingle-family houses.Theprobabilities of occupyingthedwellings of each type are considered as functions of the distancebetween theCBD and thezoneandof the mean age of the dwellings in the zone.

As

in Brusseville model, the attractivity of the potential dwelling of a certain type does not depend on individual’s currentoccupation.

The overall percentage of correct predictions given by this less detailed London modelis lower than in an Amsterdam’s one and equals 0.432.

Spatially, predictionismuch better forzonesclose totheCBD and for theoutermostsuburbs than for theintermediate zones.

A

number of attempts (Anselin and Madden, 1990;

Putman,

1990; Bertugliaetal., 1994;Tadeiand Williams,

1994)

havebeen undertaken to combine themaincomponents of the cityintheframework of onemodel. Bertugliaetal.

(1994)

present themost general formulation ofthis approach the Inte- grated Urban Model

(IUM).

They declare the following componentsas sufficientforthedescrip- tion ofurban dynamics: the housing market, the job market, the service sector, the land market, and the transportation subsystem. The state of these components is described by the following groups of variables: the numbers of population groups, the housing stock, industry, and employ- ment according to branches. Spatially, the city is divided into N zones, where individuals can live and work. The IUM operates with the flows of population between the zones,whileconsidering the costsof the trips.

Thedimensionsofthemodel’s descriptionarethe main problem of the IUM approach. In order to describetheflow ofworkers,forinstance, the model accounts for the fraction ofworkplaces in zone j occupiedby workers living in zone atthecurrent iteration, but occupied at the next iteration by workers livingin zone k. Ifthe number of spatial

zonesequalsN,then thedimensionofthisdescrip- tionhasanorderofN

(!)

andit seemsimpossibleto obtainany reasonableestimatesof theparameters.

As

a result, the IUM is a representational rather than modelingtool.

Thus,ecological modelshave asimple structure, but dealwith selected factors only, while regional models are too complex and almost always lack

parameters’

estimates.

Are

there anyintermediate approaches?CAand

MA

modelspresent^apositive response. Botharebasedonaccounting for thelocal spatialstructureof the city.

2.2 Individual-basedMicro-Modeling

Theecological and regionalapproachesutilizetop- down approaches to the complex system studies.

Within their frameworks, ^an urban system is expandedinto apredeterminednumber of compo- nents, andthisexpansionis maintainedthroughout.

Micro-approach, in comparison, is based on a

bottom-up representation of the cityasthe collec- tive ofpotentially infinite numbers of elementary units whose interactions define dynamics of the urban systematlarge.Fromthe latter point of view, theurban modelmustinclude atleast twodifferent elementary units, namely land parcelsandmigrating individuals.

The dynamics ofthecityascoverage ofparcelsis studied by means of

CA,

while individual inter- actions are considered in the framework of

MA

models.

CellularAutomataas theBackground

for

Individual-basedModeling

ThebasicfeaturesofCAmodelsareasfollows:

A

city is represented by the two-dimensional latticeof cellsCij;

Timeadvancesin discretesteps;

EachcellCij^isfoundin some statesthatbelongs to a finite setS

{sl,

s2,...,

s/};

The cells change their states according to local transitionrules.Thatis,the stateofthecellitself

and thestates ofthe cell’s neighborsatthecurrent time stepdefine the state ofthe cellat thenext timestep.

Various versions ofCAmodels includevarying in space and time form and size ofthe neighbor- hood, deterministic and stochastic transition rules, dependence of transition rules on the loca- tion of the cell withinthe lattice, the currentstate of thecell, etc.

Tobler

(1979)

^wasthefirst torecognize andstate theadvantages ofCAapproach forthe describing urban dynamics. Phipps(1989,

1992)

and Couclelis

(1985)

implemented a general CA model for the description of urban dynamics. They recognized and studied thephenomenon ofthe emergence of regular spatial structures ofcells from an initially disordered lattice. This basicproperty ofCAwas intensivelystudiedduringthe1980sforone-dimen- sionalCAmodels. Urban interpretations,however, demand two-dimensional CA; their different versions were investigated in a number of papers (Phipps, 1989; 1992; Portugali etal., 1994;Durrett and Levin, 1994; Hegselmann,

1996).

The results werequitesimilar to those obtained in one-dimen- sional models. Namely, if wesuppose that the cell changes its state towards the modal state of the neighboring cells, then CA as a whole evolves toward the persistent "segregated" state. In this state theCAconsistsofsegregateddomainsofcells, thosewithinthe domainhavingsimilar states.

Recently,anumber of operationalCAmodelsat thecity and regional levelweredeveloped. Someof these modelswellfitreal-worldcities(Battyand Xie,

1994; Itami, 1994; Benati, 1997;

Wu,

1996; White etal.,

1997).

Asanexample letusconsiderthe CA simulation ofthe city ofCincinnati (White ^et al.,

1997).

The city ^is represented in the model as an 80 x 80 latticeofcells. Cellstatesrepresent landuses andareoftwoclasses:active statesthatcanchange, namely housing,commerceand industry; andfixed states, used to represent infrastructure, i.e. rivers, railways, and roads.Althoughcellsin fixed statesdo notchange, theycanaffect thetransitionsof other cellsfrom one state toanother. Cellneighborhood

consists ofthe 112 cells lying within the circle of radius sixcells.Suchunusually large neighborhood permits a more realistic modeling of local inter- actioneffects among landuses.Inordertomakethe model operational, thetransitionrules arestochas- ticdepending upon thecurrent stateofthecelland onthecelllocation withinthe citylattice.

The simulationsconducted byWhiteetal.

(1997)

were calibrated to fit Cincinnati land-use data, beginning from 1960. Ingeneral, themodel repro- ducesmorphology ofCincinnati, both visually and in terms of the statistical measures ofspatialsimi- larity. Based on this correspondence, a series of simulationexperimentswasheldinorderto clarify the role of transportation network changes on Cincinnatidynamics.

The generalized CA approachwasimplemented in several other studies(Battyand Xie,1994; Itami, 1994;Wu, 1996; Benati, 1997; Sandersetal., 1997).

Inall ofthem,theoutcomessignificantly agreewith the dynamics of realcities.

Inadditionto its use in directCAsimulations,the representationof acityas alatticeof cells provides abasis for the

fractal

^{approach to} the description of urban morphology. The main assumption of the fractal approach is that of self-similarity of the patternincase;inotherwords, zoomingof the citystreet network, housing, etc. pattern provides geometricalstructuressimilar tothatattheprevious levelof resolution. Fromthegeographicalpoint of view, the fractal approach generalizes the classic descriptionofsettlementhierarchiesconceptualized by VonThunen andChristaller(BattyandLongley,

1994).

Self-similargeometrical patterns canbe charac- terizedbya"fractal dimension",which is a rational number that nonetheless differsfrom the standard topological dimension of one, two, or three.

A

fractal dimension is used for the analysis and comparison ofdifferent urban structures, e.g.

regionalboundaries,trafficnetworks,orresidential and industrial areas (Batty and Longley, 1994;

Frankhauser, 1994; Schweitzer and Steinbrink,

1997).

The estimatesofthe fractaldimensions for urban housing patterns fluctuate between 1.3 and

1.9,with amodalvalueabout1.7.Thefractal dimen- sions of urbannetworks

(the

topologicaldimension of whichequals

one)

fluctuatesbetween1.1and 1.9, withamodalvalue closeto 1.4.

Ifa fractal dimension characterizes urban pat- terns,then themodelthatgeneratesthegeometrical pattern ofa given fractal dimension can serve as a tool for simulating a city’s geometry. The most popularmechanismforgenerating fractalstructures is Diffusion-Limited Aggregation

(DLA)

and its generalizations. This model is an extension of a randomwalkmodelon arectangularlattice.

Fractal models produce quite likelihood urban patterns at low spatial and temporal resolution.

Frankhauser

(1994)

andSchweitzerandSteinbrink

(1997)

have imitated in this waythe rank distribu- tion of settlements of Berlin metropolis in 1945 basedonthedataon1910.BattyandLongley

(1994)

havemade athorough investigationof the fractal- generatingmodels,both atthetheoreticalleveland in relation to thegrowthofreal-worldcities.They have demonstrated, forinstance, verygood corre- spondencebetween thefractaldimensionofactual Cardiffland-use pattern and the modelsimulation

1.772versus 1.75.

To conclude, the encouraging results of the simulations of several cities by means of CA and fractal models make them serious nominees for including among the standard tools used for modeling urban infrastructure at low and inter- mediatespatialandtemporalresolution.Regarding the humancomponent, CAand fractal models do not consider it at all. Straightforward scheme for simulating social processes in thecityis provided, instead, by the multi-agent approach, presented belowin detail.

Multi-agent Simulations

of

^Urban

Population Dynamics

CA models ignoretwo basicproperties ofacityas a populated system. First, the city’s physical structure develops according to the demands of itspopulation.Second,thecity’s inhabitants,unlike elementary units ofnon-living systems, are them-

selves complex systems, whose properties can change during the course of the lifetime. Inorder toaccount forthese basicproperties,

MA

approach operateswith a two-layer model. The firstlayer the city’s housing infrastructure represents the properties of urbanhousing; thesecondlayer free human agents representsindividual citizens and reflects theirmigratorymovements (Portugaliand Benenson,

1995).

IndividualfreeagentsintheMAmodel havethe ability to estimate the state of the city on its two layers and behaveinaccordance with information regarding threelevels ofurban organization:

The individual;

The local referring to the characteristics of neighborhood andstateof theneighbors;

The global referring to the state of the wholecity.

Based ^on this information, individual agents immigrate into the city, occupy and change resi- dential locations there, and leave the city when conditionsbecome unsatisfactory.

Freeagents ^are characterized in

MA

models by theireconomicstatusandculturalidentity. Utilizing aseriesofagent-basedmodels,wewereabletoshow the emergence of different forms of cultural and economic segregation.

Moreover,

by introducing explicitlytheability oftheagentstoevolvewehave demonstratedtheemergence ofnew socio-cultural groupsinthecity space (Portugali^etal.,1994; 1997;

Benenson and Portugali, 1995; Portugali and Benenson, 1994;

1997).

In the following, these results^arereviewedand expandedin two directions.

First, thetheoretical

MA

model isimplementedin areal-worldGISenvironment. Second,the general representation ofcultural identity by means ofa

"culturalphenotype"is introducedandstudied.

2.2.2.1 Agent-basedsimulations

of

^economic

interactionsbetween individuals

In the economic version of the

MA

model, the relationships between individuals and their neighbors and neighborhood are based on those

individuals’ economic status andthevalue oftheir houses. Twoimplementations of an economic

MA

model are considered below, the abstract one, where a lattice of cells serves as thehousing layer andtheGIS-based modelwith amapof the houses and streets is taken as the background. While the abstractversion of the modellies in the main- streamofCAand

MA

simulations, the purposeof the GIS-basedversion is toaccount for the hetero- geneity of the spatial structure of the real-world city and to implement the

MA

model as an operationaltool.

Sub-model

of

^{the housing}

infrastructure

CellularAutomata representation

of

^{housing The}

infrastructure of an abstract version of the

MA

model is a squarelattice ofcells, whichsymbolize houses. Each house H_0.can be either occupied by one individual agent ^or remain empty.

A

5x 5 square with H_0. in the center is considered as the neighborhoodU(Hgj) ^ofhouse

H0..

^{Houses differ in}

their value Vj. ^{For eachtimestep, thevalueof the} house is determined anew. When an agent

A

occupies house

H0-,

^{its value V}0. is updated in accordance with A’s economic status

SA (see below)

and the average value of the neighboring houses. When a free agent leaves house

H,.7,

^and

the latter remains unoccupied, ^its value decreases ataconstant rate.

The GIS map

of

^{houses andstreets as a back-}

ground Many high-resolution GIS-based maps have recently becomeavailable. These maps make itpossible to substitute the abstractcellular space background of the model by the housing struc- ture of the real-world city. They, thus, facilitate the study of the role of housing heterogeneity and varying neighborhood structure in urban dynamics. The GIS version ofthe economic

MA

model used by us is based on a digital street and housing mapof a section of the Tel-Avivmetropol- itan area at 1:500 scale. Each houseisrepresented by two variables, namely, the value of the house and its capacity

(number

of apartments). As above, the value ofthe house is updated at every time step in accordance with the mean status of

its residents and the value of the neighboring houses.Twoamong various possibledefinitions of the neighborhoods are considered. Scenario

A

(reminiscent of the

CA-type

city) takes into accountonlythe distancesbetweenhouses;scenar- ioBaccounts for theheterogeneityinducedbythe city’s street network as well. For scenario

A,

the neighborhood

U(H)

consists of allthe housesloc- ated at aspecified distance fromH. In scenarioB, it isassumed that thedecisions madeby agentsare influenced by the condition on their side of the streetonly. In orderto implement this suggestion, theneighborhoodsofhouses located alongseveral specific streets were restrained to all those houses situatedat the sameside.

Dynamics

of

^the economic status

offree

human agents

Thedynamicsof theeconomicstatus

SA

^ofagent

^A

occupyinghouseHisdescribed inasimple logistic way:

S

^+’

^(RA. S. ^{(1 S)}

^m.

V4 ^)/{ Vt)city (6)

where

RA

^{is an individualrate}

of

^{economicgrowth}

and does not depend on t, m.

V

^{is a}

"mortgage payment"

and

(Vt}city=kz{Vltlk, ^E[1,M]}/

(M

x

M)

^{is a mean}value of housesoverthecity.

The localeconomic

information Pt

available to individual agent

A,

occupying ahouseH, is given by the economic statusof

A’s

neighbors and their houses’ values in

U(H).

^{For CA} representation of the housing

P,

^{is a mean of the status of the}

neighbors occupying the houses within U(Hiy)and the values of the unoccupied neighboring houses.

For the GIS-basedversion

P’

is an average ^{of the} meanstatus ofagents locatedinthe houseHand in the neighboring houses.

Themigrationdecisionof agent

A

dependsonthe absolute value ofthe difference

SDA

^{between A’s}

status

S

^and

^pt,

^{namely, on}

SD-

Belowwe call

SDA

^{a local}economic tension of an agent

A

atlocationH.

The average

/Vt)city

of housing values over the city gives theglobaleconomic

information

^available

to each individual agent atiteration t. According to

(6),

^it feeds back the description ofthe

agent’s

statusdynamics.

Trade

off

between migration andchanges inan individual’sstatus

According to the model’s flowchart (Fig. 1), ^at every time step, each free agent

A

in the city decides whether to move from or to stay at its present ^location. It is suggested that for an agent

A,

located currently at H, the probability of leaving its current location increases monoto- nously with an increase in an individual’s eco- nomic tensionpt._Theprobabilityofoccupyingan empty houseGwhen it isthe only possible choice decreases monotonouslywithanincrease in indivi- dual’s estimation of the economic tension at G and to repeat, does not depend on the previous

location of that individual. For details of choice process involving several vacant houses, see Portugalietal.,

(1997).

The conjunction between individual, local, and global factors can lead an individual agent

A

"to decide" not to reside within the cityin spiteof the higheconomic tension at its current location. The reasonfor this decisionmight bealack of attractive vacant houses in the city. In this case either the economic tension continues to increase, while it canbe occasionally resolved by the agent leaving the city with probability Pu. Insolvent agents, whose economic status has dropped below zero, leave the cityeventually.

Immigration

At every ^time step, a constant number of indivi- duals try to enter the city from without and to occupy a house. The economic status

SI

^and

growth rate

RI

of each immigrant I are assigned

Y’Stay

in the same house

N

Leave the city

Y N

To leave the city?

Change cultural identity

$ Change status and house value

N

T

To occupy a new house?

Y ,

_{To leave}

a house?

BEGIN

Y

# ^relevant to the cultural model only

$ relevant to the economic model only

& relevant to residents and immigrants

.

^{relevant to residents only}

FIGURE Consequenceofagent’sdecision.

randomly and independently. The distribution of immigrants’ status isnormal,withameanequalto the instantaneous agents’meanstatusover the city at previous time step and constant CV. The dis- tributionof

R

^isalso normal andindependentoft.

Global consequences

of

individuals’

economic interactions

Cellular Automata representation

of

^{housing The}

city economic structure is described in the model bymeans ofthe distribution ofhousingvalues,the distribution of the agent’sstatus and the distribu- tion ofthe status growthrate. Irrespective^{of their} initial statethese distributions converge to smooth and correlating patterns during several hundred iterations (Fig.

2(a)).

After initial period ofrapid changes, they evolve very slowly, with rich/poor domains moving slowly and stochastically throughoutthe urbanarea(Fig.

2(b)).

GIS map background For scenario

A,

the gradient of the housing values and individual status is established after hundred iterations

(Figs.

3(a)

and

3(b)).

Subsequentlyit variesslowly in the manner demonstrated for the cellular ver- sion ofthe model. Scenario B was tested against one wide street in the city. Referring to Fig. 3(c),

we see that the resulting distribution ofthe hous- ing valuesisdiscontinuous,withabrupt disparities between different sides of "Red" Street. "Red"

Street prevents penetration ^of the agents of high

(low)

economic status into the quarters oneach of its respective sides, thus constraining the variabil- ity of thesteadydistributionof thehousingvalues alongthe entirelength of the street. Comparingto CA version, the rigid street and housing patterns restrictvariabilityof thepopulationdistribution at the slowstage^ofthe city dynamics.

2.2.2.2 Individualcultural identity and simulation

of

^cultural interactions between agents

Formalrepresentationofthe non-economic features ofanindividualshoulddifferfrom one-dimensional

FIGURE 2 Economic MA-model: population distribution according to economic status and status growth rate. (See Color PlateI.)

FIGURE3(a) Initial distribution ofhouse valuesisrandom, uniform on[0,1].(SeeColorPlateII.)

quantitative representation of economic character- isticsabove. This isespeciallytrueregardingindivi- dual’s culturalidentity,which representation should capturethemulti-dimensional, nominaland quali- tative character of the latter. Suggested below

"culturalcode" isanalogoustoageneticcode,which partially pre-programs ^an individual’s behavior

when creating groups^orsocieties. Inthe genetics of qualitativefeaturesaswell in artificial lifestudies,it iscommontorepresent^anindividual’sgenotype by ahigh-dimensional binaryvector of traits(Banzhaf, 1994; Kanenko, 1995). The cultural code is also binary and, in contrast to genetic restrictions, changinginresponsetotheagent’sinteractions with

FIGURE 3(b) Distribution of house valuesat 7’: 200forscenarioA. (SeeColor PlateIII.)

the neighbors, theneighborhood, and thecityas a whole. Inconsequence, individual’s residential be- havioralsochanges.

Formal representation

of

^the cultural code and the dynamics

of

^{an agent’s cultural}identity The cultural identity of an agent

A

is described by the K-dimensional Boolean cultural code

CA (CA,

^l,CA,2, CA,3,

CA,K)

^where CA,/ {0, }, k 1,2, 3,..., K. Consequently, individuals of 2^K differentculturalidentitiesmightexist in the same city.The difference p,between agentsAand B, ac- cordingto theircultural identities is measuredby

p(CA, CB)-- Z(CA,k

^XOR

^{cu,)/.K. (7)}

k

FIGURE 3(c) Distributionof house values at T=200 for scenario B.Neighborhoods are definedas inA, but constrainedby the "Red"Street.(SeeColor PlateIV.)

Ina mannersimilar tothe economic version of the model, an agent behavesin line with the available individual, local, and global cultural information.

The representation oflocal cultural

information

^is

related to the notion of local spatial cognitive dissonance, introducedby Portugali and Benenson

(1995)

and Haken and Portugali

(1995).

Applying

their general definition to the multi-dimensional presentation of cultural identity, the local spatial cognitive dissonance

CDA

^{Of agentAis definedas}

anaverage ofthe differences between A’scultural identity and the cultural identities of itsneighbors, givenby

(7).

^Asinthe economicmodel,the higher the local dissonance of

A,

thehighertheprobability

that

A

willleaveitscurrent location and tryto oc- cupy the alternateone. Theprobability of locating at anempty houseGdecreaseswith the increase in the estimate of potential spatial cognitive ^disso- nanceatG.

The influence of the global structure of the city on anindividual’s residential behavior increases in the model with a rise in the level of residential segregation. The global cultural

information GDA

available to afreeagent

A

is,thus,determined inthe modelbythe value ofLieberson’s

(1981)

segregation index

LSx,

expressedas aprobabilityof a memberof groupXlocatedathouseHtomeet amemberof its own group within

U(H).

Visually, values of

LSx

below 0.3 correspondto a random distributionof theagents belongingto groupX,whilevaluesabove 0.8correspondto one orseveral domainsoccupied almost exclusively by these individuals. Formally, for agentsofidentityCA,

GD ^{max{0, (LS LS*)}/(1 LS*), (8)}

where

LS*

is the threshold value of LS that correspondstothe visuallysegregated pattern.

Localandglobalinformation influence an

agent’s

cultural identity in alternative ways. High local cognitivedissonance

CD

forces agent

A

tochange its cultural identity. In contrast, a high level of segregation

GD

^{ofagents having anidentity}

^CA

forces

A

to preserveits currentidentity.

An

agent A’ssensitivitytolocal cognitivedissonance

LA

^and

to global segregation

GA

(LA, GAE[0,1]) ^are propertiesinherent to

A

and independent oft.

Ifan agent

A

is forcedto occupyits currentlo- cation inspite of high cognitive dissonance, thenits cultural identitycanbechanged.This occurs in the model whenthe localtendencyof an agenttovary exceedsthe globaltendencyto preserveits current identity, that iswhen

LA. CD ^{> GA.} GD.

^{If the}

latter is true, then the probability that the ith component of

CA

^{will be changedis} proportional to the absolutevalue ofthe differencebetween the fractionof theithcomponent amongA’sneighbors and its value

(zero

^or

one)

for A. Additionally,

"mutation" of the cultural code is possible, with

probability r per component, although only one component of the cultural codecanbechanged per iteration.

Immigration

As

theeconomic versionof an

MA

model,^atevery time step a constant number of individuals try to enter the city and occupy a house there. The cultural identity of the immigrants is assigned at random, in proportion to the current fractions of agents having eachof the 2^Kpossibleidentities.

Trade

off

between migration and change

of

an individual’s cultural identity

An

inherent source ofthe cultural dynamics in the modelisthe mutationprocess that preventsitfrom becoming culturally homogeneous.

An

individual agent locatedin aheterogeneous neighborhood of non-zero dissonance, either succeeds in changing residence or fails. If it fails, the agent either changes identity towards the "modal" identity of its neighbors (Fig.

1)

or preserves its current identity due to the high level of segregation of agents of similar identity in the city. Unlike the changesintheone-dimensional economicstatus, a change in cultural identity does not necessarily decrease the cultural diversity of the city. To illustrate that, ^consider the agents located at a boundary between two segregated groups ofindi- viduals of

(0,

0,0,...,

0)

and

(1,

1, 1,...,1) ^identi- ties. It is highly probable that the identity ofan agent

(0,

0,0,...,

0)

^willchangeto one havingunit at one of the components, say, to

(1,0,

0,...,0), and thuswill differfromthe identitiesoftheagents ofboth groups.

Globalconsequences

of

individuals’cultural interactions

The aimof the culturalversion ofthe

MA

modelis to examine the process of socio-cultural segrega- tion and emergence in city, the inhabitants of which can varyin theircultural identity according tohigh numberof traits. Toqualifyasa new socio- cultural entity, agroup of individuals sharing the same cultural identity must fulfill simultaneously

twoconditions(Portugali^elal., 1997):

At the local level, most of the group members should be located within culturally uniform neighborhood;

Attheglobal level,thenumber of group members and their spatial segregation ^have to be suffi- cientlyhigh.

Our previous study demonstrates three types of persistent residentialdynamicsinthe citypopulated bytheagents,whose culturalidentityisrepresented by quantitativecharacteristic, continuouslyvarying on[0, 1] (Benensonand Portugali, 1995). ^Onetype

canbe termed a "random" city, anothera "homo-

geneous"

city, in which almost all of the agents belongtothe cultural group of either 0-or1-agents, andthe third typeis characterizedbythree coexist- ing segregated groups, whose members have close to 0-, 1- and 0.5-identities. Forthe latter case, the 0.5-group is emerging and self-organizing during the city and agents’ coevolution (Benenson ^and

Portugali, 1995; Portugali and Benenson, 1997;

Portugali et al., 1997). In the following section, the phenomenon of multiple and recurrent socio- cultural emergence is investigated based on newly introduced multi-dimensional andqualitative ^indi- vidual’s cultural code. The three-type typology ^of urban residential dynamics ^can be applied with a multi-dimensional cultural identity ^as well, and the set of parameters entailing the coexisting segregated groupsis used below to investigate the model. The threequestions respondedtoare: What are the number and level of segregation of each emerging(if

ever)

cultural entity?Arethesegroups fixed or do they vanish with time? What is their life history’? The system behavior is investigated below for the case of up to a five-dimensional cultural code.

Presentation oj the urban patterns

A city’s cultural pattern is presented by means of three kinds of maps(Fig. 4). ^{The first}represents^a

FIGURE4 Cultural MA-model: persistent city patterns.(SeeColorPlateV.)

distribution of the

agents’

cultural identity, with each identity markedbyits owncolor. Thismapis the most detailed of the three, but its method of presentationinconvenient forK

>

2 in view ofthe high numberand nonlinear ordering ofidentities.

The second type ofmap is that ofthe difference

D(CA, Co)

between the identity

CA

^{Of agent}

^A,

occupying house H, and some identity chosen apriori, say,

Co

(0, 0, 0,...,

0).

^This mapshows those effects that do not depend on K; its dis- advantage lies inthe fact that for several different identities,

CA

^{can differ equally from} the identity selected for comparison.The thirdmapisthatofa distributionofthecultural cognitivedissonance of theresidents(Portugalietal.,

1995)

and represents thedomainsofthemost intensivechangeseither in populationdistribution or in the cultural identity of the model agents. The fraction of the agents that want toleavethecity, which is defined by the overall mean value of the cultural dissonance, is used belowas anindicator of itsoverall instability.

The presented results donot depend ontheinitial distributionofthe agentsin thecity.

Model dynamics

.for

low-dimensional cultural codes

(K= 2)

The case K=2corresponds to our previous analysis of residential segregation be- tweenasmall numberofcultural groups (Portugali etal., 1994).Thecity dynamicsinthat case entails a rapid self-organization of two to four cultural identities within a few segregated patches.

Here,

theboundaries between thehomogeneous patches remain areasofinstability,withintensiveexchange of individuals (Fig.

4(a),

compare to Portugali etal.,

1994).

Letusskipthe intermediate cases ofK 3,4 and proceedtoK 5.

Model dynamics

.for

^a high-dimensional cultural code

(K= 5)

The number of possible identities for K-5 equals

25-

32. In a way similar to the case ofK= 2, the ^boundaries between the homo- geneousdomains and the heterogeneous domains, occupied by the agents of varying identities, remain areas of instability. The agents located there either try to leave their houses or change their cultural code. None of the properties ofthe specific cultural identities can be predicted in the

long run, butit is still possibleto understand and predict the following properties ofthe population distribution inthe model city:

(1)

The persistent citystructure ischaracterizedby a mixture of spatially homogeneous and heterogeneous domains. The former, whose populationsformcultural entities,coversabout halfof thecity’sareaforK 5(Fig.

4(b)).

The distribution of cultural differences p(CA,Co) between the agents withcultural code

CA

^and

the "basic"cultural identity

Co ^(0,

^{0, 0, 0,}

⁰⁾

^is

self-organizingaswell (Fig.

4(b)).

(2)

Alimitednumber of culturalentities can exist in the city simultaneously (Figs. 4(b),

5(a)).

(3)

The life-spanof a socio-culturalentityisfinite;

newly emergingentitiesreplace eachother inthe city space. About 20 percent of the entities persistinthe cityfor notlessthan 10 iterations and about 10percentexistfornotlessthan 25 iterations(Fig.

5(b)).

The globalmodel dynamicscanbe explainedon the basis ofthedistributionofcultural differences, presented in Fig.

4(b).

This distribution has two contradictory characteristics. First, the difference p(CA,

Co)

increases with theincrease in the distance ofagentA from thelocation ofthe agents having culturalcode

Co.

^Second,themulti-dimensionality and, hence, non-linear ordering of the identities implies therecurrentemergence of adjacententities

CA

^andCB,that differequally and significantly from

Co

and between themselves (p(CA, Co)p(CB,

Co)

p(CA,

CB)).

^It can be observed, for instance, at the bottom of Fig. 4(b), where the boundary between the identities, which are represented in violetandyellow(firstmapfromtheleft),isanarea ofhighdissonancebetween them (right map),while both of them differ from

Co

(middle map). ^This phenomenon impliesnon-monotonousdependence of the index of the city’s instability,represented by the mean fraction oftheagents thatwant toleave thecity,onthenumberofcoexistingentities(Fig.

6).

According to the Fig. 6, with an increase in the number of entities, the city’s instability first decreases.With a further increase in thenumberof entities, the emerging cultural entities occupy the

20

15

500 1000 1500 2000 2500

Iteration no

FIGURE5(a) Dynamicsof the number of cultural entities forK--5.

0.200 0.180 0.160 0.140 0.120 0.100 0.080 0.060

’

^0.0400.020

0.000

21 31 41 51 61 71

No of

^iterations

^(N)

81 91 101

FIGURE5(b) Survival functionfor culturalentity.

0.12

0.10

0.08

0.06

0.02

0.00

Instabilitydecreases

11 13 15

Number

of

^entities

FIGURE6 City’sinstabilityvsnumber of entities.

Instabilitydoes not decrease

17 19 21

bulkofthe territory,whiletheheterogeneousareas vanish. In such circumstances contiguity between contrastingentities is inevitable.The self-organizing boundariesbetween them thensharpenand expand and the city’s instability increases once again (Fig.

6).

In consequence, we can say that urban instabilityis limited

from

^{below,and thatthemodel}

cityitself is self-organizing and evolving towardsa criticalstructurethat preservesits internalcapacity tochanges.

To conclude, the micro-interactions at the

agents’

and houses’ levels entail recurrent self- organization ofpopulation groups at the macro- level, according to both economic and cultural characteristics. The resulting persistent residential distributiondependsonthe natureof the character- istic we are interested. Continuously varying eco- nomic characteristics induces smooth residential distributions, which is characterized by low level of instability and consequently, by slow tempo- ral changes. Qualitative and multi-dimensional

"cultural code" entails self-organization of the criticalpopulation distribution,which ischaracter-

izedby the preservationof asignificantly high level of instabilityand,therefore, by therecurrent emer- gence and disappearance of cultural groups. Con- cerning the applied aspect, the GIS-basedeconomic versionof the modelcanbeconsidered as a firststep towards the operational implementation of a

MA

simulationapproach.

BACKTO

INTEGRATION,

BASED ON THE HIERARCHY OF MODELSOF CITY COMPONENTS

Letuscomparedifferentmodel approaches accord- ing to their spatial and temporal resolutions. The spatial unit of the ecological model is the entire city. For the regional approach, a model unit represents a relatively large urban region. The spatialunitoftheCAmodelisofintermediate size anddeterminedby thedegreeof homogeneity that canbeachievedwhen partitioningacity’s territory according to a given classification of land usage.

The spatialunit of the

MA

model is the smallest,

defined by dimensions of the separate house or compactgroupof houses.

The problem ofwhich time scale fits the given approach is more complicated. In general, the model’s time scale is defined by the temporal resolutionoftheavailable parameter estimates. If, forinstance, parametersareestimatedfor monthly intervals then, obviously, the model solutions de- scribethemonthly dynamicsof the system. Conse- quently, the issueof time scaleisaproblem ofthe selection ofan appropriate temporal unit for the estimationof theparameters andtheinterpretation of the results. Regarding ecological models, selec- tionof the time unit israther arbitrary becausewe assume we can obtain a complete or almost complete descriptionof the model behavior for all possible sets ofparameters. The large number of non-uniformly interacting components of the regio- nal model make it impossible to obtain general results regardingthe solutions’ behavior.

Moreover,

the very concept ofregional modeling eventually combines processes that occur at different time scales. Forexample, yearlyto decade varyingland uses, monthlytoyearly varying inter-city migration flows, monthly varying employment market or weekly varying transportation flows are all con- sidered inthecommonframeworkof anIntegrated UrbanModel. Spatially, the statistical or adminis- trative divisions we are usually forced to utilize, provideaframeworkforaveraging ratherthan the carriers of the parameters employed in regional models. Asaresult, quantitativerelations between the parameters, partially estimated at different time resolutions or partially substituted by some

"likelihood" values, determine the behaviorof the regional model. This canmake sense in the short term, but becomespointlessifwepretend modeling asystemundergoing qualitativemodificationsand bifurcations. To avoid this discrepancy, we can constrainthe model to the processes that occurat the same time scale only. According to the syner- getics paradigm, these processes, if hypothesized properly,are not many; wecan,therefore, hope that the dimensions of the constrained model will be sufficientlylow to enableestimation of theparam-

eters and the classification of the solution. The proper^selection of the processesin regards to the time scale depends greatly onthe

system’s

specifi- city, and no other confirmation of certain prefer- ences can be obtained by any means other than theoperational"Doit!"Fromthispointofview,the results of the CA and

MA

model approaches are rather encouraging. The former utilizes land-use dynamics as a process defined by substitutions between a limited number of the parcels’ different states,which occurannuallyorperhaps rarely. The latterdealswiththe residential decisions thatpeople pose to them more frequently. Both of these approaches demand a reasonably low number of parameters and generate self-organizingbifurcative dynamics the CA model regarding the urban infrastructureand theMAmodel regardingurban residential distribution. In parallel, these approaches work with different but "adjacent"

space resolutions. Ifwe interlace the two, we can argue that the "fast" residential dynamics of the

MA

modelshouldaccountfor the outcomesof the

"slower" CA model of land-use dynamics by functioning ^as the slow control parameters that define thelocation of residential areas, number of available dwellings, etc.

An MA

model feeds back informationregarding qualitative changesin social and cultural structure of city population to the infrastructure level, since this information can influence the regulations governing transitions between land uses. It is an open question as to whether other levels of the urban hierarchy in addition tothosedefinedby land-use andindividual residential behavior should be considered sepa- rately. Trends inemployment, birthand mortality, with their characteristic time scales should be examined in this respect. In general, the develop- mentof a hierarchical structure ofmodels,eachone dealingwith urban processes at specific temporal and spatial scales, is a necessary step towards understanding and modeling urban dynamics.

Separate models, each one describing processes at similar spatial and temporal scales, should be combined in a way that the variables of the slower and spatially less-detailed models serve as