Sociodynamics Applied to the Evolution of Urban and Regional Structures

The article consists of two parts. In the first section the concepts of sociodynamics are briefly explained. Sociodynamics is a general modelling strategy for the quantitative description of dynamic processes in the human society. The central concepts of sociodynamics include transition rates depending on dynamic utilities and the master equation for the probability distribution over macrovariables. From it the set of nonlinear coupled differential equations for the quasi-meanvalues of the macrovariables can be derived.


THE MODELLING CONCEPTS OF SOCIODYNAMICS
In the last decades several interdisciplinary efforts have led to a better understanding of the interre- 85 lations between different sciences.One of these fruitful interdisciplinary branches of science is the field of synergetics which developed out of physics and treats in high generality the spatial, temporal and functional macrostructures arising in multi-component systems [1].The concepts of synergetics have been applied so far to physical, chemical and biological systems.
Since the human society is also a complex multi- component system, it was a challenge to apply the principles and the mathematical algorithms of synergetics to this object of the social sciences, too.However, there arises a fundamental diffi- culty: in contrast to comparably complex systems of physics and chemistry there do not exist fundamental dynamic equations of motion for the ele- ments of the human society, namely the individuals.
Therefore some of the mathematical concepts of synergetics could not become directly operational in the social sciences.
In spite of that dilemma we shall now see that it is possible to develop a conceptual scheme for the mathematical treatment of collective dynamic processes in the society in terms of evolutionary equations.We denote this modelling strategy which allows the application of synergetic con- cepts to the description and explanation of the macrostructure and -dynamics of the society as sociodynamics [2,3].This field therefore provides a further step into interdisciplinarity because it allows for a transfer of methods and concepts from statistical physics and synergetics to the social sciences.
So far the modelling procedures of socio- dynamics have been applied to the migration of populations [2,4], to political opinion formation [5], to dynamic phenomena in the economy [6,7] and to the evolution of regional settlement struc- tures [8,9].In this paper we present a socio- dynamic modelling approach to the evolution of urban structures (see Section 3).
We now give a brief general description of the steps of the sociodynamic modelling procedure.
Step 1 The Choice of a Configuration Space of Macrovariables Let us now consider an approximately separable sector of the society.In order to describe its state on a macrolevel we have to introduce an appro-priate set of macrovariables which expand the configuration space.The socioeonfiguration is a set of macrovari- ables most directly linked to the members of the society.It describes the distribution of attitudes and actions among the members of the subpopulations of the society with respect to the possible alternatives of a given "choice set".If n/ is the number of members of a subpopulation 7 with attitude or action i, then the socioconfiguration is the multiple of integer variables {...n/... n;... ..n...}-n.
Beyond the socioconfiguration there exists a set of material variables m depending on the sec- tor under consideration.Amounts of produced and consumed goods, prices, etc. are examples for material variables of the economic sector.In Section 3 we shall introduce variables to charac- terize the urban evolution state and they will pro- vide another example for material variables.In contrast to the socioconfiguration, the material variables are only indirectly linked to the acting and decision making members of the society.
Each set of variables (n,m) is now represented by a point in the multi-dimensional configuration space.
Step 2 Measures for the "Utility" of

Configurations
The state of the social system, which can be macroscopically characterized by the variables (n, m) is now estimated and valuated by the mem- bers of this system.
Let u/(n,m) be a measure of the (subjective) "utility" or "desirability" attributed to the state (n,m) and valuated by a member of subpopula- tion 7 who momentarily adopts the attitude i.
(This involves the simplifying assumption that this individual utility only depends on the macro- statevariables (n,m) and the status (a, i) of the individual.)Increasingly high positive (or low ne- gative) values of u/(n,m) denote an increasingly positive (negative) estimation of the situation (n, m) by the individual (a, i).
It should be mentioned, that the "utility" uff(n, m) not only includes the valuation of "material values" but also (via n) the (positive or negative) estimation of "abstract attitudes and actions" of the other members of the social system).
The choice of u/(n,m) represents the core of model building because it reflects the social psy- chology of the members of the society.
Step 3 Elementary Dynamic Processes: Transition Rates Up to now we have only considered a momen- tary picture of the social sector: We have intro- duced its macrostate (n,m) and its valuation by subjective utilities uff(n,m) of representative indi- viduals in status (a, i).Now we introduce elementary changes of the variables.If, e.g., a member of 79 changes his (her) attitude from to j, this decision induces a change of the socioconfiguration from {... nff... nj...n;...nil...} to (...(n/-l),...(n? +1)' n/...}.Similar step by step changes n k (assumed to be small but discrete) may vary the values of any of the material variables.Since the changes are in general effected by direct or indirect decisions of individuals, who act in a probabilistic manner, we describe the elementary transitions between neighbouring macrovariables from {n,m} to {n',m'} by probabilistic transition rates p{n', m', n, m}.
The crucial question now arises, which func- tional form the transition rates should have in terms of the variables (n,m) of the initial situa- tion and {n',m'} of the final situation.
We assume, that the driving force behind a transition from {n,m} to {n',m'} is the difference between the utility of.the final state {n',m'} and initial state (n,m) as seen by the actors effecting this transition.If that difference is positive, a transition is favoured, and if it is negative, a tran- sition is disfavoured.
In the simplest case, the most plausible form of p{n',m'; n,m} fulfilling all conditions is therefore p{n',m';n,m} vexp{u(n',m') u(n,m)}, where u(n, m) is chosen as a weighted mean of the utilities seen by the actors who give rise to the transition {n, m} We see that by step (3) the utilities have now obtained a dynamical meaning.They are responsi- ble for the magnitude of transition rates and thus for the dynamics of the system.They are there- fore also denoted as "dynamic utilities" or "motiva- tion potentials" to show that they are introduced into a dynamic framework which is different from the stationary (equilibrium) framework of neo- classical economics.
Step 4 Evolution Equations for the

Macrovariables
The preceding steps are now sufficient to set up a fully probabilistie description of the macrody- namies of the system.
The fundamental equation for this is the mas- ter equation for the temporal evolution of the probabilistic distribution P(n, m; t) over the variables (n,m) of the social sector under consideration.The transition rates of step (3) directly re-appear in the master equation.
This equation still comprises all effects due to stochastic fluctuations of the system.Simulta- neously it solves in principle the problem of the feedback loop between microdecisions and macro- state of a social system, since on the one side the individual decisions enter the master equation (via the utility dependent transition rates), where- as on the other hand the macrostate of the system evolves according to the master equation.This evolving macrostate in turn influences the utilities and therefore the decisions of the individuals.
It is easy to go over from the master equation to equations for the quasi-meanvalues of the macrovariables.These equations are typically a set of nonlinear autonomous coupled differential equations to which thereupon the theory of dynamical systems can be applied.
It is evident from the procedure via steps ables (those with a trend to grow) are predestined (1)-(4) that we have not made use of any to become "order parameters" dominating the "microscopic equations of motion for the indivi- dynamics of the whole system on the macroscale.dual components of the system".Nevertheless we The reason ,for this remarkable system beha- have arrived via transition rates and the master viour is that the fast variables quickly adapt equation formalism at stochastic or deterministic their values to the momentary state of the slow equations of motion for the macrovariables (n,m) variables.Since they thereupon depend on the of the social system, slow variables, the fast variables can be elimi- nated.As a consequence the slow variables alone obey a quasi-autonomous dynamics.Since 2 THE SPACE-TIME WINDOWS OF all other variables depend by adaptation on the PERCEPTION OF SETTLEMENT few slow variables which rise up to macroscopic STRUCTURES size, the latter are denoted as order parameters and determine the macrodynamics of the system.
We now apply the modelling concepts of socio- Let us now somewhat modify and generalize dynamics to the construction of models for the the slaving principle in view of its meaning for evolution of human settlements, which belong urban and regional structures and their dynamics.to the most complex space-time structures in  In settlements one can easily identify fast and the world.In settlements there exist many differ- slow processes of change and evolution: ent intertwined and interdependent organisa- The fast processes take place on the local mi- tional structures; the evolution of these structures crolevel of building sites where e.g.individual takes place on different scales.Therefore the buildings are erected or teared down, and where natural question arises whether this manifold of the local traffic infrastructure of streets, subways, structures and processes can be ordered accord- etc. is constructed.ing to some principles, for instance the space- The slow processes take place on the regional time scale or level on which they appear, macrolevel.They include the slow evolution of If this should prove true and if the relation whole settlements like villages, towns and cities between the levels could be formulated this which can be considered as population agglomwould provide the justification for considering erations of different size, density and composi- different windows of perception and for con- tion, furthermore the slow development of whole structing different, nonetheless interrelated, mod- industries.
els for each window.
The relation betwen the fast development of In the following we shall see that indeed a local microstructures and the slow development separation of levels is indicated and that it is of global regional macrostructures is rather sim- appropriate to comprehend and to connect the ple and exhibits a strong similarity to the slaving development of settlement structures on different principle: scales with separate models.Only in a final stage On the one side the fast development of local the models can and should be fused into one microstructures is driven and guided by the integrated model, quasi-constant regional macrostructure into In synergetics there exists the fruitful "slaving which it is embedded.That means the global principle" set up in high generality by Haken  [1].
regional situation serves as the environment and Verbally it can be formulated as follows: if in a the boundary condition under which each local system of nonlinear equations of motion for many urban microstructure evolves.variables these variables can be separated into On the other hand, the (slowly developing) re- slow ones and fast ones, a few of the slow vari- gional macrostructure is of course nothing but the global resultant of the many local structures of which an urban settlement is composed.However, similar to the longevity of the body of an animal, whose organs are regenerating on a shorter time scale than the life time of the whole body, the time of persistence of a regional macrostructure as a whole is much higher than the decay and regen- eration times of its local substructures.
Although this relation between urban micro- structures and regional macrostructures is rather evident it has an important consequence for model builders: one can separate to some extent the microdynamic level from the macrodynamic level and make separate adequate models for each space-time window of perception.This means in more detail: In constructing a model for the urban micro- evolution it is allowed to consider some global regional parameters (e.g.referring to the global regional population or the global regional stage of industrialisation) as given environmental condi- tions and to describe the fast local microdynamics as developing under these global conditions.
On the other hand, in constructing a model for the regional rnacroevolution it is allowed to presume that a corresponding fast microevolution takes place which adapts the local microstructures to the respective slow variables of the global de- velopment.
In view of this possibility of a separate consid- eration of the micro-and macroperspective of settlement evolution we shall present in the next sections the design principles of a micromodel for the urban and of a macromodel for the regio- nal evolution.

THE DESIGN PRINCIPLES OF A MICROMODEL OF URBAN EVOLUTION ON THE MICROLEVEL
In constructing a model for the urban evolution on the rather detailed level of individual building plots or sites we follow the general modelling strategy described in Section 1.The modelling scheme consists of the following steps: 1.A configuration space of variables characteri- zing the state of the urban system has to be set up.
2. A measure for the utility of each configuration under given environmental and populational conditions must be found. 3. Transition rates between neighbouring configurations constitute the elements of the system dynamics.The "driving forces" behind these transitions are utility differences between the initial and the final configuration.Therefore the transition rates depend in an appropriate way on these utility differences. 4. Making use of the transition rates, evolution equations for the configurations can be de- rived on the stochastic and the quasi-deterministic level as well. 5. Selected scenario simulations demonstrate the evolution of characteristic urban structures.
Step 1 The Configuration Space The city landscape is considered to be tesselated into a square lattice of plots or sites i(il,i.),j(jl, j2), where(il, i2), (jl,j2) are integer lattice coordinates.One can introduce a distance be- tween sites, for instance by the Manhattan metric d(i, j) ]il -Jl / li -j21. (3.1) The sites can either be empty or filled with differ- ent kinds of buildings, e.g.xi lodgings, Yi fac- tories and perhaps other kinds of urban uses (service stations, store houses, parks, etc.).For simplicity we consider only lodgings and fac- tories.The variables xi, yi 0, 1,2, are integers denoting the number of (appropriately tailored) building units of the corresponding kind on site i.The city configuration {x, y} {... (xi, Yi), (Xj, yj), .}(3.2) characterizes the state of the city with respect to the kind, number and distribution of its buildings over the sites.It is the purpose of the model to give a formal mathematical description of the dynamics of the city configuration.
Step 2 The Utility of Configurations The utility of a given city configuration has now to be determined.The ansatz for u(x,y) com- prises several terms designed to describe the main effects influencing this utility.
The terms contain open coefficients to be calibrated according to the concrete case.a The local term consists of the contributions of local utilities of erecting buildings on each site j" ui(x, y) u/(xj, y/) (3.3) J with u/(xj, yj)-p)x)In(x/4-1) 4-p)ln(y/4-1) 4-p)Z) ln(zj). ( The coefficients p)k) > 0 are measures of the pre- ferences to build on site j.The first two terms of (3.4) represent the increasing urban utility of site j with growing numbers x/, y/, a usefulness which however saturates if x/, yj grows to high numbers.
The third term describes the capacity constraint of site j.If z/is the empty disposable space on site j and if one unit of lodging or factory needs one unit of the disposable space, respectively, the capacity C/of site j is given by q + + (3.5) If the capacity of site j tends to be exhausted for z/= C-x/-y/ 0 the third term of the utility u/(x/,y/) approaches -.On the other hand, the first terms of u/(x/,y) are zero for x/--0 and yj 0, respectively, and approach -c for xj -1 or y/-1.As we shall see this has the consequence that states with negative values of xj or y/or with values for which xj + y/> C. can never be reached.That means, x and yj are con- fined to values xj >_ O, yj > 0 and (xj + yj) < Cj.
The value of the capacity Cj on each site j depends on how much this site is opened up for buildings.If the total urban population nc in- creases, more sites at the border of the city will be opened.In this way the size of the city area depends on its total population.We choose a Gaussian capacity distribution dZ(j, j0)] Cj Co exp 2crZ(nc) (3.6)where d(j, jo) is the Manhattan distance of site j from the central site jo and o-2(nc) is the popula- tion dependent variance: The factor Co has to be calibrated appropriately so that in the equili- brium state the population nc finds adequate to- tal numbers -]jx/and ]jYi of lodgings and factories, respectively, in the city.The interaction term describes the supportive or suppressive utility influence between buildings on different sites and j.This term is assumed to have the form u(x, y)a ij X Xj -} Z xy ai/ xiy/+ yy xx ai YiY/.i,j i,j i,j (3.7/ The signs of the coefficients decide about the interaction effect.If one chooses for instance xy ag << 0 this means that it is strongly disfavoured and not considered useful to build lodgings and factories on the same site.If, on the other hand, xy ai/ is chosen as a positive parameter for d(i, j) > do this means that lodgings on site lead to a high utility of factories on sites j at a dis- tance d(i, j) > do, and vice versa.This is a plau- sible choice since workers living in the lodgings need working places in a not too distant neigh- bourhood with d(i, j)> do.On the other hand, the choice of positive coefficients a x and a for d(i, j)< do means that it is considered useful to have further lodgings in the near neighbourhood (d(i, j) < do) of lodgings, and further factories in the near neighbourhood of factories.In this man- ner the interaction term represents the dependance of the utility of a city configuration on the location of different kinds of buildings relative to each other.
The total utility of a city configuration (x, y) is now assumed to be the sum of the two terms (3.3) and (3.7): u(x, y) UL (X, y) + uI (x, y). (3.8) Here, the simplifying tacit assumption has been made in constructing (3.8), that one objective uti- lity of a city configuration exists for all those citizens who make decisions about the development of the city.j(x) (x, y) u x).exp{ AJ x) u(x, y)} ] '(Y) (x, y) ulY) exp{AJY)u(x, y)} tearing down rates for lodgings and factories at site j with: A(X)u(x, y) u(x j+, y) u(x, y) j+/-A(Y) (X, y) U(X, yJ+) U(X, y) j+ U (3.12) Here we have taken into account that there will exist different global frequencies u!x),u!y) for building up processes and ulx),u y) ''fortearing down processes.
Step 3 The Transition Rates between Configurations The transition rates for a transition between the configuration x,y and the neighbouring config- urations {xj+,y} {..., (xj + 1; yj),...}, {x, yJ+} {..., (xj; yj + 1),...}, (3.9) must now be set up.Firstly they must be positive definite quantities.Secondly they should depend monotonously on the utility difference between the final and initial configuration, because these utility differences are the "driving forces" behind the activities effecting the transition.The simplest and mathematically most appealing form for the transition rates fulfilling these conditions is the following: j(x) (x, y) u x).exp{A u(x, y) } ] () (x, y) u y) exp{AJ+ y) u(x, y)} building up rates for lodgings and factories at site j (3.10) Step 4 Evolution Equations for Configurations The transition rates which depend on utility dif- ferences between neighboring configurations are the starting point for setting up evolution equa- tions for the configurations.Exactly speaking, the rates are probability transition rates per unit of time.The exact equation corresponding to these quantities is the master equation for the probability P(x,y;t) to find the configuration (x, y) at time t.It reads: dP(x, y; t) dt (x) t) ,()(x, y)P(x, y; t)] tJT (xJ-'Y)P(xJ-Y; "JT J --'r (x)(xj+ t) ,(x)(x, y)P(x, y; t)] + 2..,too)+ y)e(x j+ y; j+ J r (Y)(x, yJ-)e(x, yJ-; t) COjT tcojT -(Y)(x, y)e(x, y; t)] J + Z (Y)(x, y J+ ;t) ,j [coj+ )e(x, yJ+ '(y) (x, y)P(x, y; t)].J (3.13) From the master equation there can easily be derived exact equations of motion for the mean- values j(t), )j(t) of the components x, y of the city configuration, which are defined by j(t) Z xjP(x, y; t), {x,y} yj(t) Z yjP(x, y; t).{x,y} (3.14) where the bars on the right-hand side mean taking meanvalues with the probability distribu- tion P(x,y; t).The quasi-meanvalues cj(t),fj(t) .}()('(t), (t)) ()((t) (t)).
dt For unimodal probability distributions P(x, y; t) the quasi-meanvalues approximate the true mean- values; however, in the case of multimodal prob- ability distributions the quasi-meanvalues no longer approximate the meanvalues; instead they approximate the true trajectories of the evolution of the city configuration.In Figs.
and 2 we exhibit a few selected results of simulations based on the model of ur- ban evolution discussed above.

THE DESIGN PRINCIPLES OF A MODEL FOR THE REGIONAL EVOLUTION OF SETTLEMENT STRUCTURES ON THE MACROLEVEL
The "space-time window of perception" of a macromodel is open towards the more coarse- grained spatial structures and the slow temporal evolutions.On the other hand, the fast pro- cesses on each local site are averaged out here, since we look at the slow evolution on the re- gional scale only.Therefore global variables are needed which represent the regional processes and structures.
The proposed macromodel is designed accord- ing to the following principles (for details see [8, 91): 1.The economic and the population-dynamic migratory sector are integrated. 2. Populations are described by population den- sities distributed over the plane.3. The populations produce goods; the local pro- duction costs including fixed costs and pro- duction costs, hence the local individual incomes depend on the population densities.The economy is assumed to be in equilibrium with the momentary population distribution.4. The members of the population migrate be- tween different locations.Driving forces of this nonlinear migration process are income differences between locations.5.The migration leads to the formation of spatially heterogeneous population distributions; i.e. the settlements.
Let us formulate these principles in mathemati- cal form.We consider A productive populations 7, a 1,2,..., A, each producing for simplicity only one kind of commodity composed of units Ca.Furthermore we assume two service popula- tions, the landowners Pa renting premises to the producers and the transporters P dispatching the goods of the producers.Be na(x, t) the density of population 794 at posi- tion x and time and Cpa(X,t) the production density of 794, i.e. the number of units Ca pro- duced per unit of area and time.The production density is assumed to have the form t) 7 (x, t) (4.1) with the productivity factor [nc(x,t)] a t) % (4.2) The form (4.2) of ")'a expresses an "economy of scale" in the production of commodity Ca.If the productivity exponent is aa > 0, the production density grows more than proportional to n(x, t).
This will be true for many industrial goods, whereas for agrarian goods the production density grows less than proportional to n(x,t), which amounts to a productivity exponent aa < 0.
In (4.5) and (4.6) a reasonable ansatz has been made for the dependance of the fixed costs and the transport costs on the partial and total population densities na(x, t) and n(x, t), where A n(x, t) na(x, t).(4.7) a=l The fixed costs, with pa as fixed share coefficient, grow according to (4.5) over-proportionally with the total density n(x, t) of the local population, and the transport costs, with ra as transport share coefficient, are proportional to the mean transport distance da(x, t) of good Ca from the place x of production.
The fixed costs (for renting premises) and transport costs (for dispatching goods) for the producing populations 79a are simultaneously the net incomes w,(x,t)and w(x,t) for the service populations T'a and T', respectively:  The relative prices Pa of the commodity units Ca can now be determined by taking into ac- count that the goods are not only produced but also consumed by the same populations 791, 79A, 79,79 in the total area 4 under considera- tion: if cea(x,t) is the consumption density of commodity Ca, which corresponds to its produc- tion density, and which can be expressed by the local net incomes, then the equilibrium between production and consumption in the assumed closed economy of area .A can be expressed by CPa(X, t)d2x J Cct(x, t)d2x;  From (4.10) there follow the relative prices (for details see [8,9]).
Finally, the local net incomes per individual are easily obtained: w (x,t) wa (x, t) na (x, t) withc=l,...,A; ,r.(4.11)It is important to note that all economic quan- tities introduced so far, in particular the local individual net income (4.11), are expressed as functions of the population densities.
This means that the state of the simple econo- my described here (with production income consumption densities, etc.) is well defined if the population densities are known.We now as- sume that this still holds if the na(x,t) slowly evolve with time.This assumption implies that the adaptation of the economy to the momentary values of the ha(X, t) is fast and flexible enough to keep it always in momentary equilibrium with the population distribution.
We shall now see that a slow migration process governed by nonlinear migratory equations sets in if we let depend the motivations of the individuals to change their location on simple economic con- siderations.The result of the migration process is that a (perhaps initially existing) homogeneous density distribution of the populations becomes instable and that the different subpopulaions P segregate into agglomerations of different size and density.We conclude that already simple assump- tions about economic and migratory structures lead to the self-organisation of settlements.
The equations of motion for the population densities are obvious.They read dn(x', t) dt fA ra(x', x; t)na (x, t)d2x f.a r(x, x'; t)na(X', t)d2x for c 1,2, ...,A,, where the decisive quantity is d2x'ra(x', x; t) transition rate of a member of 79a from x to x' into the area element dZxt.(4.13) The form of this rate, which has been substantiated in [5], is r(x', x; t) # exp[u(x', t) u,(x, t)], (4.14)where the "dynamic utility" u(x,t), which is sometimes also denoted as motivation potential, is a measure of the attraction of location x' to a member of population 7 at time t.
A plausible assumption in the frame of our simple model is that u(x,t) is proportional to the local individual net income (x, t), i.e. us(x, t) fl(x, t), (4.15) where /3 is a sensitivity factor calibrating the strength of migratory reactions to space-dependent income variations.
If (4.15) is inserted into (4.14) and (4.12), where o(x, t) is to be expressed in terms of the population densities, Eq. (4.12) takes the form of a nonlinear integro-differential equation which can be solved numerically (see Section 5).
At the end of this short presentation of the macromodel we exhibit the interrelation of its construction elements in schematic form (Fig. 3).
A final remark should be made about the con- nection between the macromodel of Section 4   and the micromodel of Section 3. As the follow- ing simulations show the macromodel demon- strates that global population agglomerations on a regional scale can arise by migration from a hinterland into an urban area taking shape due to certain economic production laws and migration decisions.
Thereupon the micromodel considers the deci- sion mechanisms how the slowly varying total urban population exerts a population pressure which in turn leads on the local level of sites to the organisation of differentiated urban sub- structures.
We now present the result of numerical solu- tions of Eq. (4.12) in the case of only two productive populations which can be interpreted as "peasants" (c=p)and "craftsmen" (c=c).  The interrelation of the elements of the macromodel for the formation of settlement structures.craftsmen peasa FIGURE 4 Parameters: inclusion of fixed costs (p > 0) and of transportation costs (a > 0). (a) Stationary formation of more than one "town" per unit area in spatial neighbourhood, settled by "craftsmen".(b) Stationary ring-shaped rural settle- ments of "peasants" around the "towns".craftsmen peasants FIGURE 5 Parameters: inclusion of fixed costs (p > 0) and of transportation costs (a > 0).(a) Stationary formation of differentiated "town-structures" settled by "craftsmen" in each unit area.(b) Differentiated ring-shaped rural settlements of "peasants" around the "town-structures".
They are characterized by different productivity exponents (see Eq. (4.2)), namely ap -0.1 < 0; at.0.25 > 0. (4.16) Figures 4 and 5 depict nine equivalent unit areas (in which by construction the population distri- bution is periodically repeated).The densities in all figures are scaled to their maximum values (for details of the calibration of the model see [9]).

CONCLUSION AND OUTLOOK
The two models of Sections 3 and 4 have demon- strated how the concepts of sociodynamics can be applied to the modelling of evolution processes of settlements on different spatial scales.Refine- ments and extensions are possible and remain to be done.A natural task for further work consists in the concrete matching of the regional and the urban model: the regional macromodel yields the global evolution of population densities, whereas the micromodel is capable of describing the evo- lution of detailed urban structures under the in- fluence of growing population pressure.This matching problem will be treated in a forthcom- ing paper.Refined versions of the models should also take into account important empirical regu- larities of settlement evolution like the Leo Klaassen cyclic stages of urbanisation, suburba- nisation, disurbanisation and reurbanisation (see [10,11]).
obey in contrast to (3.15) self-contained autonomous equations of motion which arise from (3.16) by substituting on the right-hand side: v(x, y) ::> v((t), (t)) (3.16) which leads to dj

FIGURE
FIGURE , t) Z t(x, t).

FIGURE
FIGURE3 The interrelation of the elements of the macromodel for the formation of settlement structures.