Evolutionary Patterns of Urban Populations

-

Dimitrios S. Dendrhos and Henry Mullally*

Ebolutionary Patterns of Urban Populations

1. A MODEL OF URBAN POPULATION DYNAMICS

The following discussion is motivated in the first place by the belief that city size patterns are inherently complex, and second by a dissatisfaction with the ability of clonventional paradigms to model this complexity. As an empirical foundation, a survey of the population histories of all Standard Metropolitan Statistical Areas (ISMSAs) in the U.S. is currently in progress. For each urban area, the population in every available year over the period 1940-77 has been expressed as at portion of the national population in that year, and the records of the resultant normalized population size (NPS) have thus far been accumulated for ninety SMSAs. The patterns that emerge pose some intriguing theoretical questions, and in this paper we adopt an essentially ecological approach to resolving them. Building upon this, it is hoped that ultimately a coherent and unified explanation of such features of the urban landscape as cyclical patterns, divergent behavior, discontinuities, multiple equilibria, etc., will materialize. This diversity, which seems so central to urban events, eludes most of the current modes of analysis, such as the “new urban economics,” with its emphasis on long-run, static equilibrium.

The basic hypothesis of this work is that the growth of any city is determined by the relative economic opportunities that it offers within a competitive framework, and that this relationship can be expressed as a system of dynamical differential equations

where x denotes the city’s population, y the per capita income of its inhabitants, and the dots stand for derivates with respect to time. Eliminating time from the solutions of this system and plotting the results gives the phase portrait of the system in ( x , y) space. Such portraits can convey considerable insights as to the dynamics of the system under investigation. Of particular interest is the behavior of (1) around (and close enough to) the critical point i = y = 0. From our perspec- tive, this will tell us about the nature and stability of equilibrium populations and

*Part of this work was done under contract with the US. Department of Transportation by D. S. Dendrinos.

Dimitrios S . Dendrinos is associate professor, Graduate Program in Urban Planning, University of Kansas. Henry Mullally is assistant professor of e c m i i c s , University of Missouri, St. Louis.

00187363/81/10814328$00.50/0 GEOGRAPHICAL ANALYSIS, vol. 13, no. 4 (October 1981)

0 1981 Ohio State University Press

Dimitrios S. D d r i m s and Heny Mullally / 329

I I

Ill Limit c y c k UnPlsble lhl Unslsblr limil C Y C k from inside

(8) Stable llmll Cycle

TABLE 1. Phase Plane Portraits of Dynamical Behavior Around Singularities in A’

the paths by which such equilibria are reached. The phase portraits for the neighborhood of various singul~ties are shown in Table 1.

The system described by (1) is too general in analyzing a particular problem, such as population dynamics. Clearly the function must be specified in more detail in a manner consistent with the phenomenon of interest. Samuelson (1971) has suggested as a general model for ecological and economic dynamics a differential system derived from the prey-predator model of Volterra-Lotka. Dendrinos (1980) has proposed the following version of (1) as a model suitable for urban population analysis:

2 = x( -a, - allx + al,y)

When all = 0 we obtain a system similar to the classical Volterra-Lotka model, in which both variables oscillate indefinitely around the critical point. In our context, we interpret the parameter a,, as an indicator of the presence of urban friction of some sort, which limits the exponential growth tendency of the city. Presumably this reflects such factors as the effects of crowding. This parameter thus reflects the density-dependent nature of urban growth.

When a city changes from a form in which urban friction operates into one where it does not, then (in terms of Table 1) this means that the dynamical behavior shifts from a stable focus, case (b), to a center, case (f). Such transitions are widespread in natural systems. We have in fact observed both types of behavior in SMSA populations as it will be shown later. If we can characterize such transitions as a consequence of smooth changes in some parameter, then it is possible that cities could undergo what in mathematical terms is a Hopf bifircu- tion. The conditions under which this bifurcation takes place are given by the Hopf

330 / Geographical Analysis

bifurcation theorem. Casti (1979) provides a simple version of the theorem, which is suitably adopted below.

THEOREM. If (i) g ' (x , y , p), g 2 ( x , y , p ) are at least four times differentiable in each argument; (ii) g'(x*, y*, p ) = g 2 ( x * , y*, p ) = 0 for all values ofp; (iii)

evaluated at the critical point (x*, y*) has the jacobian 1 =

distinct complex conjugate eigenvalues A', A' such that for p > 0, Re A( p ) > 0; and if (iv) ( d / d p ) { Re A(p)} > 0 when evaluated at the point p = 0; then the equilibrium bifurcates fiom a fixed point into a closed orbii with radius proportional to Jp.

Proof: Clearly the first two conditions are satisfied by the system (2). Next, [ "* u12x* has the characteristic polynomial cosnsider that I =

1 agl/ax ag1/ay [ ag2/ax ag2/ay

1 -az,y* 0

A2 + Apx* + a12a21x*y*

with the roots

px* 2 \I( px*)' - 4aZ1al2x*y* 2 A =

Clearly when p = 0, Re A(P) = 0. Next we consider the value of Re A(p) in a neighborhood of the point p = 0. For 0 < p < j(4a2,a12x*y*/x*) it is seen that Re A(P) > 0. For 0 > p > -(4az1al,x*y*/2) we have Re A(p)i < 0. Finally, olbserve that ( d / d p ) R e A ( P ) = ( x * / 2 ) > 0 at p = 0.

Thus all the conditions of the theorem apply and we conclude that in a neighborhood of p = 0 there is an equilibrium pattern of closed orbits. In other words, the disappearance of urban friction (crowding) transforms the equilibrium pattern of urban dynamics from a stable focus to a center.

If it is a reasonable representation of urban structure, the analysis above, though remarkably simple, indicates clearly a number of features we should expect to observe in the time paths of metropolitan populations. In the first place, the model indicates that regularly oscillating behavior should occur only for a very special case --when the parameter all is zero. At all other values, it cannot occur. This implies t'hat the frictionless city must not be frequently observed. Furthermore, the transition from a mode of orbital oscillations to a convergent mode is also likely to be extremely rare, requiring as it does the existence of a city poised at the critical \due of the key parameter and subsequently undergoing the appropriate change of underlying circumstances.

In addition, for all meaningful values of the parameter representing friction, there is no possibility of limit cycle behavior, thus making such behavior impossible to observe. The particular features which the model does lead us to eKpect are those associated with the sink-spiral pattern. It is thus reasonable to expect a large number of SMSAs exhibiting damping down to a stable equilibrium point. It may well be that the time span of observations (thirty-seven years) is not sufficiently long to capture the entire pattern. What the model does indicate very unambigu- ously, however, is that movements of high amplitude should always be followed by ones of lower amplitude. Finally the model leads us to expect some SMSAs to be at a steady state. If all these features are observed, then one might be confident in asserting that the data are consistent with theory proposed.

Dirnitrios S. Dendrinos and Heny Mullully / 331

The model proposed earlier also embodies the characteristic of structural stability, meaning that its qualitative properties are not lost when subjected to small perturbations. Thus if a city with the sink-spiral pattern were subjected to some sudden shock, we would expect that its behavior will gradually and asymptotically approach the original path. Empirical evidence indeed, as it will be apparent in the next sections, does provide evidence that these qualitative properties do characterize metropolitan histories.

2. A COMPARISON WITH ALTERNATIVE MODELS

The philosophy underlying the preceding analysis is sharply at variance with the mainstream of urban economic analysis. Fundamentally, cities in this paper are perceived as nonlinear oscillators. This viewpoint takes us away from the familiar terrain of static, long-run equilibria, linear models, etc., and towards the rapidly expanding conceptual world of structural stability, resilience, bifurcation theory, and nonlinear differential systems. The new analytic world is characterized not only by its ability to handle those very features which epitomize urban phenomena, but also by its emphasis on conciseness, clarity, and qualitative insight.

Faced with the inherent complexity of cities, modelers have typically separated into two classes. On the one hand there are those such as Forrester (1971), and the largescale urban simulation efforts, who embrace complexity by building models with vast numbers of variables and equations, which then generate equilibrium time paths via computer simulation. Apart from the well-known criticisms of this work, such efforts tend to be frequently analytically intractable and yield little in the way of insight to basic urban processes.

At the other extreme lie those modelers who evade complexity by concentrating on building simple models capable of simple long-run equilibrium configurations and properties. The new urban economics typifies this approach, in which city size is envisioned as expanding smoothly to a Malthusian limit defined by an exogenous utility level. The limitations of this approach from the dynamic viewpoint have been detailed elsewhere (Dendrinos (1980)).

The approach adopted in this paper, by contrast, admits models capable of generating complexity in the phenomena of interest while retaining extreme simplic- ity in model structure. Therein lies the key property of the differential theory of dynamical systems. Differential theory emphasizes economy in the construction of models to predict a system’s behavior, in direct contrast to the largescale simulation models. Furthermore, while most systems of differential equations do not yield analytic solutions, their phase portraits are rich in qualitative insights. For this reason, Berlinski (1976) advises modelers to seek, whenever possible, systems where qualitative analysis is possible. Berlinski specifically commends the Volterra-Lotka model of interacting species in biology on the grounds that while no explicit solutions are provided, everything interesting about the general class of solutions is known, namely that they are all periodic about a center. Furthermore, these properties are inherent in the specification of the model, with the parameters defined only within broad restrictions. This approach leads naturally to such questions as to whether a dynamic system preserves its qualitative features when disturbed, that is, if it is structurally stable, as well as questions on how patterns are created, how they vanish, and so on.

In remarks directed at biologists, but bearing special pertinence for urban analysts, Pavlidis (1973) distinguishes between the modeling strategy of biology on the one hand, and much of engineering and physics on the other. In the latter fields, the important equations and variables in many problems are known, while in the former, this is rarely the case. In physics and engineering, one may frequently


start with a structural description of the system and then examine its behavior when subjected to change. In biology, by contrast, one often has to deduce structure from experimentally obtained data. This is essentially the situation concerning urban population dynamics. Given the observed behavior, what deductions can be made about the mechanisms governing urban structure?

One of the most prominent features of the data is the frequency of rhythmic fluctuations. While cities are observed to converge (in many cases) to one relatively steady state, this is not smooth asymptotic behavior which a simple logistic mechanism would explain. The cyclic nature of the phenomenon which we observe hints at some basic oscillatory mechanism, and the fact that such1 fluctuations decline in amplitude over time for a large number of cases suggests that the oscillatory mechanism is coupled with some dissipative force. Hence, ;again we find a parallel with biology, where for centuries the existence of rhythmilc fluctuations has been known. Until recently it was generally believed that these were due in most circumstances to external effects. The accumulation of clinical data now supports the existence of endogenous oscillators. Furthermore, such patterns occur in a wide variety of populations and they appear resilient to changes in their environment. The fact that the inner biological processes are not necessarily well understood has not precluded a host of rigorous studies which model oscillatory behavior (e.g., Hoppenstadt 1979; Pavlidis 1973).

While the available urban data do not reveal a uniformity of cyclic motion, this in no way vitiates the validity of models hypothesizing a fundamental oscillatory process. It is most probable that such an evidently complex phenomenon is governed by a multitude of cycles, in which case observed behavior may change drastically (Pavlidis (1973)). The analysis here is motivated by time considerations: (1) to reproduce these qualitative features which are prominent in the data and (2) tal do so using the most parsimonious differential model.

3. HISTORICAL PROFILES OF SELECTED SMSAs

For ninety metropolitan areas, the NPS for all available years in the period 1'340- 77 has been calculated.' The initial impression conveyed by the data is of the great variety of behavior among the cities. On closer inspection, however, the generic patterns which emerge are few in number, and sufficiently distinctive to provide reasonable grounds for taxonomy. Patterns repeat themselves with the frequencies expected from the theoretical discussion, and, as noted before, these configurations are familiar in the dynamics of many systems in the physical, biological, and social sciences.

The generic historical profiles, exhibited by the NPS studied, are: 1. Damped oscillatory movements (sink spiraling NPS moving towards a fixed

equilibrium), of varying frequency and amplitude 2. Steady state phases, where the NPS fluctuates around a constant value over

the entire thirty-seven-year study period 3. Relatively sharp (discontinuous) rises/falls in NPS caused by perturbations of

SMSAs 4. Structurally unstable SMSAs which, when perturbed, switched to a different

state, thus experiencing a discontinuous motion of their eqdilxium point.

'Population size for all selected SMSAs were collected using decennial census counts, as well as annual counts for the period 1960-77 whenever possible. Decennial counts are as of April 1; annual estimates are as of July 1. To our knowledge, such records have not been studied before in a systematic way. Isolated SMSAs may have been studied, but neither the universe nor a sample of them has been :malyzed from the present standpoint.

LXmitrios S. D d r i n o s and Henry Mullally / 333

Oscillatory Behavior The various types of observed oscillatory motion and their relative frequencies

are shown in Tables 2 and 3. Two significant findings emerge: (a) there are no SMSAs exhibiting source spiraling behavior; and (b) there seems that there are no SMSAs undergoing convergence towards a limit cycle. In general, the available evidence is consistent with the hypothesis that the Volterra-Lotka ecology does accurately portray urban development.2

Sink spiral behavior, the most common oscillatory movement in metropolitan historical profiles, is succeeded by a steady state when the attractor is reached (with very small fluctuations around it). There are nine SMSAs (10 percent in the sample), in the thirty-seven-year observation period, which have reached steady state. The fact that in such a small time span (thirty-seven years being a snapshot in a city’s life cycle) steady states have been observed strongly supports the existence of damped oscillators in metropolitan evolution. Figure 1 is an example of explicit damped oscillation.

Furthermore, although the sample of the ninety SMSAs used in this study is not a random one, it covers nonetheless onethird of the universe of U.S. SMSAs; and in this sample the predominant historical profile of NPS is that of oscillatory motion of the sink spiral type. From the ninety SMSAs studied, approximately sixty-four seem to exhibit such behavior.

Only three SMSAs are shown to be at their steady state throughout the study period out of which one (Syracuse, New York) is seemingly undergoing a transition phase in the seventies (Fig. 2). For the six SMSAs that have reached steady-state equilibrium within the study period, steady state seems to cover less than onethird of their observed life span. Since only six of the twenty-three SMSAs which exhibit sink spiral motion, and six out of sixty-four altogether (including those SMSAs implicitly showing sink spiral motion), it is concluded that the longest part of metropolitan evolution is spent in a motion towards a fixed point equilibrium, in a

2The set of Volterra-Lotka equations describing the predator-prey ecology is the following:

f = a ( y - Y). y = b(? - x ) y ,

where x and y are population and per capita income, respectively; f and carrying capacity and national average per capita income correspondingly.

are the urban population

Y

Y

FIG. F.l (left). Orbital Behavior of the Volterra-Lotka System FIG. F.2 (r ight) . Sink Spiral Behavior of the Volterra-Lotka System

The above formulation results in an orbital behavior, Figure F.l; if a dissipative element is introduced in any of these equations, then a sink spiral oscillatory movement is the outcome, Figure F.2. The introduction of social friction in the first equation was discussed in Dendrinos (1980).


ru~nusymptotic manner, and once there an SMSA does not remain for long unperturbed. It seems that those SMSAs that have reached such states and remuin there for relatively prolonged time periods become structurally u s t a b k , and environmental or internal perturbations cause them to switch to new states.

GBLE 2 Dynamic Patterns of Metropolitan Normalized Population Histories: A Classification Based on Ninety Selected U.S. SMSAs in the Period 1040-77 -

Type of History Number of Occurrences ~~

A. Oscillatory behavior 64 1. Sink spiral 23

a. With steady state 6 b. Towards steady state 17

2. Source spiral -

3. Orbital 1 4. Limit cycles -

5. Long-term oscillatory* 14 6. Medium-term oscillatory* 26

B Steady state’ 3 C. Perturbations 21

1. Structurally stable 15 a. Switch of long-term spirals 10 b. Switch of medium-term spirals* 5

2. Change of state 6 a. Naked discontinuity 1 b. One mode plus discontinuity§ 5 c. Hopf bifurcation -

El. Unclassified ” 2 - *.These cases have not indicated yet whether they belong to any of the categories 1-4, since the oscillatory period is greater than the study period. For their various subcate

?.It includes Syracuse, New York, which may be undergoing a phase transition from steady state to oscillatory behavior. *It includes one SMSA (Clarksville, Tenn.) which switched to a different mode. $[t includes four SMSAs (Anderson, Ind., Buffalo, N.Y., Chicago, IU., St. Louis, Ma.) uhich are switching from a steady state to an unidentified, as of 1977, mode. It also iricludes Bismarck, N.D., which was shifted from a (high frequency) sink spiral move nient to an unidentified state as of 1977. II They iiiclude ClarksviUeHopkinsville, TennesseeKentucky, and Boise City, Idaho, that ellhibit peculiar patterns that need further study to classify.

ones and their incidence, see Table 3.

These observations are of import in view of the fact that so much research and effort has been put into long-run static equilibrium analysis by the conventional ilrban economic theory; research in that field has strongly focused upon the configuration of the equilibrium form and the assumption of asymptotic movement .towards it. Little has been said about the dynamics towards equilibrium (although the conditions of existence and uniqueness, due to the particular closed form solutions sought, negated such discussion). It is suggested here, that the particular nonlinear dynamics involved in urban evolution are such that a multiplicity of equilibria (of different type) is possible and further, that it is feasible to examine their qualitative properties without having obtained closed form solutions, thus avoiding misleading results from static analysis.

A single SMSA, Rochester, New York, is showing an orbital motion through the thirty-seven-year period (Fig. 3). The existence of such movement implies strongly the operation of an orbital oscillator, a rare event in historical profflies of SMSAs. As noted previously, this phenomenon occurs in very particular circumstances and it is the result of a Hopf bifurcation in the Volterra-Lotka formulation. Rochester is our candidate for such a morphogenetic even amcng the ninety SMSAs. The Effects of Perturbation

Twenty-one SMSAs are classified as “perturbed’ in this nonrandom sample of ninety SMSAs. Because of the nature of the selection process (see Appendix note)

Dimitrios S. Dendrinos and Henry Mullally / 335

they are overrepresented. Thus, the incidence of such events must be relatively rare. These phenomena indicate that urban areas experience certain shocks (due to external or internal reasons) at certain time periods; one effect of such shocks is for NPS to switch from one oscillatory path to another qualitatively equivalent, i.e., leading to the same equilibrium point through a neighboring path. This event occurs in fifteen cases. In the remaining six cases, the NPS is tipped off to wholly new states of dynamic paths characterized by different types of equilibria. These were structurally unstable SMSAs experiencing at the time of the perturbation unstable critical points. In Figure 4, the case of Billings, Montana, is shown where the triggering action from 1964 to 1967 caused a shift from a high growth rate during the proceeding period to a milder one in the succeeding years.

TABLE 3 The Various Long- and Medium-Term Oscillations and Their Number of Occurrences in the Sample of Ninety U.S. SMSAs: 1940-77

Type Shape@) Occurrence

A.5 Long-term 14

a. Linear 5

i + . i > o , f = O / 3

i i - , i < O , f = O \ 2

i - . i > ~ , i > ~ 1 -

i i + i < O , f C O - 2

b. Convex 2

c. Concave 7

7

- f l i + i > 0, i < 0

i i + i < O , x < ~ - A.6 Medium-term 25

a. Convex w - b. Concave e 5

20

/ 1

c. With inflexion point

i - t

ii --* \ - iii --f 1 16

iv --f 4

A naked discontinuity is indicated for the Anchorage, Alaska, SMSA in Figure 5 (no data are available for the pre1970 period). It shows a 37.5 percent increase in its NPS during a seven-year period (corresponding to an annual average rate of increase of approximately 5.5 percent). The term “naked’ is used to define a relatively high exponential growth/decline rate behavior covering a relatively large time span in the observed life cycle of an SMSA. Anchorage was perturbed from an unstable equilibrium point and, as a result, it is rapidly moving to a new state, not yet reached by 1977.

Four SMSAs of varying size have a historical profile consisting of a steady state followed by a sudden decline. The relative steepness of these discontinuous motions

220

m

180

165

1M

135

311.14

3W.16

I 1940 50 60 70 SO

r G9.5

_. 441.2 _-

FIG. 1. The Tacoma, Washington, SMSA Sink Spiral Towards a Steady State. Solid points indicate actual NPS counts; years in parenthesis indicate simulated sizes through the RurigeKutta method, producing the spiral shown by the solid line. Better approximation is still feasible by depicting the second periodic movement in the seventies, through changes in the starting (1940) size.

1948 50 60 10 11 -

1940 50 60 10 11

FIG. 2 (left). The Syracuse, New York, SMSA Under Steady State (indicating the beginning of,

FIG. 3 (right). The Rochester, New York, SMSA: Exhibiting an Orbital Oscillatory Movement possibly, a phase transition in the seventies)

Dirnitrios S. Dendrirws and Henry Mullally / 337

41

44

40

36

30.9

-

--

--

..

-.

1940 50 60 65 66 10 11

83.9 -

19.32

14.14

10.16

65.58

61.0

FIG. 4. The Billings, Montana, SMSA: A Perturbation During the 1964-67 Period Shifted the Motion to Another Qualitatively Equivalent Path

varies among the four, the steepest being that of St. Louis, Missouri-Illinois, (Fig. 6) and Chicago, Illinois. Apparently, these four SMSAs are evolving towards a new equilibrium state not yet reached by 1977.

Perturbations, thus, seen to have different effects upon the various SMSAs: those that resulted in slight changes in the dynamic paths of NPS, by shifting them to neighboring ones driving towards the same equilibrium (structurally stable SMSAs); and those that resulted in qualitative change in the nature of the equilibrium state (structurally unstable SMSAs). In order for us to examine the nature of the (possible different types of) perturbations we must have a mix of these phenomena at various points in time. Unfortunately, these vents seem to have clustered around specific

_-

--

--

--

--

10 71 1940 50 64

FIG. 5. The Anchorage, Alaska, SMSA: A Naked Discontinuity


1149.56 --

1132.94 --

1116.32 --

1099.7 --

w 1940 M 60 70 80

FIG. 6. The St. Louis, Missouri-Illinois, SMSA: Undergoing a Phase Transition from a Steady State During the Period 1940-70 to a Steep Decline in NPS During the Seventies

o’bservation periods: Structurally stable SMSAs were perturbed in the sixties, whereas structurally unstable ones were perturbed in the seventies. The perturbations affecting the first type seem to have been of a shorter duration than the later case. SMSAs that were structurally stable were experiencing sink spiral behavior, whereas the unstable ones were experiencing steady state. This might point to a connection between stability and length of stay at the attractor point. Their incidence is different, too: there were many more structurally stable SMSAs perturbed than there were unstable.

Both types of SMSAs, however, are much less frequently encountered than SMSAs exhibiting some stable dynamic patterns (21 to 67 in this sample). Both seem to be evenly distributed through the regions of the nation, and over different types of SMSAs (according to size and age). The locational distribution of these SMSAs seems to indicate that such events must be connected to nationwide experienced shocks: the transitions that apparently are the causes of the perturbations in the seventies produced morphogenesis; whereas the slight shifts caused by the perturbations in the sixties were of a noncatastrophic nature. On the other hand, as stated earlier, different types of SMSAs were responding to these (different) perturbations.

Although the present report is not to address the nature of perturbations, and thus the concomitant effects of various policies and exogenous events, nonetheless, it is clear that the resiliency of certain SMSAs to such actions has been tested in the course of the thirty-seven year study period. Metropolitan robustness might have lbeen stronger during the sixties than during the seventies, assuming that the nature (of the perturbations was not significantly different.

The Apparent and Proper Motions of NPS The movement of an SMSA’s NPS is the combination of two separate move-

ments: the motion of NPS towards/around the equilibrium point and the motion

Dimitrios S . Dendrinos and Heny Mullally / 339

due to changes in the position of the equilibrium point. We shall refer to the motion towards/around the equilibrium point as the “proper” motion, and the combined movement, which includes that of the equilibrium, as the “apparent” motion. In order to distinguish the two, one must identify the movement of the equilibrium point.

As the apparent movement of NPS for the vast majority of SMSAs seems to be nonchaotic but observing rather specific general patterns, it seems that the motion of the equilibrium point of oscillating SMSAs must also obey specific patterns of movement and must not be random or chaotic. It could be either smooth or abrupt, or an identifiable combination of both. In order to study the path of the motion of the equilibrium points of NPS for the U.S. SMSAs, we must look at the nature of the various steady-state type apparent motions of NPS.

During the thirty-seven-year period, three SMSAs were found to have continu- ously experienced steady-state conditions. Steady state here is identified as a motion which has a ratio (max NPS - min NPS)/max NPS not exceeding 5 percent throughout the observation periods fluctuating around (a mean value) equilibrium value and independent of time. The sample further contains five SMSAs with steady state reached for a time period in their dynamic paths succeeded by a rapidly downwards moving NPS. In all four cases, the fluctuations around the equilibrium state seem to indicate a point attractor independent of time. Finally, the sample contains six SMSAs with downward oscillatory movement having reached the state of equilibrium, which for the remainder of the observation period stays independent of time.

On the other hand, there are five SMSAs that do show a rather linear (increas- ing/decreasing) apparent motion (i.e., slight fluctuations around a nonhorizontal line) of the NPS, possibly suggesting a continuous movement of the equilibrium point rather than a movement as interpreted here (a special part of a long-term oscillation).

Changes in the equilibrium NPS (carrying capacity) are due to shocks (endogenous as well as exogenous) of an urban economy. Technological innovations, shifts in preferences, random perturbations due to international events, public actions, etc., all affect the carrying capacity and thus the point attractor of metropolitan areas. Given that these events are random and not always congruent, they must not result in a continuous and unidirectional movement of the fixed point. It is likely that the overwhelmingly predominant smoothness observed in the dynamic paths of NPS is the result of a well-regulated motion towards a fixed point, determined by an internal oscillator defined over population and income, and rather rare discontinuities are the result of random perturbation, and adjustments of the point attractor.

4. A NUMERICAL EXAMPLE

To bring the previous, rather abstract, characterization of urban dynamics into sharper focus a series of computer simulations were run in an attempt to calibrate, for one city at least, the Volterra-Lotka system which governs its development. The Tacoma, Washington, SMSA was chosen as a good example of a metropolitan area exhibiting the symptoms of a sink-spiral pattern. A variety of methods exist for calculating numerical solutions to systems such as (2), among the most popular being the Runge-Kutta routines. These are frequently used in ecological modeling (Hall and Day 1977) and are discussed in detail in many textbooks on numerical analysis (for example, Moursund and Duris 1967). In general terms the Runge-Kutta method approximates the Taylor series expansion of a function. For a given function f( t ), if

@= F ( f ( t ) , t ) dt


exists over a given interval from t to t + At, then from Taylor's theorem

where k(At)"+' is the remainder term. The Runge-Kutta formula approximates the Taylor expansion using a one-step method which voids any of the higher deriva- tives. It is a difference approximation of the form

N

f ( t + A t ) =f(t) + b,qiAt, i = l

where

etc. The parameters a,,, b,, Ci are chosen so that the equation equals the Taylor series

expansion. The most common version is the fourth order (N = 4) Runge-Kutta method, but recently Hull, Enright, and Jackson (1976) have developed a computer mbroutine DVERK, which employs Runge-Kutta formulas of orders five and six. 'fie DVERK subroutine was used in the Tacoma simulation.

The major disadvantage of numerical methods is that the calculations must be repeated for each new parameter value or initial condition. The volume of experi- mentation can be considerable, even for such a small system as (2). Expressing median SMSA family income as a percentage of national family income for each year the system (2) can be rewritten as

3i. = x ( a ( y - 1) - p x )

Y = YY(+ x ) , (3)

where 5 denotes the carrying capacity of the SMSA, a and y are parameters reflecting the speed of adjustment of the two variables, and /3 represents urban friction.

To begin the Runge-Kutta simulation we require initial values x(O), y(O), and an appropriate choice of step size, At. For the Tacoma example x(0) is the 1940 normalized population size, 0.00137. The step size was chosen (after several trials) to be a three-month interval. The relevant per capita income data are not availabIe for 1940, and therefore numerous experimental values or y(0) were attempted, as well as for a, p, and y. While one would not expect a very precise portrayal of a nonlinear system such as (3), especially with the total absence of even a crude indication of likely values for any of the parameters, it was nevertheless possible to simulate the income/population dynamics of Tacoma in a manner which is broadly consistent with the available data, as shown on Table 4. The parameter values for the simulation are y(0) = 1.0; a = 1.12; /3 = 0.033; y = 0.003. Based on an examination of the time path of Tacoma's normalized population size, the carrying capacity was chosen to be 1.96. While it cannot be claimed that these computations give more than a faint image of the Tacoma SMSA, it does seem to replicate the

Dimitrios S . Dendrims and Heny Mullally / 341

general character of events over the chosen period. The numerical analysis does, of course, allow us to make some speculations about the long-run steady-state equilibrium towards which Tacoma is converging. In this example, 2 = y = 0 when x = 1.96, y = 1.0578.

TABLE 4 The Actual and Simulated Population and Income Levels for the Tacoma, Washington, SMSA

Actual Simulated

Year NPS Income NPS Income

1940 0.00137 - 1950 0.00180 1.1243 1960 0.00178 1.0518 1966 0.00182 - 1967 0.00194 - 1970 0.00201 1.0280 1971 0.00200 - 1972 0.00191 - 1973 0,00184 - 1974 0.00191 - 1975 0.00195 - 1976 0.00196 - 1977 0.00196 -

0.00137 Loo00 0.00191 1.0794 0.00197 1.0524 0.00189 1.0572 0.00190 1.0580 0.00195 1.0592 0.00197 1.0592 0.00198 1.0590 0.00196 1.0588 0.00198 1.0585 0.00198 1.0582 0.00198 1.0580 0.00197 1.0577

SOURCE: Data for 1940, 1950, 1980, 1970 from US. Census of Population. For other years from Current Popularion Reports, Series P-25. NOTE: Income refers to ratio of Tacoma SMSA median family income to U.S. median family income. Data from 1950, 1980, and 1970 US. Census ofPopulation.

The phase portrait of the Tacoma simulation is shown in Figure 7. The pattern is consistent with most of the features implied by the observations. Specifically, the system has a population maximum in the mid-l950s, and a local minimum in the mid-1960s. Of special significance is the fact that the steady-state median income is

Y

1.10

1.05

1.00

I

120 140 160 180

D

2 i = O

74

X=1.97 ', i I 00 220 240 I

FIG. 7. Phase Portrait of the Tacoma, Washington, SMSA Simulation. The z Count Is in NPS X lo5


at a level somewhat above the national median income. The conceptual issues suirounding this point are discussed elsewhere (Dendrinos 1980).

At this stage the numerical analysis should be interpreted within the proper context. Its primary purpose hence is heuristic. More precise and finlely calibrated methods have already been brought to bear on these problems. For one, the initial value x(0) = 0.00137 as well as the steady-state value X = 0.00197 were kept constant throughout the simulation experiments. Slight variations of these parameters, it is felt, would improve the fit. Forthcoming work will capture the numerical properties of the various urban patterns described in this work in ,a much more precise manner.

APPENDIX

The Ninety SMSAs Classified Historical population profiles were recorded for ninety selected ShlSAs covering

the period 1940-77 (Table Al). Normalized populations were constructed than for each SMSA according to its 1977 county composition. Currently, research is underway covering the universe of the U.S. SMSAs as of 1977. Needless to say that documentation of the accuracy and reliability of the July 1 estimates are beyond the scope of this paper, however, the consistency exhibited by the counts indicates that some confidence in the data is not totally unwarranted. It is recognized that the thirty-seven-year study period allows only for a glimpse at the evolution of metropolitan areas; however, in spite of such a shortness in the observation period, we are relatively confident in making certain assertions as to the evolutionary patterns of urban area, their start-up and ending conditions. No doubt, more errtended observations covering longer time periods will enable us not only to test these dynamic theories fully, but also to enrich them in view of phenomena not encountered as of now (and may be predicted by the new theories). For example, although repelling limit cycles have not been encountered in the ininety SMSAs, they are predicted by the theoretical construct proposed, under certain conditions. PL rigorous statistical analysis awaits these propositions, forthcoming in future papers.

S'ABLE ~i Classihcation of Ninety Selected US SMSAs' Normalized Population Time Senes for the Penod 1940-77

SMSA Size 10 - 5 Dynamics Code

Abilene, Tex. Akron, Ohio Albany, Ga.

Albuquerque, N.Mex .Qlexandria, La. Uentown, Pa. dtoona, Pa. Amarillo, Tex. Anaheim, Calif. Anchorage, Alaska Anderson, Ind. Ann Arbor, Mich. Anniston, Ala. Appleton, Wis. Ashville, N.C. Atlanta, Ga. Atlantic City, N.J. Augusta, Ga. Austin, Tex.

Albany, N.Y.

60.4

48.4

Sink spiral with steady state

Sink spiral with steady state 303.8 Perturbed medium-term oscil. (A6b)'

365.2 Mediumterm oscil. 185.3 Perturbed long-term oscil. (A5ai)

288.2 Long-term oscil. (A5aii)

827.6 Medium-term oscil.

64.5 Sink spiral

61.8 Long-term oscil. 72.4

83.9 Naked discontinuity 63.6 Steady state plus discontinuity

115.7 Medium-term oscil. 51.6 Sink spiral

131.9 Sink spiral 77.9 Medium-tern oscil.

841.9 Medium-term oscil. 87.1 Sink spiral

133.7 Sink spiral 215.8 Medium-term oscil.

Sink spiral with steady state

Ala C lb Ala A6civ Cla Alb C l a A5aii Ala A6ciii C2a C2b A5ci Alb Alb A6civ A6ciii Alb Alb AGciii

TABLE A1 (Continued) __

Size SMSA 10 - 5 Dynamics Code

Bakersfield, Calif. Baltimore, Md. Baton Rouge, La. Battle Creek, Mich. Bay City, Mich. Beaumont, Tex. Billings, Mont. Biloxi, Miss. Binghampton, Tex. Birmingham, Ala. Bismarck, N.Dak. Bloomington, Ind. Bloomington, Ill. Boise City, Idaho* Boston, Mass. Bradenton, Fla. Bridgeport, Conn. Brownsville, Tex. Bryan, Tex. Buffalo, N.Y. Burlington, N.C. Canton, Ohio Cedar Rapids, Iowa Champaign, Ill. Charleston, W.Va. Charleston, S.C. Charlotte, N.C. Chattanooga, Tenn. Chicago, Ill. Cincinnati, Ohio Clarksville, Tenn.

Cleveland, Ohio Colorado Springs,

Columbia, Mo. Corpus Christi, Tex. Dayton, Ohio Des Moines, Iowa Detroit, Mich. Duluth, Minn. El Paso, Tex. Erie, Pa. Flint, Mich. Fresno, Calif. Ft. Wayne, Mich. Grand Rapids, Mich. Hamilton, Ohio

Harrisburg, Pa. Hartford, Conn. yanapol is , Ind. ohnstown, Pa. ansas City, Mo.

Knoxville, Tenn. Lancaster, Pa. Lorain-Elyria, Ohio Lubbock, Tex. Macon, Ga. Rockford, Ill. Rochester, N.Y. Saginaw, Mich. San Antonio, Tex. San Diego, Calif. San Francisco Calif. San Jose, Calif. Shreveport, La. South Bend, Ind. Spokane, Wash.

Colo.

166.0 989.5 201.5 84.9 55.8

167.8 47.0 80.7

141.1 372.6 34.1 42.4 53.9 66.9

1799.0 58.1

375.3 81.6 34.1

607.7 45.6

185.4 76.5 76.1

119.9 177.1 274.8 183.5

3238.6 631.2 66.4

897.0

134.6 41.0

139.8 383.2 152.6

2020.5 122.2 201.0 125.9 238.8 217.6 172.0 266.5 115.3

197.8 483.7 529.3 123.6 597.1 206.6 160.4 123.6 92.2

111.1 124.0 450.1 104.7 471.7 777.4

1464.9 586.0 164.1 128.2 144.3

Long-term oscil. Perturbed medium-term oscil. (A6b) Long-term oscil. Mediumterm oscil. Steady state Sink spiral Perturbed long-term oscil. (A5bi) Sink spiral with steady state Lang-term oscil. Sink spiral with steady state Perturbed sink spiral (Alb) Medium-term oscil. Sink spiral with steady state

- Perturbed long-term oscil. (A5aii) Medium-term oscil. Medium-term oscil. Sink spiral Medium-term oscil. Steady state (B) plus discontinuity Medium-term oscil. Medium-term oscil. Medium-term oscil. Long-term oscil. Medium-term oscil. Long-term oscil. Long-term oscil. Sink spiral Steady state (B) succeeded by discontinuity Perturbed medium-tern oscil. (A6b) Perturbed medium-term oscil. (A6b)

Medium-term oscil. (A6b) with switch to a steady state (B)

Medium-term oscil. Medium-term oscil. Sink spiral Medium-term oscil. (A6b) with switch Sink spiral Medium-term oscil. Long-term oscil. Long-term oscil. (A5bi) with switch Medium-term oscil. Medium-term oscil.

Long-term oscil. (A5ci) with switch Sink spiral Long-term oscil. Perturbed long-term oscil. (A5ai)

with switch Sink spiral Mediumterm oscil. Medium-term oscil. Long-term oscil. Sink spiral Sink spiral Sink spiral Medium-term oscil. Perturbed long-term oscil. (A5ci) Sink spiral Medium-term oscil. Orbital oscil. Medium-term oscil. Long-term oscil. Perturbed long-term oscil. (A5bi) Medium-term oscil. Medium-term oscil. Steady state Medium-term oscil. Perturbed long-term oscil. (A5ci)

a5ci C lb a5ai A6b B Alb Cla Ala a5b11 Ala C2b A6ciii Ala

C l a A6ciii A6ciii Alb a6c1 C2b A6b A6b A6ciii a5c1 a6c1v a5c1 a5a1 Alb C2b Clb

-

- C1b

A6ciii A6ciii Alb Clb Alb A6b a5a11 Cla A6b A6ciii Cla Alb a5a1

Cla Alb A6ciii A6ciii a5b11 Alb Alb Alb A6ciii Cla Alb A6ciii A3 A6ciii a5c1 Cla A5ci.. a6c1n B a6c1v Cla


%BLE A1 (Continued) -

Size SMSA 10 - 5 Dynamics Code

-

St. Louis, Mo. 1099.7 Steady-state (B succeeded by discontinuity C2b Stockton, Calif. 141.5 Long-term oscil A5ci

Ta.coma, Wash. 196.4 Sink spiral Alb Syracuse, N.Y. 300.2 Steady state’ B

NCWE: Initially, forty SMSAs were selectively studied by the authors on the basis of their size and interesting features in their population counts time series; then, the search was extended to the first (alphabetically) fifty SMSAs in .addition to the forty. (Only the first central city and state are recorded.)

Size refers to their normalized size, as of 1977. Source: U.S. Bureau of the Census, various P-25 series population reports. Code corresponds to the “types of history” specified in Tables 2 and 3.

‘Particular feature in a long-term linearly upward movement ‘The beginning of a transition phase during the 1970s is evident.

LITERATURE CITED

Bt:rlinski, D. (1976). On System Analysis. Cambridge, Mass.: MIT Press. Casti, J. (1979). Connectioity, Complexity, and Catastrophe. New York: Wiley. Drandrinos, D. S. (1980). “A Basic Model of Urban Dynamics Expressed as a Set of Volterra-Lotka

Equations.” In Catastrophe T h e m in Urban and Transport Analysis. Report No. DOT/RSPA/DPB 25/80/20. Washington, D.C.: U.S. Department of Transportation.

Forrester, J. (1971). Urban Dynamics. Cambridge, Mass.: MIT Press. Hall, C., and J. Day (1977). Ecosystem Modeling in Theory and Practice. New York: Wiley. Hoppenstadt, F. C. (1979). Non-Linear Oscillations in Biology. Providence, R.I.: American Mathemati-

cal Society. Hull, T. E., W. H. Enright, and K. R. Jackson (1976). “User’s Guide to DVERK- A Subroutine for

Solving Non-Stiff ODES.” TR 100. Toronto: Department of Computer Science, University of Toronto. M oursund, D., and C. Duris (1967). Elematary Theory and Applications of Numerical Analysis. New

York: McGraw-Hill. P,wlidis, T. (1973). Biological Oscillators: Their Mathematical Analysis. New York: Academic Press. Samuelson, P. (1971). “Generalized Predator-Prey Oscillations in Ecological and Economic Equilibrium.”

Proceedings of the National Academy of Sciences, 68, 980-83.

Evolutionary Patterns of Urban Populations

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