- Email: [email protected]
Nonlinear discrete relative population dynamics of the U.S. regions
Nonlinear Discrete Relative Population Dynamics of the U.S. Regions* Dirnitrios
S. Dendrinosf
Urban Planning Program University Lawrence,
of Kansas Kansas 66045
and Michael
Sonis
Ear-llan University 52-100 Ramat-Gan,
Transmitted
Israel
by John Casti
ABSTRACT A new discrete relative-dynamics algorithm is tested using population interactions of U.S. regions. A K&year time period is used (1850-1983) to calibrate the log-linear model’s parameters using standard linear regression tests. We also report certain forecasts under different spatid disaggregation schemes of the U.S. regions, and under two different time-period steps (iterations) to the year 2050. AU forecasts impIy competitive exclusion for the northern regions; under certain areal disaggregations and time (iteration) steps the southern or western region also experiences conditions leading to drastic declines in their share of the U.S. population. We also report the results of certain random fluctuations on the model’s parameters and their implications regarding the dynamics of the U.S. regional population structure. The outcomes seem to be robust under these fluctuations.
INTRODUCTION Mathematical properties of fixed-point discrete maps and recent associated developments in bifurcation theory have opened up new avenues for research in the social sciences in general and spatial fields (including geography and
*Part of this research was done under contract number SE.%%16620 with the National Science Foundation, which is gratefully acknowledged, by D. S. Dendrinos. f This author wishes to thank his research assistant Mehrdad Givechi for the computer work.
APPLIED MATHEMATICS AND COMPUTATlON
25:265-285
(1988)
265
0 Elsevier Science Publishing Co., Inc., 1988 52 Vanderbilt Ave., New York, NY 10017
009&3003/88/$03.50
266
DIMITRIOS S. DENDRINOS AND MICHAEL SONIS
regional science) in particular [5, 61. These maps include the possibility of obtaining turbulent behavior in the dynamics of sociospatial stocks, involving perioddoubling cycles and deterministic chaos. Two questions have consequently emerged out of the recent analysis that are of extreme importance with regard to the relevance of these discrete maps for sociospatial dynamics, as well as for understanding sociospatial evolution over extended time horizons: first, whether or not documented empirical evidence over the past century or so and for different socioeconomic systems indicates that such systems are susceptible (at present or the recent past) to such violent dynamics (normally associated with fluids in the natural sciences); second, whether or not the future of such systems may be subject to such qualitative behavior, and/or these systems’ dynamics viewed over much more extended time frames than currently documented with census data are likely to be so. Associated with both questions is the existence of large-scale long-term forces, slaving the small-scale and medium or shortterm stock’s spatial interactions. Central in the latter issue is the form these forces take, i.e., the nature of the discrete maps and their robustness over very extended time horizons. Within the same framework, the extent of spatial dominance of these forces is an important subject for analysis; it relates to the question of the spatial disaggregation a specific discrete map might be applicable to. One of the authors [I] has demonstrated that over ranges of a century or so (1890-1980) the u&n relative population dynamics of the U.S. metropolitan sector (at the Standard Metropolitan Statistical Area level) are not temporally close to the turbulent regime of simple logistic discrete maps. The present paper is an attempt to study the U.S. discrete regional relative population dynamics, through empirical evidence employing a discrete map first proposed by the authors [14]. This recent work on nonlinear discrete relative sociospatial dynamics permits linear statistical examination of a family of algorithms depicting the delayed mechanisms in the evolution of relative population abundance in a multiple (subdivisions) spatial framework of a region (nation). Analysis presented here demonstrates that, in the cases of four-region spatial disaggregation, data from 1850 to 1980 (on a decennial step iteration) point to the existence of current long-term and large-scale forces slaving medium-term (decade in length) interactions, and giving rise to a regional population distribution pattern which has the overwhelming share of the U.S. population concentrating in the western region by 2050. When more recent annual counts (up to 1983) are added and the analysis is carried out on a per annum basis from l&50 (and by intrapolating among decennial counts for missing data), then the four-region disaggregation has the population concentrating to a very large extent in the south and to a much lesser extent in the west.
267
Nonlinear Population Dynamics
The large-scale, long-term forces shaping the short-term (annual) regional interactions push back the time horizon in which extinction takes place in ‘the northern region. North, spatially defined either as North-East and NorthCentral (in a four-region disaggregation) or in combination (in a three-region breakdown of the U.S.) always incurs competitive exclusion in population share by 2050 or shortly thereafter. Tests under conditions of random fluctuations in parameter values in all discrete maps used, under a three or four-region breakdown and with one or ten-year period iteration step, indicate that the results are quite robust at all levels of disaggregation. However, the qualitative dynamics of the U.S. regional paths differ significantly when different levels of spatial disaggregation are employed; when the three-region breakdown is juxtaposed with the four-region breakdown the regional relative population shares of south and west seem to follow different dynamic paths. These finds seem to suggest that, short of drastic changes in the environment or significant shifts in current public policies in a manner which would identify a break with past practices (which have been favoring the development of the south and west over the north during the past century), the next one hundred years or so will witness a much more pronounced frostbelt-tosunbelt movement than currently underway. in sociospatial systems, the values of model parameters are not likely to remain unchanged for prolonged time periods (a century or so). Thus, it is not expected that such forecasts will materialize exactly as predicted here. Consequently, these results may say more about the model they emerge from than about the system they intend to simulate. Whatever their worth is, it certainly lies in their ability to depict current forces at work as identified by the past performance of these models. Whether these forces will keep shaping spatial population evolution to the extent suggested, if unchecked, cannot of course be known at present. In the following sections, we first present the discrete map we employ in the analysis and then discuss the empirical findings. Conclusions are presented at the end.
THE UNIVERSAL
DISCRETE
RELATIVE
MAP
Sociospatial dynamics [5, 6, 31 is a new field in geography, regional science, and urban and regional economics. It employs the rich insights from the analytical and computational processes of discrete and continuous maps. Originally drawing from the neighboring field of mathematical ecology [4, 13, 111, and more recently from synergetics ([8,9], [15], and others), it has come of age; it has developed new mathematical models which uncover undetected previously insights-for example those of Dendrinos and Sonis
268
DIMITRIOS S. DENDFUNOS AND MICHAEL SONIS
[5], guided in their search by the path breaking work of Lorenz [lo],
May [12], Feigenbaum [7], and others. A major break with the existing literature on turbulence is that the dynamics in the new field are relative, and they can be disaggregated at will. The dynamical (qualitative and quantitative) features do, however, depend on the level of disaggregation employed. In the concluding section we elaborate on the importance of this result. The basic statement of the universal model is the following specification for the Z-location, one-stock problem:
i,j=1,2
Ai > 0, &=F,[x,(t);
h=1,2
,...) I] >o,
i = 1,2,...,
Z
i=1,2
I.
,...,
>..e, I,
Of particular interest is the log-linear specification of the F-functions in the universal discrete reIative dynamics model:
Analytical and computational properties of the above specification for I = 2, 3, 4, 5, and 10 are given in [5, 61. It is of interest to interpret the parameters and the F-functions of the above model in reference to sociospatial dynamics: the parameters Ai are those over which bifurcations occur in the state-variable (xi) space; they represent technological innovation and diffusion parameters and the effects of various environmental changes. The F,‘s are locationul advantages functions associated with the various regions (or spatial units) for the stock in question; these advantages are due to topographical features, natural-resource endowments, climate, transportation accessibility, and other locational characteristics. The ratios Fi/Fj depict the comparative advantages of region i with respect to region j, and they depend on the position one region occupies relative to another within the national (and intemationaI) space. The parameters aij are spatial elasticities of locational advantages produced in location i, in reference to the relative population abundance at location j.
269
Nonlinear Population Dynanaics
Variations on the bifurcation parameters Ai are exogenous and can be assumed to occur in a continuous or discrete manner, continuously or at random intervals over the medium term. The dynamics of the state variables xi, assumed to occur in the short term, are fast (and endogenous); the changes in the exponents qj are slow (and exogenous) and assumed to occur in the long run in a continuous or discountinuous manner. In this paper, in reference to the U.S. regional population distribution dynamics, these time horizons are specifically identified in terms of actual time periods. The parameter values depict the end result of a vast array of social, economic, political, and other variables; their combined effect is represented in the values of these phenomenological model parameters. For a more detailed discussion of these and associated issues, see [6, 41. The log-linear specifications of the F-functions resemble the CobbDouglas economic production functions found in the theory of the firm in microeconomics. Population abundances are viewed as “producing” locational advantages at any time period t. A major attribute of the above model is that by a simple logarithmic transformation it can be made linear, so that one can test its validity with existing linear regression tests. It is immediately obvious that
where Z is a “numeraire” location in reference to which all other locations’ dynamics are computed. Through the simple substitutions
Xi( t + 1) = In
q(t
+ 1)
x,(t + 1) ’
yj(t) =Inxj(t)
one obtains Xi(t+l)=ivi+&zjiyi(t), j -m
i, j=1,2
,a.*, I,
i, j =1,2 ,**., I,
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DIMITRIOS S. DENDRINOS AND MICHAEL SONIS
a linear system with potentially turbulent behavior in its original variables (xi) for particular parameter values in the Z(Z + l)-dimensional (exponent) parameter space. This paper tests observed orbits of population counts for the U.S. regions, and then it simulates its projected configuration for a 70-year time horizon. Interpretation of the transformed parameters goes as follows: the Ni’s represent technological diffusion differentials (so that if Ai < A, then N, < 0) between region i and the reference region I. If Ai > A j, then it is implied that i’s overall technological level is higher than j ‘s. This factor alone is not sufficient, however, for region i to attract higher population shares than j within a national economy. Smooth changes in these parameters may result in turbulent behavior in the universal discrete-relative-dynamics model. Finally, the exponent a i j represents the differential in comparative-advantage elasticity generated by region j’s population abundance in region i. The above model can be generalized to include J different stocks as well. The twostock, Z-location case is shown below (the reader is directed to [6] for the statement of the Z-stock, Z-location case):
A,F;
i, n = 1,2 ,..., A,>O, Z3=Z+&):h=l,2
)..., I; &):k=l,2,...,J]
i=l,2
gj(t+ l)=
I; y,(t):k=1,2
,...,
T-.-7
1,
,...,
J,
BiHj LJLH,
’
j,m=1,2 ,...,
I...> z,
>o, i=1,2
H,=H,[x,(t):h=1,2
I,
J] >O, j = 1,2 7.3.) I.
The log-linear specifications of the above are
Nonlinear
FIG. 1. Sonis 161.
%l FIG. 2.
Population
Dynumics
271
Strange containers in the three-location, one-stock model. Source: Dendrinos and
Q?_
Strange quasicontainer attractors in the three-location, one-stock model. Source: [6].
272
DIMITRIOS S. DENDRINOS AND MICHAEL SONIS
%? FIG. 3.
Global containers
in the three-location,
one-stock
model. Source:
[6].
0.5
T
0.0
0 FIG. 4.
50
25 Local and partial turbulence
in the three-location,
one-stock
model. Source: [SJ.
Nonlinear
Population
273
Dynamics
where -
mGahi,PkiyYhi, skj\ <+03,
h,i=1,2
,..., I,
k,j=1,2
,...,
1.
Among the many (not all discovered or fully explored) phenomena occurring in the various regions of the (exponent) parameter space of this discrete map, some are the following: strange containers (Figure l), strange quasicontamer attractors (Figure 2), global containers (Figure 3), and local and partial turbulence (Figure 4). These and some additional events, some stemming from the Hopf-bifurcation analogue of the continuous case, can be found in [S, 61. These phenomena will not be discussed further here, as they do not occur in the neighborhood of the parameter space found appropriate for the U.S. regional relative population distribution dynamics.
THE U.S. REGIONS: DATA AVAILABLE AND THEIR DISAGGREGATION A number of different spatial disaggregations of the U.S. regional population structure can be studied, all regions being coterminous with states. There is, however, a limitation imposed on how fine the spatial disaggregation can be, due to the constraints encountered by the available population time series from the U.S. Bureau of the Census.’ Currently, for the period 1850 to 1982, there are 14 decennial counts dating back to 1850, 40 annual counts since 1939 (not including the intervening decennial counts), and from 1920 to 1939 there are two more scattered counts for a total of 56. For all years, in this study, the latest available state population count is always used. Given these data limitations, we had three broad options open, all three partly tested for. First, use all counts on a per annum basis, intrapolating from the decennial counts for missing observations. Second, use only the 14 decennial counts, with an iteration step in the discrete map equal to one decade. Third, use only the 1939 to 1983 annual counts with a step equivalent to one calendar year in the map. The last option would limit the time into the future one could extrapolate with some confidence. Of the three options, the first and second proved to supply us with the statistically most meaningful results, for the four-region and three-region breakdowns. Consequently we report only these two tests in
‘U.S. Bureau of the Census, The Statistical Abstract of the U.S., U.S. Department of Commerce, annual.
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DIMITRIOS
S. DENDRINOS
AND MICHAEL
SONIS
this paper. Some extensions along these lines are suggested in the concluding section of this paper. Three different levels of spatial d&aggregation were considered: a fourregion breakdown of the U.S. (North-East, North-Central, South, and West’), and a three-region breakdown (where the North-East and North-Central regions are combined). A two-region split of the U.S. (where South and West are merged into one superregion) was also examined. It proved to be too aggregative, and the statistical tests were not satisfactory and thus not reported here. Naturally, there are many possible combinations which could be explored, even down to the state level, with an iteration step equal to one year. But with a decade as the time lag, one can only go as far down as the nine divisions. In this case, the statistical estimation procedure would have meant calibrating for 80 parameters (8 + 9 X 8) under 126 observations for a ratio of SO/l26 = 0.63; if a four-region disaggregation is used the ratio is 0.27, significantly favoring that breakdown. The four-region spatial disaggregation was judged to be the finest we could look at given the data limitation and the future target year chosen (2050); it is consistent with the U.S. Bureau of the Census regional breakdown, and it was perceived as the most homogeneous grouping of the U.S. regional heterogeneity. On the subject of geographic subdivision clearly more research is needed. Perturbation of the models’ parameters was carried out in order to test the robustness of the results. Random disturbances were carried out as follows: for each model specification an exogenous number of random events was chosen (ten for the one-year iteration step, and eight for the ten-year step); these events were assigned randomly to various years between 1980 and 2050. From the parameters of the models a number (equal to the number of events) are randomly selected and are perturbed in turn. Their perturbation
‘The U.S. Bureau of the Census defines the divisional composition of the four regions as follows: North-East (New England, Middle Atlantic), North Central (East North Central, West North Central), South (South Atlantic, East South Central, West South Central), West (Mountain, Pacific). It defines the state composition of the nine basic U.S. divisions as follows: New England (Maine, Vermont, New Hampshire, Massachusetts, Rhode Island, Connecticut), Middle Atlantic (New York, Pennsylvania, New Jersey), South Atlantic (Maryland, Delaware, District of Columbia, West Virginia, North and South Carolina, Georgia, Florida), East South Central (Kentucky, Tennessee, Mississippi, Alabama), East North Central (Wisconsin, Illinois, Michigan, Indiana, Ohio), West North Central (North and South Dakota, Minnesota, Iowa, Nebraska, Kansas, Missouri), West South Central (Texas, Oklahoma, Arkansas, Louisiana), Mountain (Montana, Idaho, Wyoming, Colorado, Utah, Nevada, Arizona, New Mexico), Pacific (Washington, Oregon, California, Alaska, Hawaii).
275
Nonlinear Population Dynamics
is a random number generated between 10 percent above and 10 percent below the calibrated value.
THE U.S. REGIONS:
TESTING
OF THE MODEL AND RESULTS
1.a. Four-Region Spatial &aggregation, Ten-Year Time Step Using the SPSS-x package for multiple linear regression with forced entry and default values on the thresholds PIN and POUT, on a Honeywell DPSSE system, the following results are obtained: North-East:
North-Central:
x&t
+ 1) =
0.1432x,,(t)-0~9464x,,(t)0~3450xs(t) F
-0.8312xW(t)
-“.1238
(4,101
south:
zs( t + 1) =
O.O024r,,(t)
-1.8770X&)
-0.8211Xs(t) -1.9075xW(t) -02545 ,
F(4,W
West:
XW( t + 1) =
0.0056x,,(t)
-2.3se1x&)
-1’2720xs(t) -0.9506x&)
-“.5795
F(4,10)
where the value of F(4,10) (i.e., F for four regions and lO-year time step) is simply the sum of the four numerators. For the x&t + 1) equation the multiple R is 0.945 and the value of the F statistic (16.8) is significant at the 0.0006 level; for xs(t + l), R = 0.949, and the F = 18.1 is significant at the 0.0004 level; and for x&t + l), R = 0.944, and F = 168.7 is significant below the 0.0001 level. The simulation results for
276
DIMITRIOS S. DENDRINOS AND MICHAEL SONIS
the period 1850-2050 are shown in Figure 5. On the basis of these results, the western region’s population abundance overpowers all other regions’ relative population size under current (MO- 1980) conditions. The results are mainly driven by the size of the largest exponent in the three discrete dynamic equations, this being the West’s exponent (0.5795) in the xw( t + 1) equation. Differential comparative-advantage elasticities give rise to the dynamics: the West’s total (0.5795 - 0.2545 - 0.1238 = 0.2012) is the highest among the three regions’ totals. It is noteworthy that the North-Central region’s technological diffusion level is much higher than the other two regions’, the South’s lagging far behind and being the lowest. Random perturbations of this model indicate that the results obtained are robust: the Western region is always the long-term winner in the regional competition for population share. 1.b. Four-Region Spatial Disaggregation, One-Year Time Step This case produces the following results:
x&t
+ 1) =
“&
+ 1) =
xw(t
-I- 1) =
0.7296x,,(t)
-1’0310~NC(t)0.Q176~s(t) -0.1104xW(t) -“.0216 F(431)
0.7118x,,(t)
-1.10m~NC(t) -0~-xs(t)0.8w8~W(t)
-“.0173
F’431)
0.9026x,(t)
-
1’03%Nc( t) - 0~1025xs(t)0~0g72xW( t)0’g737 F(421)
7
where the value of Fc4,1) is again the sum of the four numerators. For the x,&t + 1) equation, R = 0.999 and F = 30,854; for xs(t + l), R = 0.996 and F = 3964; and for r&t + l), R = 0.999 and F = 117,086. All three F-statistics are significant below the 0.0001 level. This is the best fit of all cases analyzed, and the simulation results are shown in Figure 6. On the basis of these results the southern region becomes dominant, whereas the western region is significantly reduced in population size but not experiencing competitive exclusion by 2050, as is the case again with the two northern regions (North-East and North-Central).
Nonlinear Population Dynamics
277
1.0 0.0 .
:
:
:
w
0.0
IL
1050
1070
1000
1010
1930
1050
1070
1lloo
2010
2030
2050
FIG. 5. Relative population distribution for four regions and ten-year iteration step. Continuous line: observed dynamics; dotted line: simulated (and forecasted) counts. The North-East (NE), North-Central (NC), and South (S) converge to competitive exclusion, whereas the West (W) attracts the overwhelming share of the US. population.
1050
1370
lW0
1310
J 1330
losO
1370
1820
2010
2030
2030
FIG. 6. Four regions, oneyear iteration step. NE and NC regions converge to competitive exclusion; W rises up to 2636 and then declines drastically, with the overwhelming share of the U.S. population concentrating in S.
278
DIMITRIOS S. DENDRINOS
AND MICHAEL SONIS
The South’s dominance is due to the sum of its exponents ( - 0.1104+ 0.8998+ 0.0972 = O&366), which is higher than any other region’s sum of differential comparative-advantage elasticities. The South’s technological diffusion level is still ranked lowest, but the West now has exceeded that of the North-Central region. A particularly interesting cyclical event is projected in this run: The West initially grows in share quite steeply and at some times even exceeds the South’s share of the U.S. population; however, the South’s comparative advantages by 2050 overwhelm those of the West, forcing it into drastic declines thereafter. Random perturbations of this model’s parameters indicate that the results are not robust. The West and South end up sharing the U.S. population. The North always experiences competitive exclusion. 2.a.
Three-Region
Spatial L&aggregation,
Ten-Year Time Step
Now the results are as follows:
North:
4+1)=&g. south: xs( t + 1) =
0.0236x,(t)
-z~oso7xs(t) - 1’4444xW(t) -“‘= F(3,10)
West: 0.1747x,(t) x,(t
a)
=
-2.8440"s(t) F’32
-0~1Qg5xw(t)0~7072
10)
where the value of F(3,1”) for this three-region, ten-year step is now the sum of the three numerators. For xs we have R = 0.930, F = 19.2 (significant at 0.0063 level); for xw we have R = 0.992, F = 187.96 (below 0.0001 level); the calibration phase and the corresponding projections are shown in Figure 7. As in the previous cases, the results obtained-competitive exclusion for the North and overwhelming concentration in the West- are due to the relative magnitudes of the a i j’s, that is, the differentials in comparative-advantage elasticities generated by region f’s relative population size on region i. Here the North’s exponent ( - 2.0807) is negative and larger by far in absolute value than the South’s ( - 1.4444) and the West’s (0.1530) in the South’s dynamics [ x,(t + 1) versus xs( t ), xw(t ), xN( t )]; similar is the case in the West’s dynamics, where now the West’s exponent (0.7072) is positive.
Nonlinear PopulationDynamics
279
.’ : : .:.
‘.
1
;
‘.
, ‘.
:
FIG. 7. Three regions, ten-year iteration step. N and S converge to competitive exclusion; W absorbs the population.
A medium-run qualitative event is depicted in the projections: around the year 2010 the three regions obtain almost a uniform population distribution. This result may be used as a check on the validity of these models’ projections. Random fluctuations in this model’s parameters indicate robustness of the results, as the western region comes up the winer in the interregional population competition. 2.b. Three-Region Spatial Disaggregation, One-Year Time Step Here the results obtained have the following form: 1
%J(t+ 1)= F(3,1)’ xs(
x,(t
t +
1) =
+ 1) =
0.9469x,(t)
-1.0316rs(t)0.9594XW(t)0.M)27 F(3.1)
1.2156x,(t)
>
-1.0810XS(t)0.22181W(t)0.gs~ F(34
>
where the value of Fc3*‘) is again the sum of the three numerators.
280
DIMITBIOS S. DENDRINOS AND MICHAEL SONIS
0.0 1350
1570
1390
1010
1930
1050
1970
1ss0
2010
2030
2050
FIG. 8. Three regions, one-year iteration step. Results are similar to Figure 7, except that now the decline of the South has been moderated, and the North’s abandonment extended beyond 2050.
For xs we have R = 0.995, F = 4404; for xw we have R = 0.999, F = 137,971; both F’s are significant at a level below 0.0001. The high exponent for the West (0.9853) and its high technological diffusion coefficient (1.2156) pull population into this region (Figure S), although again (as in case 1.b) only the North’s relative population size tends to competitive exclusion, approaching it well beyond 2050. Of all the projections, this one seems the mot likely. Even in this case, if by 2650 the U.S. population reaches 400 million (not unlikely according to the U.S. Bureau of the Census projections3), then the West will contain about 320 million people, and the density will be approximately 180 persons per square mile, By comparison, the 1983 population density of Japan was close to 125 persons per square mile, that of India 86, that of the Netherlands 136, and that of the U.S. about 10. Bangladesh at the same time was experiencing the worlds highest population density, about 5365 persons per square mile.
3U.S. Department of Commerce, Bureau of the Census, Statistical Abstmct of the U.S., 1975, Figure l-1, p. 4, Series I, II, II-X, III. It is projected according to Series I that by 2050 the U.S. population will be 500 million; according to Series II, 325 million; so that 400 million falls in between.
Nonlinear Population LIynumics
281
11
1050
1070
1090
FIG. 9.
1010
1030
The the-region
1050
1970
lsw
ZOlO
ZWO
ZOO0
timeseries linear regression model.
Model perturbations seem to indicate some degree of robustness in this case: the West’s population share of 2050 exceeds that of the South, whose exceeds that of the North; no region experiences competitive exclusion. Similarly to case 2.a, the three regions seem to experience uniform population distribution around 2000, slightly earlier than the previous model’s projections indicate. Having presented the tests from the discrete maps, we now supply results from running simple time-series regressions on the threeregion breakdown with 56 observations. They provide an alternative with which one can compare the nonlinear model specifications. The following models were found to apply (Figure 9):
XN( T) = 3.09931so that at T = 2050 we have x,(T (significant at < 0.0001 level). south:
O.O0131T,
= 2050) = 0.4138,
xs( T) = 0.69547 - 0.000194T,
R2 = 0.343, F = 289.9
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DIMITRIOS S. DENDRINOS AND MICHAEL
so that at T = 2050 we have (significant at 0.0008 level). West:
SONIS
x,(T= 2050) = 0.2978, R2 = 0.434, F = 12.5
xW(T) = - 2.79437 + O.O015T, so that at T = 2050 we have x,(T = 2050) = 0.2806, (significant at < 0.0001 level).
R2 = 0.973, F = 1970
The low level statistics for the southern region are due to the almost horizontal regression line. In all, the statistics are lower than in the nonlinearly specified model, although within acceptable ranges. Projecting to 2050 and summing the three regions’ shares, one obtains a total (0.9922) very close to satisfying the conservation condition (rN + xs + x w = 1.00). This cannot hold, however, for extended time horizons. It is this constraint which ultimately limits the use of this linear model, in spite of its good performance over a 7@year time horizon (1980-2050) and its calibration period (18.50-1983). A noteworthy observation from the three-region breakdown history in relative populationdynamic paths over the 1850-1983 period is the following: during this time span a host of events took place nationwide and in each region-significant technological change, drastic socioeconomic shifts, significant environmental fluctuations (for example, a number of wars), etc. In spite of all these factors, the paths of the three regions as recorded indicate very smooth, nonrandom motion. It may be that by using the relative lens one sees certain constants in population dynamics not observable under the absolute lens.
CONCLUSIONS
AND SUGGESTIONS
FOR FUTURE
RESEARCH
All four log-linear models (and the linear one as well) embody unstable equilibria in the relative population dynamics of the U.S. regions. Competitive exclusion in one or more regions is the outcome around 2050, when two regional (spatial) disaggregation schemes and two time scales for each scheme are tested using 130 years of observation to calibrate the models’ parameters. Tests employing the annual step show the South absorbing the vast share of the U.S. population, with the West anticipated by 2050 to hold on to about a &percent share of the population (1.b) under a four-region breakdown; when the one-year step is used, under the three-region spatial disaggregation of the U.S., the West is found to have a share by 2050 of about 0.848, and the South approximately 0.149 leaving the North with a 0.003 share (2.b).
Nonlinear Population Dynamics
283
Statistics using the one-year step size were better than those obtained from using the ten-year time lag. Both were far superior to using simple linear time-dependent regression models for the various regions. When the decennial step size is employed, the West absorbs almost all of the U.S. population by 2040 (l.a), whereas both the North (in the NorthEast-North-Central spatial disaggregation) and the South experience competitive exclusion. When a three-region breakdown is used (with the North as the sum of North-East and North-Central) a decennial step has the West by 2050 accumulating a 0.989 share of the U.S. population, the rest being located in the South, while the North exhibits competitive exclusion. The fact that the oneyear step size result in a better fit (and more likely projections) than the ten-year step size may suggest that the forces shaping regional population distributions manifest themselves in regional interactions spanning a very short time span. In spite of their internal variety and complexity, comprising many sociological, political, economic, and other components, a simple one-year-step dynamic interdependence on their relative population stock siLe is enough to depict their effect. Along the same lines, simple difference equations and a very narrow band in their parameter. space accurately replicate the effect of these forces over extended time horizons: more than a century (1850-1983). In turn this may suggest the presence, over this time span, of very strong and selective processes operating in the dynamics of relative spatial population distribution. These processes, depicted by these simple dynamical equations, could be view-ed as a “code” for the development schemes of these relative population distributions. Experiments employing random fluctuations in these parameter values (checking the dynamic stability of the code) result in qualitatively similar results in most cases, thus indicating that the dynamic instability characterizing regional population interactions at these levels of spatial disaggregation is qualitatively robust. Different spatial disaggregations for the northern region alone result in forecasts which have qualitatively significantly different implications for the southern and western regions (identical spatially under the two areal disaggregation schemes); and different real time scales involved in the iterations of the discrete map result in qualitatively different forecasts. These two findings point to the existence of robustnessin model, beyond the robustness found in the parameters of all specific models tested. The different spatial and temporal scales of analysis imply different spatial and temporal forces at work at any time period. Which one of these forces (i.e., which model from the family of models presented) may prevail in the future is at present unknown. Another interesting finding is that various models (with different spatial and temporal scales) of one family (the log-linear), as well as linear models
284
DIMITRIOS S. DENDRINOS AND MICHAEL SONIS
(involving only time dependence), produce acceptable (and indeed good) fit over an extended time period (in this instance a l%@year horizon). All have different forecasting implications. The uneasy realization that in social sciences different theories can be tested on identical data sets and found acceptable is confirmed by the experiments presented in this paper. Consequently, these results cannot and should not be treated as forecasts; the nonlinear dynamics of sociospatial systems imply that only qualitative statements can be made regarding alternative scenarios of future sociospatial configurations, not precise quantitative statements [4]. However, these simulations provide some strong evidence regarding the agglomeration and deglomeration forces present over the past 13@year period in the U.S., shaping the current regional allocation of population patterns. The nature of the large-scale, long-term forces slaving either the ten-year or one-year interactions among the U.S. regions, along the same lines as speculated in previous work by Dendrinos (with Mullally) [4] for urban agglomerations, may consist of an aggregate economicdemographic force operating over the nation’s landscape within time frameworks of one century or so. Per capita income and relative population interactions seem to depict the effect of this force at the nine-region spatial disaggregation [2]. Further work is needed to identify this force and its effects among different levels of regional spatial disaggregation. As suggestions for future research two kinds of tests immediately come to mind: first, to test the state composition of the subdivisions, at the margin (by moving one or two states from each subdivision to another); second, to randomly select states in each subdivision and test the model’s validity. The first test will test the hypothesis of homogeneity within subdivisions (i.e., whether or not these particular subdivisions are meaningful for testing an otherwise acceptable model). The second test will test whether this discrete map is meaningful altogether. Although the originally stated system of the Z-location, one-stock model can produce turbulence, the range of parameter values depicting current regional relative population distribution in the U.S., and extended over more than one century of evolution, is far from the ranges of such behavior. Thus, the allocation patterns of relative population in the U.S. are calm and not expected to produce chaotic motion at these levels of spatial disaggregation. However, much finer levels of regional breakdown (well below the nine-division level [2]) cannot be excluded from doing so. What is deduced is in fact quite unsettling, because it indicates that regional spatial dynamics are unstable. Only one stock (relative population) was considered in this model. Clearly, regions contain many stocks: wealth, built capital stock, natural resources, and others. Lack of data inhibits one from incorporating these stocks into the
Nonlinear Population Dynatnics
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dynamics analyzed at present. Future time series may allow one to examine the hypothesis that the dynamics of these stocks depend on, and do not shape, population dynamics.
REFERENCES 1
2 3
4 5 6 7 8 9 10 11 12 13 14
15
D. S. Dendrinos, Turbulence and fundamental urban/regional dynamics, presented at the Annual Meeting of the American Association of Geographers, Washington, Apr. 1983, and at the Theoretical Task Force Meeting on Dynamic Analysis of Spatial Development, IIASA, Luxenburg, Austria, Oct. 1984; Proceedings, to appear. D. S. Dendrinos, Regions, antiregions and their dynamic stability: The case of the U.S. (1929-79), 1. Regional Sci. 24(1):65-84 (1984). D. S. Dendrinos, Ecological Studies of Urban Nonlinear Dynamics, Report SES-82-16620, National Science Foundation, Decision and Management Science Division, 1985. D. S. Dendrinos (with H. Mullally), Urban Euolution: Studies in the Mutkmutical Ecology of Cities, Oxford U.P., 1985. D. S. Dendrinos and M. Sonis, The onset of turbulence in discrete relative multiple spatial dynamics, J. App. Math. Cotnp. 22:25-44 (1987) D. S. Dendrinos and M. Sonis, Turbulence and Sock-Spatial Dynamics: Toward a Structural Theory of Social Systems Euolutim, Springer, 1988. M. J. Feigenbaum, Quantitative universality for a class of non-linear transformations, 1. Statist. Phys. 19:25-52 (1978). H. Haken (Ed.), Chaos and Order in Nature, Series on Synergetics, Springer, 1981. H. Haken, Synergetics: An Introduction, 3rd ed., Series on Synergetics, Springer, 1983. E. N. Lorenz, Deterministic non-periodic flows, I. Atmospheric Sci. 20:130-141 (1963). R. L. May, Stability and Complexity in M&l Ecosystems, Princeton U.P., 1974. R. L. May, Simple mathematical models with very complicated dynamics, Nature 261:459-467 (1976). R. H. McArthur, Geographical Ecology, Harper and Row, 1972. M. Sonis and D. S. Dendrinos, A discrete relative growth model: Switching, role reversal and turbulence, in P. Friedrich and I. Masser (eds.) International Perspectives of Regional Decentralization, Nomos, Baden Baden, 1987. W. Weidlich and G. Haag, Quantitative Sociology, Springer, 1983.
S. Dendrinosf
Urban Planning Program University Lawrence,
of Kansas Kansas 66045
and Michael
Sonis
Ear-llan University 52-100 Ramat-Gan,
Transmitted
Israel
by John Casti
ABSTRACT A new discrete relative-dynamics algorithm is tested using population interactions of U.S. regions. A K&year time period is used (1850-1983) to calibrate the log-linear model’s parameters using standard linear regression tests. We also report certain forecasts under different spatid disaggregation schemes of the U.S. regions, and under two different time-period steps (iterations) to the year 2050. AU forecasts impIy competitive exclusion for the northern regions; under certain areal disaggregations and time (iteration) steps the southern or western region also experiences conditions leading to drastic declines in their share of the U.S. population. We also report the results of certain random fluctuations on the model’s parameters and their implications regarding the dynamics of the U.S. regional population structure. The outcomes seem to be robust under these fluctuations.
INTRODUCTION Mathematical properties of fixed-point discrete maps and recent associated developments in bifurcation theory have opened up new avenues for research in the social sciences in general and spatial fields (including geography and
*Part of this research was done under contract number SE.%%16620 with the National Science Foundation, which is gratefully acknowledged, by D. S. Dendrinos. f This author wishes to thank his research assistant Mehrdad Givechi for the computer work.
APPLIED MATHEMATICS AND COMPUTATlON
25:265-285
(1988)
265
0 Elsevier Science Publishing Co., Inc., 1988 52 Vanderbilt Ave., New York, NY 10017
009&3003/88/$03.50
266
DIMITRIOS S. DENDRINOS AND MICHAEL SONIS
regional science) in particular [5, 61. These maps include the possibility of obtaining turbulent behavior in the dynamics of sociospatial stocks, involving perioddoubling cycles and deterministic chaos. Two questions have consequently emerged out of the recent analysis that are of extreme importance with regard to the relevance of these discrete maps for sociospatial dynamics, as well as for understanding sociospatial evolution over extended time horizons: first, whether or not documented empirical evidence over the past century or so and for different socioeconomic systems indicates that such systems are susceptible (at present or the recent past) to such violent dynamics (normally associated with fluids in the natural sciences); second, whether or not the future of such systems may be subject to such qualitative behavior, and/or these systems’ dynamics viewed over much more extended time frames than currently documented with census data are likely to be so. Associated with both questions is the existence of large-scale long-term forces, slaving the small-scale and medium or shortterm stock’s spatial interactions. Central in the latter issue is the form these forces take, i.e., the nature of the discrete maps and their robustness over very extended time horizons. Within the same framework, the extent of spatial dominance of these forces is an important subject for analysis; it relates to the question of the spatial disaggregation a specific discrete map might be applicable to. One of the authors [I] has demonstrated that over ranges of a century or so (1890-1980) the u&n relative population dynamics of the U.S. metropolitan sector (at the Standard Metropolitan Statistical Area level) are not temporally close to the turbulent regime of simple logistic discrete maps. The present paper is an attempt to study the U.S. discrete regional relative population dynamics, through empirical evidence employing a discrete map first proposed by the authors [14]. This recent work on nonlinear discrete relative sociospatial dynamics permits linear statistical examination of a family of algorithms depicting the delayed mechanisms in the evolution of relative population abundance in a multiple (subdivisions) spatial framework of a region (nation). Analysis presented here demonstrates that, in the cases of four-region spatial disaggregation, data from 1850 to 1980 (on a decennial step iteration) point to the existence of current long-term and large-scale forces slaving medium-term (decade in length) interactions, and giving rise to a regional population distribution pattern which has the overwhelming share of the U.S. population concentrating in the western region by 2050. When more recent annual counts (up to 1983) are added and the analysis is carried out on a per annum basis from l&50 (and by intrapolating among decennial counts for missing data), then the four-region disaggregation has the population concentrating to a very large extent in the south and to a much lesser extent in the west.
267
Nonlinear Population Dynamics
The large-scale, long-term forces shaping the short-term (annual) regional interactions push back the time horizon in which extinction takes place in ‘the northern region. North, spatially defined either as North-East and NorthCentral (in a four-region disaggregation) or in combination (in a three-region breakdown of the U.S.) always incurs competitive exclusion in population share by 2050 or shortly thereafter. Tests under conditions of random fluctuations in parameter values in all discrete maps used, under a three or four-region breakdown and with one or ten-year period iteration step, indicate that the results are quite robust at all levels of disaggregation. However, the qualitative dynamics of the U.S. regional paths differ significantly when different levels of spatial disaggregation are employed; when the three-region breakdown is juxtaposed with the four-region breakdown the regional relative population shares of south and west seem to follow different dynamic paths. These finds seem to suggest that, short of drastic changes in the environment or significant shifts in current public policies in a manner which would identify a break with past practices (which have been favoring the development of the south and west over the north during the past century), the next one hundred years or so will witness a much more pronounced frostbelt-tosunbelt movement than currently underway. in sociospatial systems, the values of model parameters are not likely to remain unchanged for prolonged time periods (a century or so). Thus, it is not expected that such forecasts will materialize exactly as predicted here. Consequently, these results may say more about the model they emerge from than about the system they intend to simulate. Whatever their worth is, it certainly lies in their ability to depict current forces at work as identified by the past performance of these models. Whether these forces will keep shaping spatial population evolution to the extent suggested, if unchecked, cannot of course be known at present. In the following sections, we first present the discrete map we employ in the analysis and then discuss the empirical findings. Conclusions are presented at the end.
THE UNIVERSAL
DISCRETE
RELATIVE
MAP
Sociospatial dynamics [5, 6, 31 is a new field in geography, regional science, and urban and regional economics. It employs the rich insights from the analytical and computational processes of discrete and continuous maps. Originally drawing from the neighboring field of mathematical ecology [4, 13, 111, and more recently from synergetics ([8,9], [15], and others), it has come of age; it has developed new mathematical models which uncover undetected previously insights-for example those of Dendrinos and Sonis
268
DIMITRIOS S. DENDFUNOS AND MICHAEL SONIS
[5], guided in their search by the path breaking work of Lorenz [lo],
May [12], Feigenbaum [7], and others. A major break with the existing literature on turbulence is that the dynamics in the new field are relative, and they can be disaggregated at will. The dynamical (qualitative and quantitative) features do, however, depend on the level of disaggregation employed. In the concluding section we elaborate on the importance of this result. The basic statement of the universal model is the following specification for the Z-location, one-stock problem:
i,j=1,2
Ai > 0, &=F,[x,(t);
h=1,2
,...) I] >o,
i = 1,2,...,
Z
i=1,2
I.
,...,
>..e, I,
Of particular interest is the log-linear specification of the F-functions in the universal discrete reIative dynamics model:
Analytical and computational properties of the above specification for I = 2, 3, 4, 5, and 10 are given in [5, 61. It is of interest to interpret the parameters and the F-functions of the above model in reference to sociospatial dynamics: the parameters Ai are those over which bifurcations occur in the state-variable (xi) space; they represent technological innovation and diffusion parameters and the effects of various environmental changes. The F,‘s are locationul advantages functions associated with the various regions (or spatial units) for the stock in question; these advantages are due to topographical features, natural-resource endowments, climate, transportation accessibility, and other locational characteristics. The ratios Fi/Fj depict the comparative advantages of region i with respect to region j, and they depend on the position one region occupies relative to another within the national (and intemationaI) space. The parameters aij are spatial elasticities of locational advantages produced in location i, in reference to the relative population abundance at location j.
269
Nonlinear Population Dynanaics
Variations on the bifurcation parameters Ai are exogenous and can be assumed to occur in a continuous or discrete manner, continuously or at random intervals over the medium term. The dynamics of the state variables xi, assumed to occur in the short term, are fast (and endogenous); the changes in the exponents qj are slow (and exogenous) and assumed to occur in the long run in a continuous or discountinuous manner. In this paper, in reference to the U.S. regional population distribution dynamics, these time horizons are specifically identified in terms of actual time periods. The parameter values depict the end result of a vast array of social, economic, political, and other variables; their combined effect is represented in the values of these phenomenological model parameters. For a more detailed discussion of these and associated issues, see [6, 41. The log-linear specifications of the F-functions resemble the CobbDouglas economic production functions found in the theory of the firm in microeconomics. Population abundances are viewed as “producing” locational advantages at any time period t. A major attribute of the above model is that by a simple logarithmic transformation it can be made linear, so that one can test its validity with existing linear regression tests. It is immediately obvious that
where Z is a “numeraire” location in reference to which all other locations’ dynamics are computed. Through the simple substitutions
Xi( t + 1) = In
q(t
+ 1)
x,(t + 1) ’
yj(t) =Inxj(t)
one obtains Xi(t+l)=ivi+&zjiyi(t), j -m
i, j=1,2
,a.*, I,
i, j =1,2 ,**., I,
270
DIMITRIOS S. DENDRINOS AND MICHAEL SONIS
a linear system with potentially turbulent behavior in its original variables (xi) for particular parameter values in the Z(Z + l)-dimensional (exponent) parameter space. This paper tests observed orbits of population counts for the U.S. regions, and then it simulates its projected configuration for a 70-year time horizon. Interpretation of the transformed parameters goes as follows: the Ni’s represent technological diffusion differentials (so that if Ai < A, then N, < 0) between region i and the reference region I. If Ai > A j, then it is implied that i’s overall technological level is higher than j ‘s. This factor alone is not sufficient, however, for region i to attract higher population shares than j within a national economy. Smooth changes in these parameters may result in turbulent behavior in the universal discrete-relative-dynamics model. Finally, the exponent a i j represents the differential in comparative-advantage elasticity generated by region j’s population abundance in region i. The above model can be generalized to include J different stocks as well. The twostock, Z-location case is shown below (the reader is directed to [6] for the statement of the Z-stock, Z-location case):
A,F;
i, n = 1,2 ,..., A,>O, Z3=Z+&):h=l,2
)..., I; &):k=l,2,...,J]
i=l,2
gj(t+ l)=
I; y,(t):k=1,2
,...,
T-.-7
1,
,...,
J,
BiHj LJLH,
’
j,m=1,2 ,...,
I...> z,
>o, i=1,2
H,=H,[x,(t):h=1,2
I,
J] >O, j = 1,2 7.3.) I.
The log-linear specifications of the above are
Nonlinear
FIG. 1. Sonis 161.
%l FIG. 2.
Population
Dynumics
271
Strange containers in the three-location, one-stock model. Source: Dendrinos and
Q?_
Strange quasicontainer attractors in the three-location, one-stock model. Source: [6].
272
DIMITRIOS S. DENDRINOS AND MICHAEL SONIS
%? FIG. 3.
Global containers
in the three-location,
one-stock
model. Source:
[6].
0.5
T
0.0
0 FIG. 4.
50
25 Local and partial turbulence
in the three-location,
one-stock
model. Source: [SJ.
Nonlinear
Population
273
Dynamics
where -
mGahi,PkiyYhi, skj\ <+03,
h,i=1,2
,..., I,
k,j=1,2
,...,
1.
Among the many (not all discovered or fully explored) phenomena occurring in the various regions of the (exponent) parameter space of this discrete map, some are the following: strange containers (Figure l), strange quasicontamer attractors (Figure 2), global containers (Figure 3), and local and partial turbulence (Figure 4). These and some additional events, some stemming from the Hopf-bifurcation analogue of the continuous case, can be found in [S, 61. These phenomena will not be discussed further here, as they do not occur in the neighborhood of the parameter space found appropriate for the U.S. regional relative population distribution dynamics.
THE U.S. REGIONS: DATA AVAILABLE AND THEIR DISAGGREGATION A number of different spatial disaggregations of the U.S. regional population structure can be studied, all regions being coterminous with states. There is, however, a limitation imposed on how fine the spatial disaggregation can be, due to the constraints encountered by the available population time series from the U.S. Bureau of the Census.’ Currently, for the period 1850 to 1982, there are 14 decennial counts dating back to 1850, 40 annual counts since 1939 (not including the intervening decennial counts), and from 1920 to 1939 there are two more scattered counts for a total of 56. For all years, in this study, the latest available state population count is always used. Given these data limitations, we had three broad options open, all three partly tested for. First, use all counts on a per annum basis, intrapolating from the decennial counts for missing observations. Second, use only the 14 decennial counts, with an iteration step in the discrete map equal to one decade. Third, use only the 1939 to 1983 annual counts with a step equivalent to one calendar year in the map. The last option would limit the time into the future one could extrapolate with some confidence. Of the three options, the first and second proved to supply us with the statistically most meaningful results, for the four-region and three-region breakdowns. Consequently we report only these two tests in
‘U.S. Bureau of the Census, The Statistical Abstract of the U.S., U.S. Department of Commerce, annual.
274
DIMITRIOS
S. DENDRINOS
AND MICHAEL
SONIS
this paper. Some extensions along these lines are suggested in the concluding section of this paper. Three different levels of spatial d&aggregation were considered: a fourregion breakdown of the U.S. (North-East, North-Central, South, and West’), and a three-region breakdown (where the North-East and North-Central regions are combined). A two-region split of the U.S. (where South and West are merged into one superregion) was also examined. It proved to be too aggregative, and the statistical tests were not satisfactory and thus not reported here. Naturally, there are many possible combinations which could be explored, even down to the state level, with an iteration step equal to one year. But with a decade as the time lag, one can only go as far down as the nine divisions. In this case, the statistical estimation procedure would have meant calibrating for 80 parameters (8 + 9 X 8) under 126 observations for a ratio of SO/l26 = 0.63; if a four-region disaggregation is used the ratio is 0.27, significantly favoring that breakdown. The four-region spatial disaggregation was judged to be the finest we could look at given the data limitation and the future target year chosen (2050); it is consistent with the U.S. Bureau of the Census regional breakdown, and it was perceived as the most homogeneous grouping of the U.S. regional heterogeneity. On the subject of geographic subdivision clearly more research is needed. Perturbation of the models’ parameters was carried out in order to test the robustness of the results. Random disturbances were carried out as follows: for each model specification an exogenous number of random events was chosen (ten for the one-year iteration step, and eight for the ten-year step); these events were assigned randomly to various years between 1980 and 2050. From the parameters of the models a number (equal to the number of events) are randomly selected and are perturbed in turn. Their perturbation
‘The U.S. Bureau of the Census defines the divisional composition of the four regions as follows: North-East (New England, Middle Atlantic), North Central (East North Central, West North Central), South (South Atlantic, East South Central, West South Central), West (Mountain, Pacific). It defines the state composition of the nine basic U.S. divisions as follows: New England (Maine, Vermont, New Hampshire, Massachusetts, Rhode Island, Connecticut), Middle Atlantic (New York, Pennsylvania, New Jersey), South Atlantic (Maryland, Delaware, District of Columbia, West Virginia, North and South Carolina, Georgia, Florida), East South Central (Kentucky, Tennessee, Mississippi, Alabama), East North Central (Wisconsin, Illinois, Michigan, Indiana, Ohio), West North Central (North and South Dakota, Minnesota, Iowa, Nebraska, Kansas, Missouri), West South Central (Texas, Oklahoma, Arkansas, Louisiana), Mountain (Montana, Idaho, Wyoming, Colorado, Utah, Nevada, Arizona, New Mexico), Pacific (Washington, Oregon, California, Alaska, Hawaii).
275
Nonlinear Population Dynamics
is a random number generated between 10 percent above and 10 percent below the calibrated value.
THE U.S. REGIONS:
TESTING
OF THE MODEL AND RESULTS
1.a. Four-Region Spatial &aggregation, Ten-Year Time Step Using the SPSS-x package for multiple linear regression with forced entry and default values on the thresholds PIN and POUT, on a Honeywell DPSSE system, the following results are obtained: North-East:
North-Central:
x&t
+ 1) =
0.1432x,,(t)-0~9464x,,(t)0~3450xs(t) F
-0.8312xW(t)
-“.1238
(4,101
south:
zs( t + 1) =
O.O024r,,(t)
-1.8770X&)
-0.8211Xs(t) -1.9075xW(t) -02545 ,
F(4,W
West:
XW( t + 1) =
0.0056x,,(t)
-2.3se1x&)
-1’2720xs(t) -0.9506x&)
-“.5795
F(4,10)
where the value of F(4,10) (i.e., F for four regions and lO-year time step) is simply the sum of the four numerators. For the x&t + 1) equation the multiple R is 0.945 and the value of the F statistic (16.8) is significant at the 0.0006 level; for xs(t + l), R = 0.949, and the F = 18.1 is significant at the 0.0004 level; and for x&t + l), R = 0.944, and F = 168.7 is significant below the 0.0001 level. The simulation results for
276
DIMITRIOS S. DENDRINOS AND MICHAEL SONIS
the period 1850-2050 are shown in Figure 5. On the basis of these results, the western region’s population abundance overpowers all other regions’ relative population size under current (MO- 1980) conditions. The results are mainly driven by the size of the largest exponent in the three discrete dynamic equations, this being the West’s exponent (0.5795) in the xw( t + 1) equation. Differential comparative-advantage elasticities give rise to the dynamics: the West’s total (0.5795 - 0.2545 - 0.1238 = 0.2012) is the highest among the three regions’ totals. It is noteworthy that the North-Central region’s technological diffusion level is much higher than the other two regions’, the South’s lagging far behind and being the lowest. Random perturbations of this model indicate that the results obtained are robust: the Western region is always the long-term winner in the regional competition for population share. 1.b. Four-Region Spatial Disaggregation, One-Year Time Step This case produces the following results:
x&t
+ 1) =
“&
+ 1) =
xw(t
-I- 1) =
0.7296x,,(t)
-1’0310~NC(t)0.Q176~s(t) -0.1104xW(t) -“.0216 F(431)
0.7118x,,(t)
-1.10m~NC(t) -0~-xs(t)0.8w8~W(t)
-“.0173
F’431)
0.9026x,(t)
-
1’03%Nc( t) - 0~1025xs(t)0~0g72xW( t)0’g737 F(421)
7
where the value of Fc4,1) is again the sum of the four numerators. For the x,&t + 1) equation, R = 0.999 and F = 30,854; for xs(t + l), R = 0.996 and F = 3964; and for r&t + l), R = 0.999 and F = 117,086. All three F-statistics are significant below the 0.0001 level. This is the best fit of all cases analyzed, and the simulation results are shown in Figure 6. On the basis of these results the southern region becomes dominant, whereas the western region is significantly reduced in population size but not experiencing competitive exclusion by 2050, as is the case again with the two northern regions (North-East and North-Central).
Nonlinear Population Dynamics
277
1.0 0.0 .
:
:
:
w
0.0
IL
1050
1070
1000
1010
1930
1050
1070
1lloo
2010
2030
2050
FIG. 5. Relative population distribution for four regions and ten-year iteration step. Continuous line: observed dynamics; dotted line: simulated (and forecasted) counts. The North-East (NE), North-Central (NC), and South (S) converge to competitive exclusion, whereas the West (W) attracts the overwhelming share of the US. population.
1050
1370
lW0
1310
J 1330
losO
1370
1820
2010
2030
2030
FIG. 6. Four regions, oneyear iteration step. NE and NC regions converge to competitive exclusion; W rises up to 2636 and then declines drastically, with the overwhelming share of the U.S. population concentrating in S.
278
DIMITRIOS S. DENDRINOS
AND MICHAEL SONIS
The South’s dominance is due to the sum of its exponents ( - 0.1104+ 0.8998+ 0.0972 = O&366), which is higher than any other region’s sum of differential comparative-advantage elasticities. The South’s technological diffusion level is still ranked lowest, but the West now has exceeded that of the North-Central region. A particularly interesting cyclical event is projected in this run: The West initially grows in share quite steeply and at some times even exceeds the South’s share of the U.S. population; however, the South’s comparative advantages by 2050 overwhelm those of the West, forcing it into drastic declines thereafter. Random perturbations of this model’s parameters indicate that the results are not robust. The West and South end up sharing the U.S. population. The North always experiences competitive exclusion. 2.a.
Three-Region
Spatial L&aggregation,
Ten-Year Time Step
Now the results are as follows:
North:
4+1)=&g. south: xs( t + 1) =
0.0236x,(t)
-z~oso7xs(t) - 1’4444xW(t) -“‘= F(3,10)
West: 0.1747x,(t) x,(t
a)
=
-2.8440"s(t) F’32
-0~1Qg5xw(t)0~7072
10)
where the value of F(3,1”) for this three-region, ten-year step is now the sum of the three numerators. For xs we have R = 0.930, F = 19.2 (significant at 0.0063 level); for xw we have R = 0.992, F = 187.96 (below 0.0001 level); the calibration phase and the corresponding projections are shown in Figure 7. As in the previous cases, the results obtained-competitive exclusion for the North and overwhelming concentration in the West- are due to the relative magnitudes of the a i j’s, that is, the differentials in comparative-advantage elasticities generated by region f’s relative population size on region i. Here the North’s exponent ( - 2.0807) is negative and larger by far in absolute value than the South’s ( - 1.4444) and the West’s (0.1530) in the South’s dynamics [ x,(t + 1) versus xs( t ), xw(t ), xN( t )]; similar is the case in the West’s dynamics, where now the West’s exponent (0.7072) is positive.
Nonlinear PopulationDynamics
279
.’ : : .:.
‘.
1
;
‘.
, ‘.
:
FIG. 7. Three regions, ten-year iteration step. N and S converge to competitive exclusion; W absorbs the population.
A medium-run qualitative event is depicted in the projections: around the year 2010 the three regions obtain almost a uniform population distribution. This result may be used as a check on the validity of these models’ projections. Random fluctuations in this model’s parameters indicate robustness of the results, as the western region comes up the winer in the interregional population competition. 2.b. Three-Region Spatial Disaggregation, One-Year Time Step Here the results obtained have the following form: 1
%J(t+ 1)= F(3,1)’ xs(
x,(t
t +
1) =
+ 1) =
0.9469x,(t)
-1.0316rs(t)0.9594XW(t)0.M)27 F(3.1)
1.2156x,(t)
>
-1.0810XS(t)0.22181W(t)0.gs~ F(34
>
where the value of Fc3*‘) is again the sum of the three numerators.
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DIMITBIOS S. DENDRINOS AND MICHAEL SONIS
0.0 1350
1570
1390
1010
1930
1050
1970
1ss0
2010
2030
2050
FIG. 8. Three regions, one-year iteration step. Results are similar to Figure 7, except that now the decline of the South has been moderated, and the North’s abandonment extended beyond 2050.
For xs we have R = 0.995, F = 4404; for xw we have R = 0.999, F = 137,971; both F’s are significant at a level below 0.0001. The high exponent for the West (0.9853) and its high technological diffusion coefficient (1.2156) pull population into this region (Figure S), although again (as in case 1.b) only the North’s relative population size tends to competitive exclusion, approaching it well beyond 2050. Of all the projections, this one seems the mot likely. Even in this case, if by 2650 the U.S. population reaches 400 million (not unlikely according to the U.S. Bureau of the Census projections3), then the West will contain about 320 million people, and the density will be approximately 180 persons per square mile, By comparison, the 1983 population density of Japan was close to 125 persons per square mile, that of India 86, that of the Netherlands 136, and that of the U.S. about 10. Bangladesh at the same time was experiencing the worlds highest population density, about 5365 persons per square mile.
3U.S. Department of Commerce, Bureau of the Census, Statistical Abstmct of the U.S., 1975, Figure l-1, p. 4, Series I, II, II-X, III. It is projected according to Series I that by 2050 the U.S. population will be 500 million; according to Series II, 325 million; so that 400 million falls in between.
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11
1050
1070
1090
FIG. 9.
1010
1030
The the-region
1050
1970
lsw
ZOlO
ZWO
ZOO0
timeseries linear regression model.
Model perturbations seem to indicate some degree of robustness in this case: the West’s population share of 2050 exceeds that of the South, whose exceeds that of the North; no region experiences competitive exclusion. Similarly to case 2.a, the three regions seem to experience uniform population distribution around 2000, slightly earlier than the previous model’s projections indicate. Having presented the tests from the discrete maps, we now supply results from running simple time-series regressions on the threeregion breakdown with 56 observations. They provide an alternative with which one can compare the nonlinear model specifications. The following models were found to apply (Figure 9):
XN( T) = 3.09931so that at T = 2050 we have x,(T (significant at < 0.0001 level). south:
O.O0131T,
= 2050) = 0.4138,
xs( T) = 0.69547 - 0.000194T,
R2 = 0.343, F = 289.9
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DIMITRIOS S. DENDRINOS AND MICHAEL
so that at T = 2050 we have (significant at 0.0008 level). West:
SONIS
x,(T= 2050) = 0.2978, R2 = 0.434, F = 12.5
xW(T) = - 2.79437 + O.O015T, so that at T = 2050 we have x,(T = 2050) = 0.2806, (significant at < 0.0001 level).
R2 = 0.973, F = 1970
The low level statistics for the southern region are due to the almost horizontal regression line. In all, the statistics are lower than in the nonlinearly specified model, although within acceptable ranges. Projecting to 2050 and summing the three regions’ shares, one obtains a total (0.9922) very close to satisfying the conservation condition (rN + xs + x w = 1.00). This cannot hold, however, for extended time horizons. It is this constraint which ultimately limits the use of this linear model, in spite of its good performance over a 7@year time horizon (1980-2050) and its calibration period (18.50-1983). A noteworthy observation from the three-region breakdown history in relative populationdynamic paths over the 1850-1983 period is the following: during this time span a host of events took place nationwide and in each region-significant technological change, drastic socioeconomic shifts, significant environmental fluctuations (for example, a number of wars), etc. In spite of all these factors, the paths of the three regions as recorded indicate very smooth, nonrandom motion. It may be that by using the relative lens one sees certain constants in population dynamics not observable under the absolute lens.
CONCLUSIONS
AND SUGGESTIONS
FOR FUTURE
RESEARCH
All four log-linear models (and the linear one as well) embody unstable equilibria in the relative population dynamics of the U.S. regions. Competitive exclusion in one or more regions is the outcome around 2050, when two regional (spatial) disaggregation schemes and two time scales for each scheme are tested using 130 years of observation to calibrate the models’ parameters. Tests employing the annual step show the South absorbing the vast share of the U.S. population, with the West anticipated by 2050 to hold on to about a &percent share of the population (1.b) under a four-region breakdown; when the one-year step is used, under the three-region spatial disaggregation of the U.S., the West is found to have a share by 2050 of about 0.848, and the South approximately 0.149 leaving the North with a 0.003 share (2.b).
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Statistics using the one-year step size were better than those obtained from using the ten-year time lag. Both were far superior to using simple linear time-dependent regression models for the various regions. When the decennial step size is employed, the West absorbs almost all of the U.S. population by 2040 (l.a), whereas both the North (in the NorthEast-North-Central spatial disaggregation) and the South experience competitive exclusion. When a three-region breakdown is used (with the North as the sum of North-East and North-Central) a decennial step has the West by 2050 accumulating a 0.989 share of the U.S. population, the rest being located in the South, while the North exhibits competitive exclusion. The fact that the oneyear step size result in a better fit (and more likely projections) than the ten-year step size may suggest that the forces shaping regional population distributions manifest themselves in regional interactions spanning a very short time span. In spite of their internal variety and complexity, comprising many sociological, political, economic, and other components, a simple one-year-step dynamic interdependence on their relative population stock siLe is enough to depict their effect. Along the same lines, simple difference equations and a very narrow band in their parameter. space accurately replicate the effect of these forces over extended time horizons: more than a century (1850-1983). In turn this may suggest the presence, over this time span, of very strong and selective processes operating in the dynamics of relative spatial population distribution. These processes, depicted by these simple dynamical equations, could be view-ed as a “code” for the development schemes of these relative population distributions. Experiments employing random fluctuations in these parameter values (checking the dynamic stability of the code) result in qualitatively similar results in most cases, thus indicating that the dynamic instability characterizing regional population interactions at these levels of spatial disaggregation is qualitatively robust. Different spatial disaggregations for the northern region alone result in forecasts which have qualitatively significantly different implications for the southern and western regions (identical spatially under the two areal disaggregation schemes); and different real time scales involved in the iterations of the discrete map result in qualitatively different forecasts. These two findings point to the existence of robustnessin model, beyond the robustness found in the parameters of all specific models tested. The different spatial and temporal scales of analysis imply different spatial and temporal forces at work at any time period. Which one of these forces (i.e., which model from the family of models presented) may prevail in the future is at present unknown. Another interesting finding is that various models (with different spatial and temporal scales) of one family (the log-linear), as well as linear models
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DIMITRIOS S. DENDRINOS AND MICHAEL SONIS
(involving only time dependence), produce acceptable (and indeed good) fit over an extended time period (in this instance a l%@year horizon). All have different forecasting implications. The uneasy realization that in social sciences different theories can be tested on identical data sets and found acceptable is confirmed by the experiments presented in this paper. Consequently, these results cannot and should not be treated as forecasts; the nonlinear dynamics of sociospatial systems imply that only qualitative statements can be made regarding alternative scenarios of future sociospatial configurations, not precise quantitative statements [4]. However, these simulations provide some strong evidence regarding the agglomeration and deglomeration forces present over the past 13@year period in the U.S., shaping the current regional allocation of population patterns. The nature of the large-scale, long-term forces slaving either the ten-year or one-year interactions among the U.S. regions, along the same lines as speculated in previous work by Dendrinos (with Mullally) [4] for urban agglomerations, may consist of an aggregate economicdemographic force operating over the nation’s landscape within time frameworks of one century or so. Per capita income and relative population interactions seem to depict the effect of this force at the nine-region spatial disaggregation [2]. Further work is needed to identify this force and its effects among different levels of regional spatial disaggregation. As suggestions for future research two kinds of tests immediately come to mind: first, to test the state composition of the subdivisions, at the margin (by moving one or two states from each subdivision to another); second, to randomly select states in each subdivision and test the model’s validity. The first test will test the hypothesis of homogeneity within subdivisions (i.e., whether or not these particular subdivisions are meaningful for testing an otherwise acceptable model). The second test will test whether this discrete map is meaningful altogether. Although the originally stated system of the Z-location, one-stock model can produce turbulence, the range of parameter values depicting current regional relative population distribution in the U.S., and extended over more than one century of evolution, is far from the ranges of such behavior. Thus, the allocation patterns of relative population in the U.S. are calm and not expected to produce chaotic motion at these levels of spatial disaggregation. However, much finer levels of regional breakdown (well below the nine-division level [2]) cannot be excluded from doing so. What is deduced is in fact quite unsettling, because it indicates that regional spatial dynamics are unstable. Only one stock (relative population) was considered in this model. Clearly, regions contain many stocks: wealth, built capital stock, natural resources, and others. Lack of data inhibits one from incorporating these stocks into the
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dynamics analyzed at present. Future time series may allow one to examine the hypothesis that the dynamics of these stocks depend on, and do not shape, population dynamics.
REFERENCES 1
2 3
4 5 6 7 8 9 10 11 12 13 14
15
D. S. Dendrinos, Turbulence and fundamental urban/regional dynamics, presented at the Annual Meeting of the American Association of Geographers, Washington, Apr. 1983, and at the Theoretical Task Force Meeting on Dynamic Analysis of Spatial Development, IIASA, Luxenburg, Austria, Oct. 1984; Proceedings, to appear. D. S. Dendrinos, Regions, antiregions and their dynamic stability: The case of the U.S. (1929-79), 1. Regional Sci. 24(1):65-84 (1984). D. S. Dendrinos, Ecological Studies of Urban Nonlinear Dynamics, Report SES-82-16620, National Science Foundation, Decision and Management Science Division, 1985. D. S. Dendrinos (with H. Mullally), Urban Euolution: Studies in the Mutkmutical Ecology of Cities, Oxford U.P., 1985. D. S. Dendrinos and M. Sonis, The onset of turbulence in discrete relative multiple spatial dynamics, J. App. Math. Cotnp. 22:25-44 (1987) D. S. Dendrinos and M. Sonis, Turbulence and Sock-Spatial Dynamics: Toward a Structural Theory of Social Systems Euolutim, Springer, 1988. M. J. Feigenbaum, Quantitative universality for a class of non-linear transformations, 1. Statist. Phys. 19:25-52 (1978). H. Haken (Ed.), Chaos and Order in Nature, Series on Synergetics, Springer, 1981. H. Haken, Synergetics: An Introduction, 3rd ed., Series on Synergetics, Springer, 1983. E. N. Lorenz, Deterministic non-periodic flows, I. Atmospheric Sci. 20:130-141 (1963). R. L. May, Stability and Complexity in M&l Ecosystems, Princeton U.P., 1974. R. L. May, Simple mathematical models with very complicated dynamics, Nature 261:459-467 (1976). R. H. McArthur, Geographical Ecology, Harper and Row, 1972. M. Sonis and D. S. Dendrinos, A discrete relative growth model: Switching, role reversal and turbulence, in P. Friedrich and I. Masser (eds.) International Perspectives of Regional Decentralization, Nomos, Baden Baden, 1987. W. Weidlich and G. Haag, Quantitative Sociology, Springer, 1983.