It is also the case that single-center cities are becoming a thing of the past; they are becoming more and more “multi-center” in character. Industrial estates, medical facilities, shopping malls, and business parks are being built (or are relocating) to regions away from the city. This of course creates demand for decentralized travel networks, as has been discussed in Chapter 7. Thus the next step in modeling the growth of cities will be to develop approaches that incorporate the multi-center structure in a way that invites more mathematical analysis.
*where now b may be of either sign.
Chapter 17
THE AXIOMATIC CITY
In this chapter we try to be a little more formal by defining axioms for an equilibrium model of a circular city. By using the word “equilibrium” here, we mean that the “forces” attracting people to live in the city are balanced at every point by the “forces” that “repel” them. Of course, there are no forces in the physical sense of that word, but by analogy with the balance between gravity and pressure gradients in stars it is possible to suggest certain forms of “coercion” that persuade individuals to live exactly where they do. Let’s get started.
X = ρ(r): CIRCLES IN THE CITY
We make the following assumptions (or equivalently, define the following axioms):
I. The population is distributed with circular symmetry in the plane, with radial “civic” population density ρ(r).
II. There is an inward cohesive force inducing citizens to relocate nearer the city center; this is because of the desirability of being near work and shopping locations, with lower transportation costs, etc.
III. There is an outward dispersive force, specifically a housing pressure gradient, inducing citizens to relocate farther from the city center; this is because of higher rental and housing costs in the central regions, etc. This assumption may not be entirely realistic for many modern cities (but we proceed with it nonetheless).
IV. The civic mass M(r) is the total amount of “civic matter” interior to r.
V. The cohesive pressure C(r) is similar to a two-dimensional “gravitational attraction” in the plane. In a similar manner, it can be written in terms of the civic mass and distance from the city center.
VI. The housing pressure P(r) has a power-law dependence on the population density, namely P(r) = A(ρ(r))γ where A and γ are positive constants. (Recall that this power law dependence was mentioned in Chapter 3.)
VII. The city is said to be in equilibrium if at each point within it the two opposing pressure gradients balance one another. This means that there is no net inducement for the populace to move elsewhere (clearly unusual in practice!).
The model we shall develop is essentially a two-dimensional version of the pressure balance equation for stellar equilibrium, as suggested above. Who would have thought that two such different topics could be so intimately connected? Suppressing our astonishment, let’s combine these assumptions in mathematical form by referring to Figure 17.1. The civic mass is readily seen to be
where ξ is a dummy radial variable. Consider a small change δC in C corresponding to a small increment δM in a radial direction; from Figure 17.1 we have that δM ≈ ρδr. Since we are assuming a gravity-type attraction, we posit that
Figure 17.1. Schematic geometry for the civic mass integral in equation (17.1).
In this equation, the “civic coercion” constant k and α are also positive constants; for strictly two-dimensional “gravitational attraction” α = 1, but we shall keep it arbitrary for most of this chapter. Assuming further that C(r) is differentiable, we find that, using the standard limiting procedure, that
The civic pressure gradient is from the higher housing-cost pressures in the central city regions to the lower pressure regions farther out. The cohesive pressure is higher in these external regions, and so the corresponding pressure gradient is in the opposite direction. For a city in equilibrium, therefore, the following equation must be satisfied:
From equations (17.2) and (17.3) we find that
There are several general properties of our “city” that can be established from this equilibrium equation. For example, it follows trivially that the rental gradient P′(r) vanishes at any noncentral location where the population density ρ(r) = 0. In particular, the following exercise is left to the reader:
Exercise: Prove the following results:
(a) The central rental gradient P′(0) is always zero at an equilibrium provided α < 2. (Hint: use L’Hôpital’s rule.)
(b) The quantity decreases monotonically outward for r > 0.
(c) For all r, and
(d) If the rental P(r) = 0 for some radius r = R, then the central rental P(0) satisfies
Further results can be found by using the following definition of a mean rental (r):
Exercise: Prove that if the central rental P(0) ≥ 0, then and hence that
(Hint: integrate by parts twice.)
Now we proceed to look at some more specific city models. We set γ = 2 in “Axiom” VI and write equation (17.4) in terms of the population density ρ(r) to obtain
If we perform a differentiation and define β2 = A/πk, then we have
This expression can be rearranged in the form
This equation is related to the well-known Bessel equation of order , namely.
The constant may take on any value, but is often an integer. A bounded solution to this equation satisfying the condition y′(0) = 1 is y = Jv(x), where Jv(x) is a Bessel function of the first kind of order . Consequently, the corresponding solution of equation (17.6) can be shown to be
In this expression ρ(0) is the central population density and = |(1 − α)/(3 − α)|. It will be left as an exercise for the reader to verify this result. Note that if α = 3, the original differential equation (17.6) reduces to Euler’s equation, solutions for which may be found by seeking them in the form ρ(r) ∝ rm and solving the resulting quadratic equation for m. Only solutions for which the real part of m is positive will be appropriate for a city containing the origin, but for an annular city, all solutions are in principle permitted.
Exercise: Verify by direct construction that (17.8) is a solution to (17.6).
(Hint: let ρ(r) = r(1−α)/2z(r) and then make a change of independent variable, t = r(3−α)/2.)
Such idealized “cities” can be referred to as “Bessel cities” for obvious reasons. When α = 1, the solution (17.8) reduces to the simple form ρ(r) = ρ(0) J0(r/β), and when α = 2 the solution is . From the definition of housing rental the corresponding expressions for P(r) are proportional to the squares of these solutions; they are illustrated in Figure 17.2. The model for α = 2 has the disadvantage that, unlike the case for α = 1, the gradients ρ′(0) and P′(0) at the center are not zero. While this is not a major problem, this model does not have these “nice” properties enjoyed by the other one.
Figure 17.2. P1(r) (solid line) is the square of the solution (17.8) for α = 1, ρ0 = 1, and β = 1 (for simplicity). P2(r) (gray line) is the square of the solution (17.8) for α = 2, ρ0 = 1, and β = 1.
For Bessel cities, both the density and rental are oscillatory and vanish infinitely often as the distance from the center increases indefinitely. Furthermore, the model predicts that the population density can become negative. Obviously this cannot be the case in reality, so it makes sense to define a finite Bessel city by truncating the model at the first zero of the corresponding Bessel function. It is interesting to note that in his α = 1 models, Amson (1972, 1973) identifies the positive but ever-decreasing peaks in the Bessel function as “satellite town belts” surrounding a central city, with the regions of negative density identified as greenbelt regions. If the first zero of J0(r/β) occurs at r = R1, the area of this can be called the central area, with central population M(R1). The zeros of Bessel functions are tabulated and available online. Since the first zero of J0(x) occurs when x ≈ 2.405, it follows that R1 ≈ 2.405β = 2.405(A/πk)1/2. Hence the central area is
From this sim
ple result we see that it depends directly on the rental coefficient A and inversely on the coercion coefficient k, but is independent of the central density ρ(0). It can also be shown that for given values of A and k the central rental P(0) varies as the square of the central population.
Chapter 18
SCALING IN THE CITY
X = r(n): SIZE OF THE CITY
No, by “scaling in the city” we don’t mean what Spiderman does in his various movie adventures . . . There is, according to Brakman et al. (2009, available in Oxford’s online resource center),
a remarkable regularity in the distribution of city sizes all over the world, also known as the “Rank-Size Distribution.” Take, for example, Amsterdam, the largest city in the Netherlands and give it rank number 1. Then take the second largest city, Rotterdam, and give it rank number 2. Keep on doing this for those cities for which you have data available, possibly selecting only cities exceeding a certain minimum size. If you calculate the natural logarithm of the rank and of the city size (measured in terms of the number of people) and plot the resulting data in a diagram you will get a remarkable log-linear pattern, this is the Rank-Size Distribution. If the slope of the line equals minus 1, as is for example approximately the case for the USA, India, and France, the relationship is known as Zipf’s Law.
This means that the largest city is always about twice as big as the second largest, three times as big as the third largest, and so on, in approximate inverse proportion to its rank. Mathematically, if we rank cities from largest (rank 1) to smallest (rank N) to get the rank r(n) for a city of size n, then log r(n) = log A + alogn, or equivalently r(n) = Ana, where the parameters A and a are chosen to fit the data. This power law is called an allometric relationship (allometry is the study of the change in proportion of parts of an organism as a consequence of growth). Let’s elaborate a little on this idea of allometry. If two quantities are related by a power law, y ∝ xa say, then a is called the scaling exponent. If a = 1 the quantities exhibit isometric scaling, for example, change of size without change of proportion (this is also referred to as geometric similarity). Allometry is often expressed in terms of a scaling exponent based on the mass M of the object of interest. Thus an isometrically scaling object (such as a cube of side L) would have all volume-based properties change in proportion to L3, or equivalently as mass to the first power, that is, V ∝ M; all surface area-based properties change in proportion to L2, or M2/3, and all length-based properties change in proportion to L, or M1/3. Another example worth mentioning is Kleiber’s law: an organism’s metabolic rate is proportional to M3/4; whereas breathing and heart rates are both proportional to M1/4 (see West 1999 for more details).
In 1949, Harvard linguist George Zipf proposed [31] that city sizes (along with many other things) follow a special form of the distribution where a ≈ −1. This has become known as Zipf’s law: the frequency of cities within a given size is inversely proportional to their rank. However, as pointed out by Batty and Longley [1], most of the work on city-size distributions neglects any spatial structure that exists within cities. By this they mean that cities as measured by populations or incomes, among other measures, are considered as points with their sizes reflecting the competition between cities as opposed to competition within the city. In other words (and as illustrated by Zipf’s law) there are a small number of large cities and a large number of small ones because there are not the resources and demand to sustain many large cities. Zipf’s law is empirical in that it has been observed in data taken from a wide range of cities in space and time, but so far no entirely satisfactory theoretical explanation has been found for it. Such an explanation must surely establish the basis for the observed organizational principles that appear to be replicated across such wide spatial and temporal scales.
But it does seem that similar principles must be at work within cities, based on microeconomic factors such as population density, rent, employment, transportation costs, and so on. Of interest then, in this context are the constraints that geometry imposes on “density” and “nearness” in a city; a plausible approach by virtue of the fact that sizes of structures indirectly reflect population and employment “volumes.” In particular, as buildings grow in size, their shape must change to enable them to function efficiently, and the scaling exponent a is related to the governing allometric “law.” Examples include the cost of heating (or cooling) a building; crudely speaking, heat loss will be proportional to the surface area of the building, but the amount of heat required to maintain equable temperatures will depend essentially on the volume of the building. Natural lighting provides another illustration; this will depend on the surface area of the building, but since the area changes with size more slowly than does the volume, the shape of the building must change as it grows to accommodate the requisite increase of natural illumination. In fact, it has been argued that cities yield some the best examples of fractals (see below). It is possible to fit power laws and allometric scaling relations to several geometrical properties of buildings—perimeter, area, height, and volume—using a large database of buildings in Greater London, which contains some 3.6 million “building blocks”!
So what might be these basic scaling laws for cities? We mention perhaps just enough to whet the reader’s appetite for more advanced discussions of this topic. (The reader may wish to consult Appendix 9 for a short introduction to fractals.) Modifying somewhat the notation in that Appendix to fit the present context, we identify the number of parts composing an object, their total length and area at a given scale a as N, L, and A, respectively. Specifically, they are defined by the relations
TABLE 18.1.
Estimated fractal dimension D for several cities
City
D
Beijing (1981)
1.93
Berlin (1945)
1.69
Boston (1981)
1.69
London (1981)
1.72
Los Angeles (1981)
1.93
Melbourne (1981)
1.85
Mexico City (1981)
1.76
Paris (1981)
1.66
Rome (1981)
1.69
Tokyo (1960)
1.31
If A(R) is the area of the object, considered to be constant regardless of the scale of resolution, then it is reasonable to define the density as
If we allow the scale to become finer and finer, so that as a → 0 (assuming 1 < D < 2), then N(a) → ∞, L(a) → ∞, A(a) → 0, and ρ(a) → 0. In particular, the perimeter becomes infinite as the scale becomes finer. This is an important characteristic (among others) associated with fractal behavior.
In their book Fractal Cities [1], Michael Batty and Paul Longley tabulate the estimated fractal dimension D for 28 cities. Table 18.1 contains some of these estimates (to three significant figures). There is also some evidence to suggest that fractal dimension has a tendency to increase as a city grows; Batty and Longley note that from 1820 to 1939 the estimates of D increase from 1.32 to 1.79, though it did drop slightly to 1.77 in 1962 and farther still to 1.72 in 1981.
It will be interesting to see how this relatively new subject of fractal cities itself “evolves” over time. Some indications are already at hand, as discussed in the section below.
X = N(t): LARGER CITIES ARE “FASTER”
In a fascinating paper published in 2007 [32], the first sentence reads “Humanity has just crossed a major landmark in its history with the majority of people now living in cities.” The title of this paper, by Bettencourt et al., was “Growth, Innovation, Scaling, and the Pace of Life in Cities.” One major finding was that many diverse properties of city life, designated here by y(t), are simple power law functions of population size N(t), that is,
In other words, they “scale” as power laws. Examples of y(t) are total electrical consumption, gross domestic product, and number of gasoline stations. In one sense, this is hardly surpr
ising; these and many other properties will in general increase with population size. Nevertheless, the remarkable fact appears to be that for quantities representing wealth creation and innovation, the power law exponent β ≈ 1.2. β > 1 implies increasing returns; β < 1 implies economies of scale. This latter situation occurs for city infrastructure, where β ≈ 0.8. Furthermore, β ≈ 1 is associated with individual human needs (jobs, housing, water consumption), so that these quantities tend to be directly proportional to population size.
Understanding the implications of these power laws is crucial because, as the authors point out, “a major challenge worldwide is to understand and predict how changes in social organization and dynamics resulting from urbanization will impact the interactions between nature and society.” There is a balance between the innovative and destructive aspects of city living, however. In addition to providing large-scale social services, education, and health care, for example, cities are the main source of crime, pollution, and disease in society.
X and the City: Modeling Aspects of Urban Life Page 14
