where = N − c/k, we see that this has been encountered above; equation (15.11) is just (15.8) with a reduced carrying capacity (we assume that c ≤ Nk). The solution (15.9) still applies with this new horizontal asymptote replacing the earlier one. Nevertheless, this reduction is obviously helpful in the relentless fight against foul vermin! In terms of the original capacity N, the maximum growth rate is k(N − c/k)2/4 and occurs at x = (N − c/k)/2.
Returning to the constant reduction rate, we now have a governing equation
To study this, we write the quadratic polynomial on the right-hand side of (15.12) as Pc(x) in the form
Now Pc(x) possesses real zeros if and only if N2/4 > c/k, that is, if c < kN2/4. Then these zeros occur at
Clearly, Pc(x) > 0 when x− < x < x+ as opposed to the case for c = 0, that is, where P0(x) > 0 when 0 < x < N. Another way of thinking about this is by noticing that the right-hand side of equation (15.12) is just the original quadratic function kx(N − x) “lowered” by the amount c. This is illustrated in Figure 15.2 for x in units of N (or equivalently, N = 1) and k = 1, c = 0.15.
Recalling that Pc(x) = dx/dt, we can easily see from the figure that for c ≠ 0 (dotted curve), dx/dt < 0 for both x < x− and x > x+. This means, according to this simple model, that the population of rats, bedbugs, or whatever it is declines to zero or to x+. Obviously the former is the more desirable of the two situations. Furthermore, if c > kN2/4 then dx/dt for all x-values and the vermin population declines continuously to zero. This might reasonably be termed “overkill”!
Figure 15.2. Right-hand side of equation (15.12) for c = 0 and 0.15 (solid and dotted curves respectively).
Exercise: Show that the solution of equation (15.12) is given by
and d is the difference of the zeros of Pc(x), namely, x+ − x−. When c = 0, x+ = N, and x− = 0 also verify that result this reduces as it should to the solution (15.9). The graphs of x(t) in units of N are shown in Figure 15.3 for both c = 0 (solid curve) and c = 0.15 (dashed curve), with k = 1 and x0 = 0.25. Each exhibits the classic “logistic curve” sigmoid (or stretched S) shape.
X = ∑N: HOW MANY PEOPLE HAVE LIVED IN LONDON?
Well, I was one of them, but enough about me. Suppose that at time t1 the population was N1, and at a later time t2 the population was N2. Assuming that the growth was exponential, the annual growth rate r can then be determined from these data using the equation
Figure 15.3. Solution curves from equation (15.15) for c = 0 and 0.15.
At any time t during the interval [t1, t2] the population was N1er(t−t1) and the total number of “person-years” from t1 to t2 is therefore
Hence the number of person-years lived is, from (15.16),
The left-hand side, expressed in person-years, can be converted to persons by dividing by a time factor. This should be an average life expectancy, but that is very difficult to determine, even if known in principle, owing to gender, regional, and temporal variations. For example, extracting possibly relevant (i.e., Northern European) data from a table in the Encyclopedia Britannica (1961), we have:
Humans by Era
Average Lifespan at Birth (years)
Upper Paleolithic
33
Neolithic
20
Bronze Age and Iron Age
35+
Medieval Britain
30
Early Modern Britain
40+
Early Twentieth Century
30–45
In view of these “data” we shall take an average of r−1 = 40 years for the range 1801–2006 considered in Table 15.1, and estimate the quantity
TABLE 15.1
for a range of data points. Summing all these from the table we have the estimate of
Clearly this is a rough-and-ready approach, dependent on the width of the time intervals and the reliability of the data. The result is not particularly sensitive to the choice of life expectancy, but it is sensitive to the number of intervals chosen in the overall time frame. The more intervals we choose, assuming the data are available, the more different the cumulative population will generally be. To see this, just take a single interval from 1801 to 2006. We take t1 = 1801 and t2 = 2006, and N1 = 9.59 × 105, N2 = 7.66 × 106. Using equation (15.19), we find that Ntot (now equal to ΔN of course) ≈ 16.7 million. If we were to go back to the years t1 = 1000 and t2 = 2600 as before, then N1 ≈ 104 and N2 is again as before. The corresponding value of Ntot is now 290 million for the same value of r. Obviously the value for London’s population in the year 1000 is somewhat suspect, but this is just to illustrate the point regarding the interval width. The average lifetime was probably lower in medieval times, so if we set r−1 = 30 yr then Ntot increases to 386 million.
Nevertheless, this is an interesting application of some simple mathematics, and is probably far more reliable than estimates of all the people who have ever lived (Keyfitz 1976)!
Chapter 16
GROWTH AND THE CITY
Patterns of the distribution of populations around city centres are extremely variable. Clearly city development depends on constraints imposed by features of the local geography such as lakes, coastlines and mountain ranges. Rivers have played an important role as centres of attraction and indeed one of the first known cities, Babylon, was located at the junction of the Tigris and Euphrates rivers. The sizes of population centres also exhibit great variability, with numbers from just a few to ten or more million.
—A.J. Bracken and H.C. Tuckwell [30]
X = Ntot and X = ρ(r): SIMPLE URBAN GROWTH MODELS
In light of the above comments, it is somewhat surprising that fairly general quantitative patterns of urban population density ρ (population per unit area) can nevertheless be formulated for single-center cities. As far back as 1951 Colin Clark, a statistician, compiled such data for 20 cities, and found that ρ declined approximately exponentially with distance from city centers (though “city center” is not always easily defined in practice; central business district is perhaps a better term). If we restrict ourselves to the special but important case of circular symmetry, wherein ρ = ρ(r), r being the radial coordinate, then naturally we expect that ρ′(r) < 0, but also perhaps that the density drops off more gradually as r increases, that is, ρ″(r) > 0. Furthermore, we might anticipate that this density profile gets flatter as the “boundary” of the city is approached. If we consider the simplest possible model of exponential decline, the suggested conditions for ρ are satisfied, so let us consider
A and b being positive constants; in fact from (16.1), A = ρ(0) ≡ ρ0 and b = −ρ′(0)/ρ0. Of course, in practice, urban population growth is a dynamic process, so to be more general we should permit A and b to be time-dependent (and this we shall do shortly). In fact, several values of A and b reported for London over a period of 150 years are listed below, along with correspondingly less information for three other cities, Paris, Chicago, and New York. Before analyzing these data, we derive an expression for the population.
The total metropolitan population in a disk of radius R0 is given by the integral
(This is called the “civic mass” in Chapter 17.) An estimate of the total population in a spatially infinite city(!) can be found from the limiting case
This is not such an unreasonable result as might be first thought, because the population density (16.1) is decreasing exponentially fast. Obviously the result is formally unchanged if A and b are time-dependent; for example, if the data justified it we could choose both parameters to be linearly decreasing functions of time. This will no doubt vary from city to city and we do not pursue it here.
In Tables 16.1–16.4, A is expressed per square mile in thousands, and b in (miles)−1. Cities have two fundamental modes of growth: up and out! Of course, in practical terms growth is usually a combination: there are ultimate limits on both of them. The parameter A is a measure of “up” in the sense of central density; b is a measure of how far “out” growth occurs—the smaller b is, the
more decentralized the city becomes—this is often referred to as “urban sprawl.” Notice that the total population is proportional to A but inversely proportional to the square of the parameter b. The expression for Ntot based on equation (16.3) is generally of the right order of magnitude, particularly for London, but it does overestimate populations in later years, particularly for Chicago and New York. This partly because of the uncertainty of densities at small values of R, and also because cities are decidedly not circularly symmetric (as the first paragraph in this section implies)! Political and not just natural boundaries also play a role in urban development.
Generally, the value of the central density ρ(0) = A decreased, and so did b, meaning that the population of London became more decentralized. This trend is not always realized (at least, according to the limited data listed in Montroll and Badger (1974)). The corresponding data for Chicago, Paris, and New York, are shown in Tables 16.2–16.4.
Notice the precipitous drop in both parameters for London and Rome following the end of World War II.
TABLE 16.1.
Estimated population of London in the period 1801–1951 (based on equation (16.3))
TABLE 16.2.
Estimated population of Chicago in the period 1880–1950 (based on equation (16.3)
TABLE 16.3.
Estimated population of Paris in the period 1817–1946 (based on equation (16.3))
TABLE 16.4.
Estimated population of New York in the period 1900–1950 (based on equation (16.3))
According to Montroll and Badger (1974), the British archaeologist Sir Leonard Woolley estimated that the ancient city of Ur had an average density of 125,000 people per square mile at the height of its mature phase, around 2000 B.C. The density per square mile for parts of fourteenth-century Paris was 140,000, as was true for parts of London in 1700. By 1900, parts of New York’s Lower East Side had reached densities of 350,000 per square mile, but even this is small compared with many non-Western cites. If A increases while b remains constant, this can be accomplished quite easily over time. Thus parts of Hong Kong reached densities of about 800,000 per square mile; such high densities correspond to about one person for every four square yards!
Exercise: Verify this last statement.
One problem with the density profile (16.1) is that it cannot reproduce the so-called “density crater” for the resident population in a large metropolitan area. This phenomenon means that the maximum population density occurs, not in the central region but in a ring surrounding the city center. This can be accomplished with the function
where b > 0, c >, and the maximum density occurs at radius r = b/2c. (Note that b is now of opposite sign to its counterpart in (16.1).) A graph of the normalized density function ρn(r) = ρ(r)/ρ0 is shown in Figure 16.1 for the simple choice of b = c = 1.
Equation (16.4) could represent the density of a city with an extensive central business district. Let’s investigate the properties of this profile in some detail. The maximum density is
Figure 16.1. Form of the normalized population density function (16.4).
We suppose that there is a fairly well defined “perimeter” of the urban area at r = rp, for which ρ(rp) = ρp. Solving equation (16.4) for rp yields the result (since ρ0 > ρp for all realistic models of cities):
It is possible to classify these urban density profiles in terms of the magnitude and sign of the parameter combination Specifically, the sequence “youth, early maturity, late maturity, old age” is characterized by Newling (1969) as corresponding to the β-intervals* (−∞, −1), (−1, 0), (0,1), and (1, ∞). This can be seen in a qualitative manner from Figure 16.1 by mentally imagining these four stages of development to be, successively, the curve to the right of the ordinates r = 2, 1, 0.3, and 0. But from where do these intervals for β come? It’s all to do with points of inflection. Simply put, points of inflection in the graph of ρ(r) occur when there is a change of concavity (if those points are in the domain of the profile). From equation (16.4) such points occur at
It can be seen that if b < −, i.e., β < −1, then there are no points of inflection and ρ(r) decreases monotonically from the central business district outward (youthful city). If − < b < 0, (−1 < β < 0) or 0 < b (0 < β < 1) there is a single point of inflection (in early and late maturity, respectively). Finally, if b > (β > 1) there are two such points and the full density crater profile is evident (aging city).
There can be variations on this theme using the concept of a “traveling wave of metropolitan expansion.” We can illustrate the stages of city development by positing a traveling pulse of “shape” ρ(r) moving outward with speed v, that is,
This can be interpreted as a point of constant density moving out radially with speed v. In principle v may itself be a function of time; in any case, as the city develops in time, the shape of the density profile may change in accordance with the above equation.
Returning to the basic form (16.4), we now examine the density profile when some or all of the parameters A(= ρ0), b, and c may be functions of time. In view of the spatial behavior exhibited by ρ(r) from (16.4), that is, initially increasing to a maximum, then decreasing monotonically, Newling (1969) chose the same temporal form for ρ0(t):
ρ0 now being a constant. Furthermore, he chose b to be a linear function, b(t) = b0 + gt, where g and c are constants. From equation (16.4) (noting again the change in sign for b from equation (16.1)) it can be seen that b = ρ′(0)/ρ(0), the prime referring to a spatial derivative. Thus
Proceeding as with the derivation of equation (16.6) gives the corresponding result for the radius of the urbanized area (now at time t) as
The “speed” of urban expansion can be then defined as rp(t)/t for t > 0. This type of model has also been used in ecological contexts, studying the spread of animal and insect populations (as well as diseases) in so called reaction-diffusion models.
Now let’s use the density profile (16.4) to compute some overall population levels. As in equation (16.2), we may compute the total metropolitan population in a disk of radius R0. It is given by the integral
Changing the variable of integration to R = r − b/2c enables us to write N(R0) as the sum of two integrals, namely,
where the limits are now α = b/2c and β = R0 − b/2c, and the (indefinite) integrals are
Noting that the second of the (definite) integrals can be expressed in terms of the error function
and the expression for the total population within a disk of radius R0 can be written as
Noting that erf(∞) = 1, we may take the formal limit of this equation to find that in the limiting case of an infinitely large city with this density profile,
Exercise: Verify equations (16.12) and (16.13).
Some closing comments are in order. In the final model above, the parameter c was taken to be constant. Note that it plays a similar role in equation (16.4) as does b in equation (16.1), notwithstanding the quadratic term: it serves as a measure of the decentralization of the city. As we noted from the tabulated data in Tables 16.1–16.4), b and similarly c in fact tend to decrease over time, corresponding to a “flattening” of the density profile as the city expands radially. This phenomenon may well be a result of improvements in the urban transportation systems (see Chapter 7). Also, Newling’s four stages of city development are based on a progression of the parameter β in time from negative to positive values; this may not occur in general and so raises a question as to the efficacy (at least quantitatively) of this sequence of stages. However, the tabulated data indicate that the maximum densities were indeed higher in the nineteenth than the twentieth century, so in that sense, the four stages may well be qualitatively correct.
More realistic and advanced models should (and do) include several other factors that generalize those implicit in the above formulations. One such a factor might be temporal variations in population density, allowing for the possibility of saturation; if so, a logistic model might suffice here. Inclusion of dif
fusion (spread into neighboring regions) would then have the density satisfying a partial differential equation of the “reaction-diffusion” type, at least in a continuum model (see Adam 2006, chap. 14). A highly populated inner core will give rise to congestion, and this will be proportional to the local density within a given region. Another factor, also neglected here, is the effect of (local) immigration and emigration. One might expect a considerable influx of immigrants in a central district undergoing rapid growth, and the reverse for inner city districts that have become “blighted.”
X and the City: Modeling Aspects of Urban Life Page 13
