Figure 14.1. Cartesian road “grids” in rectangular (shown here, square) and circular cities.
It may be of some interest to take the mathematical limit as n → ∞, in the above equation and calculate the average length for a continuum of roads—perhaps more appropriate for ducks on a circular pond (though possibly a good description of the morning commute!). In practice, it is less messy just to set up the integral for the average value of the function L(x) = 2(r2 − x2)1/2, that is,
This is a standard integral, and can be evaluated with a trigonometric substitution to find that
Hence, proceeding as before, setting πr/2 = kA1/2 yields . It would appear that this is indeed the limit of the decreasing sequence of k-values suggested by equation (14.1).
Exercise: (a) Verify this result for ; (b) find k for n = 100.
Now we proceed to evaluate and k for a rectangular town center. We suppose that the longer sides are parallel to the x-direction, for convenience. The sides are of length a, and b > a. There are n + 1 N-S equally spaced roads (including the sides) and, similarly, m + 1 W-E equally spaced roads. We assume the N-S and W-E road spacing is the same, Δ, say. The average distance traveled (again, from side to opposite side, for simplicity) is
If we let b = (1 + α)a, this can be rearranged to give
The parameter α is a measure of how “rectangular” the town center is; the smaller the value of α the closer the center is to being square. Furthermore, since we wish to write and A = ab = (1 + α)a2, it follows that
This can be simplified still further by noting that, since a = mΔ and b = nΔ,
Substituting for n we find that, after a little rearrangement,
Figure 14.2 shows excellent agreement for bounds on k(α) when compared with the range quoted by Smeed. In passing, it is worthwhile to note that the topics discussed here are related to the probabilistic one of determining the average distance between two random points in a circle. A simple derivation of one such result can be found in Appendix 7. There has been much in the mathematical literature devoted to this problem, and it has been adapted by several theoretical urban planning groups to model optimal traffic routes between centers of interest.
Figure 14.2. Proportionality parameter k (see equation (14.5)) as a function of α (m = 10 here).
X = Ti: Question: What difference does a beltway make?
A beltway (or ring road in the UK) is a highway that encircles an urban area so that traffic does not necessarily have to pass through the center. A driver wishing to get to the other side of the city without going through the center might be well advised to use this alternative route. However, I’ve heard it said regarding the M25 motorway (a beltway around outer London) that it can be the largest parking lot in the world at times! Nevertheless, in my (somewhat limited) experience, a combination of variable speed limits and traffic cameras usually keeps the traffic flowing quite efficiently via a feedback mechanism between the two.
As always in our simple models, circular cities will be considered to be radially symmetric; that is, properties vary only with distance r. The city is of radius r0 and v(r) is the speed of traffic (in mph) at a distance r from the city center, 0 ≤ r ≤ r0. V0 is the speed of traffic around the (outer) beltway from any starting point A on the perimeter. For simplicity we will assume that V0 is constant and that v(r) is a linearly increasing function; as we will see, even these simplistic assumptions are sufficient to provide some insight into the potential advantages of a beltway. Thus v(r) = ar + b, a ≥ 0, b ≥ 0. A driver wishes to travel from point A to point P(r0, θ), (see Figure 14.3) both situated on the perimeter road. We shall suppose that there is also a circular “inner city” beltway a distance r1 < r0 from the center O, along which the constant speed is V1 ≤ V0. She then has three choices: (1) to drive right into the city center and out again to P, the path ADOFP; (2) to drive around the outer beltway along path ABP, and (3) to take an intermediate route using the inner beltway along the path ADEFP. We shall calculate the times taken along each route under our stated assumptions.
Figure 14.3. Radially symmetric velocity contours in a circular city.
For the first choice, noting that v = dr/dt, we have that
For the second route,
Finally, for the third route, involving some travel around the inner beltway,
Obviously this last case is intermediate between the other two in the sense that as the inner beltway radius r1 → 0, T3 → T1, and as the outer radius r1 → r0, T3 → T2. It is interesting to compare these travel times; so let us examine the first two cases and ask when it is quicker to travel along the outer beltway to P, that is, when is T2 < T1? This inequality can be arranged as
Since f1(0) = f2(0) = 0, and , it follows that the graphs of these two functions will intersect at some radius, rc say, if and only if , i.e., if 2V0 > (π − θ)b. If rc < r0 then there will be an interval (0, rc) for which it is quicker (in this model) to travel along the outer beltway to P. If rc > r0 then the curves do not intersect and it is always quicker to use the outer beltway.
Chapter 15
SEX AND THE CITY
We can think about the growth of cities in several ways, none of them prurient, despite the title of this chapter. Perhaps the most obvious one is how the population changes over time; another is how the civic area changes; yet another might be the number of businesses or companies in the city; and so on. These statistical properties are generally referred to as demographics, and they can include gender, race, age, population density, homeownership, and employment status, to name but a few. In this chapter we shall focus attention on some the simplest possible models of population growth in several different contexts, ending with some using “real” data from a bygone era. We’ll start with exponential growth.
X = P(t): MATHEMATICAL MODELS OF POPULATION GROWTH
In the November 4, 1960, issue of the journal Science, a paper [28] was published with the following provocative title: “Doomsday: Friday 13 November, A.D. 2026.” Beneath this was the sentence: At this date human population will approach infinity if it grows as it has grown in the last two millennia.
The authors presented a formula for the population P in terms of positive constants A, b, and tD (to be defined below) as
In this equation, the subscript D stands for “Doomsday”! We proceed to derive more simply a form of this result to illustrate the principle behind it. But first, a caveat. In what follows, differential calculus is used. So what’s the problem with that? Well, in so doing, we are making the implicit assumption of differentiability, and hence continuity of the population of interest, whether it be humans, bedbugs, or tumor cells. But the populations are discrete! In these models, there are always an integral number of people, bedbugs, and so on. Calculus is strictly valid only when there is a continuum of values of the variables concerned and the dependent variables are differentiable functions. In that sense continuum models can never be totally realistic, even when there are billions of individuals (such as the number of cells in a tumor). Nevertheless, if there are sufficiently many individuals, we can justifiably approximate the properties of the population and its growth using calculus. Further informal discussion of one aspect of this “discrete/continuum” problem can be found in Appendix 8. Now we can begin!
Everyone has heard of “exponential growth” but not everyone knows what it means. It applies to (here, differentiable) functions P(t) for which the growth rate is proportional to P, or equivalently, the “per capita” growth rate is a constant, k. In mathematical terms
If we know that at some time t = 0 say, the population is P0, then the solution of equation (15.1) is
If k < 0, then P decreases exponentially from its initial value of P0. Such exponentially decreasing solutions are used in many algebra books to illustrate the phenomenon of radioactive decay. If k > 0 we see an immediate problem: the solution grows without bound as t → ∞. This is reasonably supposed to be unrealistic, because if P represents the population of a city, country, or the world, there is
not an unlimited supply of resources to maintain that growth. In fact, the English clergyman Thomas Malthus (1766–1834) wrote an essay in 1798 stating that “The power of population is indefinitely greater than the power in the earth to produce subsistence for man.” This has been paraphrased to say that populations grow geometrically while resources grow arithmetically. Not surprisingly, exponential growth is sometimes referred to as “Malthusian growth.”
Let’s modify equation (15.1) just a little, and examine the innocuous looking first-order ordinary differential equation
(with P(0) = P0 as before). For reasons discussed below we will restrict the parameter ε to the range −1 < ε < ∞. This simple looking nonlinear ordinary differential equation has some surprises in store for us, and in fact is sometimes referred to as the “Doomsday” equation. The first thing to do is integrate it to obtain
where C is a constant of integration. This can be found immediately by setting t = 0 in equation (15.4). After a little rearrangement, the solution to equation (15.3) is found to be
with the restriction on t being
Equation (15.5) is in precisely the form stated at the beginning of this subsection.
Exercise: Derive the solution (15.5).
Before we examine some of the implications of this solution, and restrictions on it (remember: every equation tells a story), let’s reexamine equation (15.3) and plot the quantity k−1dP/dt = P1+ε ≡ ri, i = 0, 1, 2, as a function of P for the three cases ε > 0, ε = 0, and −1 < ε < 0, respectively. This last inequality ensures that the population growth rate for this case is not declining, since
In Figure 15.1 these values correspond to the top, middle, and bottom curves. The case ε = 0 (dashed line) clearly corresponds to the equation for exponential growth we have already seen (equation (15.2)). The upper curve corresponds to the arbitrary (and for illustrative purposes, rather large) value for ε of 0.5.
Figure 15.1. Curves proportional to the growth rates for super-exponential, exponential, and sub-exponential growth (from equation (15.3)).
The lower curve has ε = −0.5. Now let’s consider a much smaller but positive value of ε, 0.05 say. Then the growth is just slightly “super-exponential” with the right-hand side of equation (15.3) increasing in proportion to P1.05. From equation (15.5) the solution is
This solution is undefined (“blows up”!) at time
in whatever units of time are being used (usually years, of course). This doomsday time is the finite time for the population to become unbounded. Thus if we take the current “global village” world population of 7 billion (as declared on October 31, 2011) to be P0, and, for simplicity, an annual growth rate k = 0.01 yr−1 (just 1% per year) then for the (admittedly arbitrary) value ε = 0.15, equation (15.7) gives
This is the doomsday time for our chosen value of ε! Note that if ε ≤ 0 the problem does not arise because there is no singularity in the solution. The population will still tend to infinity, but will not become infinite in a finite time, as is predicted for any ε > 0, no matter how small. The original “doomsday” paper [28] (and others listed in the references) should be consulted for the specific demographic data used.
Despite such a projection, I thought there was no suggestion that the world population is actually growing according to equation (15.3) until I read a paper [29] published in 2001. The first two sentences from the abstract of that paper may serve to whet the reader’s appetite for further discussion later in this book (Chapter 18): “Contrary to common belief, both the Earth’s human population and its economic output have grown faster than exponential, that is, in a super-Malthusian mode, for most of known history. These growth rates are compatible with a spontaneous singularity occurring at the same critical time 2052±10 signaling an abrupt transition to a new regime.” Doomsday again?
Exercise: Obviously the choice ε = 0 in equation (15.3) results in the standard exponential growth of P as noted above. Using a limiting procedure in equation (15.5), show that
Exercise: Another model, the “double exponential” model of bounded population growth, is defined by
where P∞ is the asymptotic population (approached as t → ∞), 0 < b < 1, and 0 < a < 1.
(i) Show that
(ii) A town population starts at P(0) = 80,000 and has an upper bound P∞ = 200,000. It is known that after 10 years the population has risen to 150,000. Determine the constants a and b.
(iii) Find the population after 15 years.
X = x(t): ANOTHER SIMPLE NONLINEAR MODEL—THE LOGISTIC EQUATION
Now we shall discuss a population model that predicts a definite limit on growth. Let’s examine the logistic differential equation
both k and N are positive constants. We have changed the dependent variable from P to x to reflect the fact that these models can apply to many different things. N is often called the carrying capacity in ecological contexts and the saturation level in chemistry. It can be used to describe, in a very simple manner, the spread of diseases or rumors in a closed community of population N, or a cultural fad, or indeed the spread of anything that can be spread by casual person-to-person contact, from the common cold to word-of mouth advertising about a new brand of coffee! (See Appendix 8.) The variable x(t) describes the number of individuals with the disease, or who have heard the rumor, read the ad, and so on, and for our purposes the most relevant range of values for x is 0 < x < N. The case x > N is also easily accommodated and will be discussed below. Equation (15.8) is separable, and using partial fractions we can integrate it to obtain
D being a constant of integration to be determined. Hence
where we have defined K = eD. The general solution x(t) is therefore
Now rumors don’t start and diseases don’t spread without someone to initiate them, so we define the initial value of x to be x(0) = x0 > 0. This enables us to find K in terms of the (in principle known) quantities x0 and N. Thus
On rearranging the final result, the solution becomes
Check that the initial condition is satisfied, and note also that limt→∞x(t) = N. This means that x = N is a horizontal asymptote; if nothing else changes, then the population x approaches the limiting value N from below (in this case) as time increases. Note that the solution (15.9) is still valid if x0 > N; now the solution decreases monotonically from its initial value toward the asymptote from above. This case is meaningless in the present context, though it can apply to a simple model of overpopulation when the natural resources in a nature reserve, say, become compromised, perhaps by pollution or an environmental disaster. Then the normally sustainable population can no longer be supported, and the numbers decrease toward the new carrying capacity.
Further information about the evolution of x(t) may readily be found from examining both its population growth rate and its per capita growth rate using equation (15.8). The latter rate is just
and clearly in the interval 0 < x < N this is a linearly decreasing function. This means that the per capita growth rate slows uniformly from a maximum near kN (when x is small) toward zero as the carrying capacity (or saturation level) is approached. On the other hand, it can be seen from equation (15.8) that the population growth rate dx/dt as a function of x is a downward facing parabola with intercepts at x = 0, N and a maximum of kN2/4 at x = N/2.
X = ugh!: BED BUGS (OR RATS) IN THE CITY
We can now change the context somewhat from the spread of rumors and diseases to a city-related version of harvesting or fishing. Suppose that a city has a problem with vermin—whether of the four-legged, six-legged (bed bugs!), or winged variety (e.g., pigeons), and a program is introduced to reduce the population of these unwanted “critters.” How might we incorporate this “vermin reduction program” into the equations we have been discussing above? We can do this mathematically in two straightforward ways, depending on how the program is administered: by including a constant reduction rate or including a term proportional to the existing vermin population. Were we to replace “vermin” by “fish” then w
e would note that the former more appropriately describes a fishing effort in which the rate at which fish are caught on each fishing foray is the same. In the latter case the catch rate is proportional to the population, and the number of fish caught will thus diminish or increase as the population does. Taking the latter case first because it is mathematically simpler, and possibly more relevant to a town or city with a vermin problem, we can modify equation (15.8) as
Here the term cx represents the vermin reduction rate (in units of e.g. rats/week). The constant c can be thought of as the per capita reduction rate (rat?) resulting from the program. Noting that equation (15.10) may be rewritten as
X and the City: Modeling Aspects of Urban Life Page 12
