16

The Great Explosion of 2023

Our previous chapter concluded on a note depicting a very alarming situation: (1) a 1990 world population of over five billion people and a population doubling time of around 17 years and (2) a projection to a year-2000 population of over eight billion and a doubling time of around 12 years. Clearly, it is urgent that something be done about this serious crisis of rampant growth of world population.

However, we need to look at the data more closely. Before we try to resolve this incredible crisis, perhaps it is necessary to examine how serious it really is. So we return to the equation that describes so-called coalition or hyperbolic growth:

where N is the population at time t and a is the growth coefficient; N₀ is the population at time t = 0. We obtained the values a = 0.002675 per year and based on t = 0 in the year 1650.

Some populations are listed in table 16.1, computed from equation (16.1), for various years commencing with 1650 and ending with 2023. Magnitudes of observed populations are also shown and the differences between computed and observed populations are indicated. Several important facts can be seen in the table:

TABLE 16.1

For the years 1650 through 1980 there is surprisingly close agreement between computed and observed populations. This is apparent in the right-hand column of the table.

For 1990, however, there is a difference of +0.488. That is, the computed population is nearly 500 million more than the observed. This is a very interesting and important point. We shall come back to it later.

For the years 2000 and beyond, computed populations rise to ridiculous values: nearly 52 billion in 2020 and over 245 billion on, say, January 1,2023.

Okay, here it comes. Doomsday is established by setting the denominator of equation (16.1) equal to zero. This gives t_e = 1/a = 373.832, where t_e stands for explosion time. Since t = 0 corresponds to 1650, then explosion year is 2023.832.

So 2023 will be the year of the great explosion. Both Keyfitz (1968) and von Foerster et al. (1960) arrive at about the same conclusion. Specifically, on November 1, 2023 there will be an infinite number of people in the world and the doubling time will have shrunk to zero.

Interesting. Now back to reality. Of course, the world's population is going to continue to increase in the years to come. However, obviously the population will not become infinite. Inevitably and increasingly there will be forces that will slow down, terminate, or even reverse the growth rate.

These forces imposing limitations to population growth are comprised of agricultural, demographic, economic, environmental, scientific, sociological, and technological components. Collectively, they specify and emphasize the finite capacity of the world for maintenance of a human population.

Apparently, population crowding effects began to appear, on a worldwide basis, during the decade of the 1980s. We noted the very important fact that the computed population for 1990 was almost 500 million more than the observed population. It is remarkable that after a million years of human population increases, crowding effects and limitations-to-growth factors made their first appearances only quite recently.

Long ago it was found that an appropriate way to slow down exponential or Malthusian types of growth phenomena was to attach a crowding or finite-resources term to the growth equation. This turned a skyrocketing exponential growth into a stabilized logistic growth. It seems logical to modify our hyperbolic growth equation in a similar fashion.

To follow the approach of Austin and Brewer (1971), we arbitrarily attach a growth limitation term to the equation for unrestrained explosive hyperbolic growth. That is,

where the quantity in parentheses requires that the growth rate, dN/dt, become zero when N = N_*. This quantity, N_*, is the carrying capacity, that is, the maximum population the earth is able to sustain. The solution to equation (16.2) is

which we will call the modified coalition growth equation. This expression is not as formidable as it might appear; indeed, it is quite easy to handle. We note that if N_* approaches infinity, that is, resources are sufficiently large to support an infinite population, equation (16.3) reduces to (16.1) as expected.

Unfortunately, (16.3) is in the form that says “t is a function of N” and not “N is a function of t” as we would prefer. This is not a problem; we still know how the world's population N increases with time t.

At this point we shift the time origin, t = 0, from the year 1650 to a more recent date: 1980. Also, with computations involving populations and growth rates through 1990, a value of N* is obtained. These calculations provide the following: a = 0.0303, , and with t = 0 corresponding to 1980.

Populations calculated from equation (16.3) are listed in table 16.2. We compute a 1995 world population of about 5.73 billion and a year 2000 population of around 6.22 billion. These values are in good agreement with those given by the United Nations (1993).

A plot of equation (16.3) is shown in figure 16.1, in which we note several things:

In the years to come, the world's population certainly continues to increase. However, because of the stabilizing effect of the crowding term in (16.2), the explosive increase of uncontrolled hyperbolic growth is avoided.

As shown in figure 16.1, the rate of population increase reaches a maximum value in 2004 when the population is 6.667 billion people. The corresponding annual increase is 101 million people. This maximum occurs when the growth curve passes through the so-called inflection point, as shown in the figure. After that, the rate begins to decline and eventually becomes zero.

TABLE 16.2

Population of the world, N, in billions Computed from modified coalition growth equation.

FIG. 16.1

Population of the world with projections leading to (a) infinite population in 2023 and (b) stabilized population of /V* = 10.0 billion.

One thing needs to be emphasized. Projections of future populations are not made on the basis of simply fitting curves to observed data. Demographic analyses of trends of the world's population are done on a country-to-country basis and take into account such things as age distribution, life expectancy, migration, mortality, and fertility of a particular nation's population. World trends are established as the aggregation of country and regional trends. For those interested in these topics, a good place to start is the interesting article by Coale (1974). More advanced treatments of the subject are provided by Keyfitz (1968), by Pollard (1973), and by Song and Yu (1988).

Complementing precise demographic analyses are mathematical frameworks such as our modified coalition growth model. These models can answer a lot of questions.

PROBLEM 1. The growth rate, dN/dt, is a maximum when its derivative, d²N/dt², is equal to zero. Utilizing equations (16.2) and (16.3), show that the coordinates and slope of the growth curve at the inflection point are

PROBLEM 2. In the previous chapter we established that during the period from 1,000,000 B.C. to 1950, the number of person-years, M, was about 1,867 billion. By exact, numerical, or graphical integration of equation (16.3), confirm that the value of M between 1950 and 2050 will be about 625 billion.

We have taken a fairly good look at the world's population. Here is a point on which to conclude. From equation (16.2) we determine that during 1995 the world's population increased by about 95 million people. If these 95 million were to comprise an entirely new country, where would it rank, in 1995 population, among the world's nations? The answer is number 11, after China, India, United States, Indonesia, Brazil, Russia, Pakistan, Japan, Bangladesh, and Nigeria. Just think: an enormous new country every year!