## Appendix D

Population Growth and Lily World

The mathematical details that underscore the topics of population growth, the estimation of future populations, and the determination of the Earth's carrying capacity at a specific epoch are highly complex, and the full details are not what we need to worry about here. The researchers who study such topics, however, usually analyze the predictions of their models in terms of the so-called r- and K-processes (although other names have been used to describe the behaviors observed).

An r-process, also called a Malthusian process, describes the growth in population numbers when there are absolutely no checks on how many individuals the environment can support. Under these circumstances the population grows exponentially, with a growth rate r such that at time t the population P(t) = P(0) exp(r t), where P(0) is some initial population at a reference time t = 0. Provided r > 0, then the population must always increase, and as t ) 1, so the P(t) ) 1, with the population becoming ever larger. Clearly no such population can really exist; there has to be a point at which the population exceeds the carrying capacity, and at this moment the population tends to crash catastrophically, and, rather than the apparently utopian P(t) as t )i, it is found that P(t) ) 0 as t )i, indicating that extinction has occurred.

The classic example of an r-process growth strategy is that associated with algae blooms such as what is known as a Red Tide. In these circumstances, so many bacteria form in estuarine regions that the water is literally turned red, the characteristic color of the dinoflagellates producing the bloom. Such algae blooms thrive until the entire regional food and oxygen supply is depleted, at which point the algae population dies off in a terminal Malthusian meltdown. Under r-process growth strategies, the population invariably follows a boom to bust to extinction pathway.

In contrast to the r-process, the K-process, or logistic model, as it is often called, describes the population growth when the carrying capacity K is taken into account. In this situation, as the population increases and approaches the carrying capacity, so the growth rate decreases, and after some finite time TK the population reaches and stays at its maximum possible value, with P(t) = K for all t > TK. Under the K-process, a population need not necessarily crash or undergo a Malthusian meltdown—provided the carrying capacity truly remains constant with time, a situation that need not necessarily apply.

By way of illustrating how the r- and K-processes work, let us consider ''Lily World.'' There isn't much to Lily World; it is a square, 1-km-sided lake with a surface area of 1 km2 and a depth that we needn't worry about. Upon Lily World grow square water lilies, each of which, when fully formed, has a surface area of 1 cm2 (Figure D.1). All that a new lily needs in order to thrive in this imaginary (and certainly idealized) world is enough room to grow to its full size—that is, to 1 cm2.

Now, the ratio of the area of a single water lily to the area of the entire lake is 10~10; in other words, a fully covered Lily World can support up to a 10 billion water lilies. This is Lily World's carrying capacity. In the case of Lily World we know what the carrying capacity is from the very outset (for Earth this quantity is still a Figure D.1. The kingdom of Lily World is a 1-km-sided square of water upon which identical 1-cm-sided square lilies can grow. It is a finite world that can support at any one instance a maximum of 10 billion lilies.

Figure D.1. The kingdom of Lily World is a 1-km-sided square of water upon which identical 1-cm-sided square lilies can grow. It is a finite world that can support at any one instance a maximum of 10 billion lilies.

debated issue), and the question that now arises is how long it will take before the carrying capacity is exceeded. Let us begin by setting up a time step of, say, one day such that we can express the growth rate in Lily World as the number of lilies that are produced and die per day.

As a first example, let us assume that each lily once formed lives forever (i.e., the death rate is zero per day). Under these circumstances, the population will just keep growing until the carrying capacity of Lily World is reached, and then the birth rate will switch to zero births per day. We can now ask how long might it take for the birthrate of lilies to go to zero? The answer to this question is simply the length of time it takes to generate 10 billion lilies (assuming we start with just one lily), since at this point the total area of all the lilies will equal that of Lily World—1 km2. If we assume that the number of lilies doubles every day, then the population will increase in the following manner: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 ... and so on, such that after T days the population will be 2(t~1'. With this doubling birthrate, the carrying capacity of Lily World will be exceeded after just 33 days, since 233 = 8.6 billion, and 234 = 17 billion. It is incredible to think that given a population that doubles every day a single 1-cm square water lily, which could be comfortably held in the palm of one's hand, can within the time span of about a month become a legion 10-billion strong covering an entire 1 km2 of water. To go from something so small and harmless to a population that literally throttles its world is sobering, and it is representative of a key problem that humanity must, in the very near future, deal with.

One important point to note from the population numbers for days 33 and 34 is that the buildup to exceeding the carrying capacity is exceptionally rapid. The population of lilies does not creep up to its limit. It literally crashes right through it. If we look at the area occupied by the lilies compared to the carrying capacity [i.e., the ratio A = N(T) x 1 cm /1km2, where N(T) is the number of lilies on day T], then on day 33, A = 0.43, on day 34, A = 0.86, and on day 35, A = 1.72. In this case, we see the remarkable situation that just 2 days before the final collapse of Lily World, the area occupied by the lilies is less than half of the carrying capacity. One day before the collapse (day 32), 14% of Lily World is still open for colonization. On day 34, if it were actually possible, the number of lilies would occupy an area nearly twice as large as the carrying capacity of Lily World. After 40 days, again if possible given the initial assumptions, the combined area of lilies would cover a 110 km2 lake.

If, as opposed to doubling every day, the number of lilies trebled every day, then the carrying capacity of Lily World would be exceeded after just 22 days, and if the number of lilies doubled every fifth day (say), then the carrying capacity would be exceeded after some 170 days. By now, the end result should be clear and obvious, and as Thomas Malthus so aptly stated the situation in 1798, ''The increase of population is necessarily limited by the means of subsistence.''

Let us try another model. Clearly, as we assumed earlier, lilies do not live forever, and so let us introduce a death rate. Again, let us take an idealized death rate (given that such a terminal experience can be idealized) such that after a finite number of days a lily simply disappears, leaving its previously occupied area ready for a new lily to grow into. Keeping the growth rate to be the same as in our earlier example, such that the population on day T is equal to twice that of the population alive on day T— 1, and also imposing a finite lifetime of (say) 8 days, what now is the time to reach the carrying capacity of Lily World? The day-by-day increase in the number of lilies will now proceed in the following manner: 1, 2, 4, 8, 16, 32, 62, 128, 256, 510, 1016 ... and so on. We can see in this sequence that the effect of the death rate kicks in on day 10, when the number of lilies is 510 rather than 512, when the death rate is zero. So, we can see that the population of lilies is increasing more slowly, but the inevitable still occurs, and the carrying capacity of Lily World is exceeded after just 35 days. This result shows that if lilies live for 8 days then Lily World lasts just 2 days longer than the utopian case when lilies never die.

Again, if we look at the area ratio of lilies to the carrying capacity, then on day 35, A = 0.78, and on the day 36, if further growth was allowed, A = 1.55. Once again, the approach to the carrying capacity is extremely rapid, and the population of lilies on day 35 could be excused (if they were sentient) from thinking that anything about their future was amiss, since some 23% of Lily World would still be open for colonization. If the lifespan of an individual lily is reduced to say 4 days, then the population increases as 1, 2, 4, 8, 16, 30, 56, 104, 192, 352, 644 ... , and the carrying capacity is exceeded between days 39 and 40, extending the lifetime of Lily World by about a week compared to the infinite lily lifetime calculation. The shorter the lifetime of the lilies, so the greater is the amount of time required to approach the carrying-capacity limit, but the point is, the population always reaches the carrying-capacity limit if the birthrate is greater than the death rate.

As a final example, let us consider the situation where the birth rate of lilies varies according to the following rule: When A < 1, then Births (T) = (1 - A) [2*Births(T—1)-Deaths(T—1)] + A*Death-s(T— 1). This rule says that when the total area of lilies is very much less than the carry capacity of Lily World (this is the A << 1 condition), then the number of births at a given time T is determined by the number of lilies alive at time T— 1 minus the number of lilies that will have died since time T— 1. In addition, while A < 1, the amount by which the number of lilies increases in each time interval is weighted by the factor (1— A), and this indicates that when A is small this factor will have virtually no effect, but as A gradually increases so the weighting term decreases.

If we think a little more about how this rule is going to work, we can see that there are two limits. To begin with, when A is near zero, the birth rate increases as described in our second example, where the number of lilies doubles on each time step and each lily lives for a fixed number of days. As the carrying capacity of Lily World is approached, however, so A ) 1, and the birthrate will begin to approach an equilibrium with the death rate; in other words, the population is no longer increasing but simply maintaining a dynamic balance, with births and deaths marching on, adding to and extracting lilies from the total population exactly in step. In this fashion, although each lily still only lives for a finite amount of time, Lily World will last forever, supporting lilies at its maximum capacity (corresponding to A = 1).

If we now go back to the calculation in which each lily lives for 4 days, then the population initially increases in exactly the same manner as 1, 2, 4, 8, 16, 30, 56, 104, 192, 352, 644 ... Rather than exceeding the carrying capacity between days 39 and 40, however, with the new birthing rule the population levels off after day 40 and remains constant for all times T > 40 days. Rather than Lily World lasting for just an extra week due to the finite lifetime of each lily, it now lasts indefinitely. The manner in which the ratio of the total Figure D.2. The variation of the ratio of the total area of lilies to the carrying capacity (A) for the various scenarios described in the text. The time interval shown is from day 25 onward, since for all time steps prior to the 25th day the area ratio is essentially zero. The carrying-capacity limit occurs at A =1, and if the population exceeds this limit then collapse in inevitable (the collapse is not shown in the figure). The D labels indicate the lifetime of each lily: D = 1 corresponds to an infinite lifetime, while D = 8 indicates a lily lifetime of 8 days and D = 4 indicates a lily lifetime of 4 days. The D = 4+ limit curve corresponds to the situation where the birthrate approaches the death rate as A approaches unity.

Figure D.2. The variation of the ratio of the total area of lilies to the carrying capacity (A) for the various scenarios described in the text. The time interval shown is from day 25 onward, since for all time steps prior to the 25th day the area ratio is essentially zero. The carrying-capacity limit occurs at A =1, and if the population exceeds this limit then collapse in inevitable (the collapse is not shown in the figure). The D labels indicate the lifetime of each lily: D = 1 corresponds to an infinite lifetime, while D = 8 indicates a lily lifetime of 8 days and D = 4 indicates a lily lifetime of 4 days. The D = 4+ limit curve corresponds to the situation where the birthrate approaches the death rate as A approaches unity.

area of lilies compared to the carrying capacity of Lily World (the A term) varies with time is illustrated in Figure D.2. The lines labeled D = i, 8, and 4 in the figure correspond to r-process or Malthusian growth, while the curve labeled D = 4+ limit is a K-process, or logistic model growth.

What are the lessons to be learned from the idealized workings of Lily World? Perhaps the first lesson is that Thomas Malthus, writing in 1798, was exactly right, and that in a world with finite resources the population cannot increase indefinitely. There is a limit to the population that any finite world can support (irrespective of any increase in the food yield per acre of land bioengineering might produce), and if the population exceeds that limit, then it is doomed. Second, we find that if the number of births over a given time interval is greater than the number of deaths in that same time interval, then the carrying capacity will always be exceeded (sooner or later), and the population must necessarily crash. A stable, long-lived population of lilies that fully utilizes the available carrying capacity, however, can be produced, but this situation requires that the rate of increase in the population be carefully controlled so that the net growth rate (i.e., the number of births minus the number of deaths in any given time interval) goes to zero as the population approaches the carrying limit.

In Lily World the carrying capacity (by construction) was known from the very outset. The problem in the real world is that there is no consensus between researchers as to what human population the Earth can reasonably carry. The only point and it is a key point that all researchers do agree upon is that the Earth's carrying capacity has almost certainly been reached, and indeed, many researchers also believe that it was actually breached some time ago. Irrespective of whether we are talking about the Earth or a terraformed Mars or Venus, or even a Ceres world, the message for humanity is the same: utopias (if one dares to use such a word) can come about, but they are fragile, finite, and subject to change. Most importantly, however, it is abundantly clear that no one world (utopian or otherwise) can support for very long an unconstrained increase in its population.