Population growth is not accurately modelled by an exponential function over the long term.  Often, those unfamiliar with biology assume that since an exponential function was attributed to biological population growth in some textbook or was observed for some time in nature, the world is on the verge of catastrophe.

The usual function that is assumed to represent a population over time is the simple exponential function (don't worry if you aren't mathematically-inclined):

where

• t = number of time intervals
• P = initial population at time t=0
• e = the natural growth constant, e (irrational, about 2.718)
• r = the growth rate of the population over one time interval, expressed in decimal

                    This function looks like the following in general:

Figure 1

In other words, the simple exponential population model says that a population will double in size over a constant period.  That is to say, for example, two people make four people, four people make eight people, eight people make 16 people, etc.

In fact, this function does model populations in environments with unlimited resources, such as...  Hm.  If there were unlimited resources somewhere, we could all have as much as we wanted and there would still be plenty left.  There are, however, some situations where there are effectively unlimited resources, such as a petri dish with a single bacterium.  In such a situation, exponential growth can continue, unimpeded, for many generations.  However, as the bacteria consume the resources available, the total population will stabilize.  Due to the conservation of mass, a bottle with a few bacteria and some bacteria food (whatever those guys eat) at the bottom will not fill to the brim with bacteria (assuming that bacteria have roughly the same density as the food they eat).

Therefore, that old tricky question, "If a bottle of bacteria started with a single bacterium at 11:00 and was full at 12:00, when was the bottle half full?" is purely academic as it ignores resources.  The bottle would have to have been full of whatever bacteria eat at 11:00 for the bottle to be full of bacteria at 12:00.  See, for example, this page.

Making these two simple mistakes (assuming an exponential growth rate and unlimited resources) has led Dr. Albert A. Bartlett to make a crazy video that suggests that, for example, a 5% growth rate in a city today will lead to that city's population doubling in 14 years (due to the rule of 70).

Fortunately for those living in reality (i.e., not Dr. Al), the population growth function is not 

but rather

Where

• t, P, e, and r are as previously defined
• K = the carrying capacity of the environment

This is known as the logistic function and is commonly used in modelling biological populations [3].

Which, in general, looks like the following figure:

Figure 2

In practical terms, this means that population increases sharply for a time, but is limited by the carrying capacity, K.  Notice how the red line approaches 1 (in figure 2, K=1).

That ends up looking a lot less scary than the original pure exponential function.  The true, logistic population growth modelling function has a limit as t tends to infinite: K.  In formal mathematical terms, it is the limit

Population growth can be made to look scary by focusing on the exponential area of the logistic function (roughly -4 to 1 in the above graph).  Doing so produces graphs that look exponential and threaten to reach infinity in short order.  For example, the following graph looks like it predicts that the world population is on the verge of exploding (or "going straight up"):

Figure 3 (See [6])

On the other hand, looking at population predictions over the next 40 years shows the logistic tendency of population growth (in millions) (note logarithmic scale):

Figure 4  (See [2])

Notice how figure 4 shows a part of what figure 2 looks like.  This is not the first time that the population has come to some levelling maximum carrying capacity.  Until 1000BC the population of the earth was stable at roughly 5 million.  In other words, the carrying capacity, K, was 5 million.  The population increased until around 1BC to 250 million (K=250 million).  The next increase began around 800AD and has continued until the present day with the exception of the Black Death and the Plague.  Over the history of the world, human population has generally been limited by a K bound because the exponential function takes over when K is increased.  Why has the the value of K remained constant over long periods and then increased?  In a word, technology. 

When technology (especially agricultural technology) improves, the K bound increases.  During the period between when the K bound is first increased and when the new K bound is hit, an exponential growth occurs for a fraction of that period.  The Green Revolution (no, thankfully that is not an environmentalist movement; on the contrary, it was true progress) has increased our K bound to some new figure (K is around 10 billion in figure 4).

Figure 5 (See [5])

Estimates of where the population of the world will level off are largely estimates of where K will stop, and these estimates are likely unreliable (estimating the limits of technology is a shaky business).  Other factors (such as fear-mongering biological-model-incompetent Physics PhDs) may stop us short of actually reaching K.  One thing is almost sure if we hit K again: we will face the same non-catastrophe as we did all the previous times.  As figure 5 shows, our rate of increase is tapering off since about 1965.  In the grand scheme of things, this probably means we are running up against a carrying capacity.  It is possible that this may be an artificially imposed carrying capacity (for example, we may be selfish and not want to pay for kids, and now that we have the means to easily not have them, we may simply be having fewer).  This seems to be the case, because populations are declining in more industrialized countries rather than in less industrialized countries [1] [7].  Any artificial carrying capacity will be lower than an environmental carrying capacity, so our current predicament is particularly non-threatening.

That same Dr. Al mentioned previously is fond of saying, "The greatest shortcoming of the human race is our inability to understand the exponential function."  I would say in reply that the greatest shortcoming of Dr. Al is his inability to understand the logistic function.  Dr. Al is also fond of asking "Can you think of any problem in any area of human endeavour on any scale, from microscopic to global, whose long-term solution is in any demonstrable way aided, assisted, or advanced by further increases in population, locally, nationally, or globally?"  Yes, Dr. Al: having children.

In other (highly relevant) news, Thomas Sowell just came out with a new book, Intellectuals and Society, and I just bought it.

References

[1] World population. (2010, January 24). In Wikipedia, The Free Encyclopedia. Retrieved 12:23, January 24, 2010, from http://en.wikipedia.org/w/index.php?title=World_population&oldid=356054928

[2] Data from http://esa.un.org/unpp/ as found on [1]

[3] Logistic function. (2010, January 24). In Wikipedia, The Free Encyclopedia. Retrieved 12:37, January 24, 2010, from http://en.wikipedia.org/w/index.php?title=Logistic_function&oldid=355416051

[4] Population growth. (2010, January 24). In Wikipedia, The Free Encyclopedia. Retrieved 01:42, January 24, 2010, from http://en.wikipedia.org/w/index.php?title=Population_growth&oldid=355779197

[5] International Data Base (IDB) (2010, January 24) In US Census Bureau,  Retrieved 01:42, January 24, 2010, from http://www.census.gov/ipc/www/idb/worldpop.php 

[6] Overpopulation. (2010, January 24). In Wikipedia, The Free Encyclopedia. Retrieved 01:55, January 24, 2010, from http://en.wikipedia.org/w/index.php?title=Overpopulation&oldid=356071654

[7] Total fertility rate. (2010, January 24). In Wikipedia, The Free Encyclopedia. Retrieved 02:01, January 24, 2010, from http://en.wikipedia.org/w/index.php?title=Total_fertility_rate&oldid=356064014

Notes

http://www.voxeu.org/index.php?q=node/4508

food to population

introduce malthus

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