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Linear Growth Versus Exponential Growth
Logistic Growth Versus Exponential Growth
Compound Growth Versus Exponential Growth
Exponential Brownian Motion
The Mechanism Of Population Doubling
Generations and Population Doublings
What Is Exponential?

External Links:
Malthusian Growth Model - by Steve McKelvey.
Mathematical Modelling in a Real and Complex World - by the Connected Curriculum Project
Exponential Growth and The Rule Of 70
- by EcoFuture 

Wikipedia - Malthusian Catastrophe

Mathematical Modelling - Malthusian Growth and Population - Joseph M. Mahaffy
Mathematical Modelling - Logistic Growth  - Joseph M. Mahaffy

Ecology and Evolution: Biology 301 (Exponential and Logistic Growth)-- Wethey

Population Growth Models - Logistic Growth. Cindy Klevickis
Exponential and Logistic Growth
- Dept. of Entomology, Virginia Tech, Blacksburg, VA. By Alexei Sharov

Logistic Growth versus Exponential Growth (and Couttsian Growth)

Introduction

In this article I will examine two fundamental models for growth - logistic growth and exponential growth. What are they? What are the distinguishing features of each?

Let us take a starting figure of 1 million in 1700, and apply a positive annual growth rate for both models:

  Start 1725 1750 1775 1800 1825 1850 1875 1900 1925 1950

Exponential Growth

1 2 4 8 16 32 64 128 256 512 1024
  Start 1750 1800   1850 1890 1900 1950 1960    
Logistic
Growth
1 2 4   20 50 60 130 150    
  Start 1750 1775 1815 1830 1860 1890 1930 2000 ?  
Actual Growth 1 2 4 8 16 32 64 128 256 512  

Table A: Exponential Growth prediction, Logistic Growth prediction and Actual Growth of US population

In the Exponential Growth prediction, I am taking Benjamin Franklin's 1751 observation regarding the 25 year doubling period of the population of the USA. This equates to an average constant growth rate of 2.8% per annum. 

For the Logistic Growth prediction (again of the population of the USA), I am drawing on Figure 5.5  from How Many People Can The Earth Support by Joel E. Cohen (1995). His sources are Pearl and Read (1920), who assumed an upper asymptote (carrying capacity) of 197.27 million, US Bureau Of The Census (1974) and United Nations (1993).  

The Actual Growth of the population of the USA is based on figures from "Atlas Of World Population History" by Colin McEvedy and Richard Jones. Both the Exponential Growth and Logistic Growth models appear to have failed in their predictions, compared to reality, though this graph appears to indicate a closer fit to reality for the logistic growth model: 

Graph A: Population Model Predictions and Actual Growth Of Population of USA From 1700

Logistic Growth - a failed model?

In How Many People Can The Earth Support, Joel L. Cohen (1995) provides a brief history of failed attempts to fit the actual growth of human populations. This of course starts with Pierre Francois Verhulst, a professor of mathematics in Brussels, Belgium (who coined the term 'logistic curve' in 1845):

"Verhulst fitted the logistic curve to three censuses of the American population. His predictions  of future American growth did not turn out well."

In 1920 it was the turn of biometry and vital statistics professor Raymond Pearl and his colleague Lowell J. Reed, who unwittingly repeated Verhulst's efforts of 80 years earlier (see Table A above). Writing during the Great Depression, in 1924 they went on to predict an upper limit for the Earth's human population of 2 billion people - this was passed in 1930! A later attempt by Pearl and associate Sophia Gould in 1936 then estimated an upper limit of 2.6 billion - this was passed by 1955.

Cohen mentions a 1992 report (Tuckwell and Koziol) that promoted the use of the logistic model for population growth predictions around the world. The model was initially successful, but failed following a baby boom post World War II. The report states "...Thus there are apparently logistic-type regimes which persist until some major event occurs." As Cohen (1995) puts it:

"That is, the logistic curve works until it doesn't.

The principal assumption of the logistic curve, that the population growth declines in a straight line as the population increases, is flatly contradicted by the increasing population growth rate throughout most of the history of world population up to 1965...."

Logistic Growth - still a preferred model

Despite these failings, most articles that I have read on the subject prefer the logistic model to the simple exponential model (see links above, or search the internet for the words logistic and exponential). Here's a classic example from the Wikipedia entry for Malthusian Catastrophe which argues:

"...no strong evidence that the human population—nor any real population—actually follows exponential growth. In plant or animal populations that are claimed to show exponential growth, closer examination invariably shows that the supposedly exponential curve is actually the lower limb of a logistic curve, or a section of a Lotka-Volterra cycle. Also, examination of records of estimated total world human population shows at best very weak evidence of exponential growth..."

At least the simple exponential growth model (the Malthusian Growth Model) is a complete model. Yet in this Wikipedia article it is suggested that it's better to use "the lower limb of a logistic curve" or "a section of a Lotka-Volterra cycle".  This is worse than Cohen suggests. Cohen argued that proponents of the failed logistic curve model only admit to failure when some major event occurs. This Wikipedia article argues that, if we select bits of logistic curves, or sections of cycles, then we should be able to make the growth model fit the facts nicely!

So why the preference for the (failed) logistic model? The main argument seems to be the apparent lack of a feedback mechanism to restrict growth in the exponential growth model. The Malthusian Growth Model for exponential growth is invariably explained as flawed because a model using a constant rate of growth recognises no limit to growth. That's ironic, because Malthus was the first to use this version of his model to explain that such growth would therefore trigger checks on population to prevent such unrestrained growth (refer Malthus - An Exponentialist View). As Dennett (1995) puts it:

""It was Malthus who pointed out the mathematical inevitability of such a crunch in any population of long-term reproducers - people, animals, plants... 

So the normal state of affairs for any sort of reproducers is one in which more offspring are produced in any one generation than will in turn reproduce in the next. In other words, it is always crunch time."

As I explain below, the dismissal of the Malthusian Growth Model reflects a lack of understanding of Exponential Growth which has two basic models - Constant Rate, and Variable Rate. The other obvious failure is to understand that both models accommodate positive population growth and negative population growth.

For the record, whilst I disagree with Cohen's assessment of the exponential growth model (because he ignores Variable Rate exponential growth and negative population growth - see Paul R. Ehrlich and the prophets of doom - An Exponentialist View), I tend to agree with his assessment of the flawed logistic growth model. In fact, as I explain below, any set of population growth figures produced via a logistic growth model can be derived via the Couttsian Growth Model (variable compound interest rates, including negative and positive). 

Exponential Growth - Constant Rate

For a starting population of 100 at a 1% exponential growth rate, 980 years of growth would look like this:

Exponential Growth

100 200 400 800 1600 3200 6400 12800 25600 51200 102400 204800 409600 819200 1638400
Years 0 70 140 210 280 350 420 490 560 630 700 770 840 910 980

Table B: Starting population of 100 at 1% exponential growth rate

When graphed, this appears as follows:

Graph B: Starting population of 100 at 1% exponential growth rate

For a starting population of 100 at a 2% exponential growth rate, a 490 years of growth would look like this:

Exponential Growth

100 200 400 800 1600 3200 6400 12800 25600 51200 102400 204800 409600 819200 1638400
Years 0 35 70 105 140 175 210 245 280 315 350 385 420 455 490

Table C: Starting population of 100 at 2% exponential growth rate

When graphed, this appears as follows:

Graph C: Starting population of 100 at 2% exponential growth rate

I have taken the liberty of using the Rule Of 70 to quickly extrapolate my results for exponential growth (Note: The Rule of 70 is only useful for growth rates between negative 7% and positive 7%). Approximate as this method is, the sheer power of exponential growth is revealed. For an even more powerful example, look at the relative "stability" of the population growing at 1% when compared with that growing at 2%:

Graph  D: Comparison of populations doubling every 35 years, and every 70 years

Malthus took great care to provide examples from around the world of negative population growth (today called exponential shrinkage). As explained further in Negative Growth - Constant Rate, a population which maintains a constant rate of exponential shrinkage is subject to population halving. Hence, the limit to growth problem does not arise if a population is subject to a constant rate of negative growth. The problem here is one of extinction!

The Malthusian Growth Model is only in trouble if the constant rate of growth is positive. It is in trouble because, as Malthus himself helped to point out (see Malthus - An Exponentialist View), any population which sustains positive population growth could fairly quickly cover the Earth - clearly an absurdity! However, if the upper growth limit is reached, simply invoke a negative rate of growth to reduce the population below the limit to growth. The negative rate of growth could be a constant rate, even the same absolute value as the previous positive rate used to reach the limit to growth. This is roughly what Malthus expected. 

Although Malthus never proposed the idea of a lower growth limit, it is easy to see that if an arbitrary upper limit to growth is set, and an arbitrary minimum limit is also set, then the population can oscillate between constant positive and constant negative rates of growth. 

Exponential Growth

100 200 400 800 1600 3200 6400 12800 6400 3200 1600 800 400 200 100
Years 0 35 70 105 140 175 210 245 280 315 360 395 430 465 500

Table D: Starting population of 100 at 2% exponential growth rate. Assume 100 is minimum population, and 12800 is maximum population. If population hits 12800, negative growth used. If population hits 100, positive growth is used. This population is always growing (or shrinking) exponentially at 2%.

Graphed, it looks like this:

Graph  E: Malthusian Growth Model flipping from constant positive 2% to constant negative 2%

This example shows that it is possible to use the Constant Rate version of the Malthusian Growth Model and maintain a state of perpetual exponential growth. Okay, so it requires the constant rate to switch between positive and negative rates, but it's much closer to what Malthus proposed than he is usually given credit for. Most writers limit their arguments to dismissing Malthus on the grounds that a population cannot grow endlessly at a constant positive rate of growth. Rather than dismissing Malthus, they prove their own inability to appreciate the true nature of the Malthusian Growth Model (so I've given it a new name, the Couttsian Growth Model). Now, pure constant rate exponential growth is explored via the Malthusian Growth Model, and variable rate exponential growth (specifically catering for both positive and negative rates) is explored via the Couttsian Growth Model.

Perhaps the arbitrary nature of the upper and lower limits in this example highlights the similarly arbitrary nature of a carrying capacity in the logistic model. Fortunately, we don't need to worry about exponential growth at a constant rate (negative and / or positive). Malthus actually had something far more realistic, and universal in its application.

Exponential Growth - Variable Rate

The true nature of the exponential growth of populations does not reveal itself until variable rates of growth are considered. In fact, Logistic Growth uses variable rates of growth which are inexorably tied to the limit to growth.

However, I am interested in exponential growth at variable rates. How does it work in the real world? I have explored this real-life nature of population doubling (and halving) in the article The Scales Of 70, which provides a simple model for analysing historical population growth examples and predicting future population growth.

 I find that population doubling and halving are adequate to explain almost all cases of actual exponential growth. The Couttsian Growth Model itself is universal in its application across all time for all replicator populations. Should you require a greater degree of granularity than is provided through the use of the New Malthusian Scale (and its reliance on population halving and population doubling) then I recommend the use of the Scales of e.

For those that doubt the assertion that variable rates of growth result in exponential growth, consider the following example. For a starting population of 100 at exponential growth rates between 1% and 2%, 700 years of growth could look legitimately like this:

Exponential Growth

100 200 400 800 1600 3200 6400 12800 25600 51200 102400 204800 409600 819200 1638400
Years 0 70 105 150 200 250 300 370 405 440 500 570 605 665 700

Table E: Starting population of 100 with periods of variable rates of exponential growth between 1% (doubling time 70 years) and 2% (doubling time 35 years)

Graphed, it looks like this:

Graph  F: Population Doubling Between 35 and 70 years

It looks remarkably exponential in nature, doesn't it?

Remember, using the Rule Of 70 the population doubling period at 1% is 70 years and at 2% it is 35 years. You will note that the population growing at variable rates between 1% and 2% (inclusive) always doubles somewhere between 35 years and 70 years. 

Exponential Growth (Variable Rate) - Universal Laws 

This leads to the following universal laws:

The following universal laws defined by the Couttsian Growth Model apply for both constant and variable rates of growth:

Here is Malthus' argument for exponential population growth (A Summary View - 1830):

"The immediate cause of the increase of population is the excess of the births above deaths; and the rate of increase, or the period of doubling, depends upon the proportion which the excess of the births above the deaths bears to the population."

The opposite case can be used to provide an argument for exponential population shrinkage:

"The immediate cause of the decrease of population is the excess of the deaths above births; and the rate of decrease, or the period of halving, depends upon the proportion which the excess of the deaths above the births bears to the population."

Together, these two statements lay the foundations of the Couttsian Growth Model which alone can explain the results shown in Table E. Perhaps it would be simpler all round if we called such real-world growth Malthusian. For a real-world example, see Total Midyear World Population for 1950 to 2050 AD from the US Census Bureau.

Malthus is so clearly right that I wonder why it is still necessary to try and persuade people (including many professional demographers and evolutionists) that variable growth rates result in exponential growth. Still, after so long, people continue to refute Malthus' claim. 

Producing Logistic Growth figures via the Couttsian Growth Model

Taking another look at the Logistic Growth prediction from Table A, how can the refined version of the Malthusian Growth Model  - the Couttsian Growth Model - produce the same growth? Table F below slots in some dates borrowed from the actual growth of the American population (really, any ascending dates that fit within the logistic growth dates below will do):

  1700 1750 1800 1815 1830 1850 1860 1890 1900 1930 1950 1960 2000

Variable Exponential Growth

1 2 4 8 16 20 32 64 75 128 130 150 256
Average annual growth rate since last double   1.40 1.40 4.67 4.67   2.33 2.33   1.75     1.00
Variable Doubling Period   50 50 15 15   30 30   40     70
Logistic
Growth
1 2 4     20   64 75   130 150  

Table F: Starting population of 1 million at variable rates of exponential growth between 1% and 5%. Dates in red "borrowed" from Actual Growth of American population. Bolded population figures show doubling series

Note that it doesn't really matter what the actual variable growth rates are between doublings, so long as the average is the same as that listed in Table F. 

Graphed, it looks like this (the two are identical):

Graph  G: Variable Exponential Growth can match ANY Logistic Growth curve, without the need to factor in shaky assumptions on fixed limits to growth

Although variable exponential growth (Couttsian Growth) can match the growth reflected in any Logistic Growth curve, the reverse is not true. The Demographic History Of Egypt provides a classic example. The Couttsian Growth Model accommodates such growth, whereas the Logistic Growth Model cannot explain it at all.  This demonstrates the universal applicability of the Couttsian Growth Model, and the ultimate failure of the logistic growth model.

Couttsian Growth Model - The Challenges

Despite the fact that the Couttsian Growth Model represents a universal law of nature for all replicator populations, and is invaluable for explaining any historical scenario of population growth for any species, how does it help us to predict the future course of population growth for a population? After all, a good scientific theory does not just explain historical or current facts, but has the power to predict future developments.

Of course, as stated, we have the universal laws stated above. More than that, we can use The Scales Of 70 to fairly easily and accurately predict when a population will halve or double just by adding up the rates of growth and counting how many rates we have added together. 

The challenge is to predict what those growth rates are going to be, particularly for the human race. Some discussions on that point can be found in Human Replicators - An Exponentialist View.  What we do know is that as limits to growth are approached, either replication rates must fall or death rates must rise (or a combination of the two), thus lowering the overall growth rate (and perhaps turning the rate negative to produce shrinkage). Whatever happens, another period of exponential growth at a variable rate (positive or negative) can be applied to our population. Hence, the Couttsian Growth Model accommodates growth within limits.

Perhaps the greatest challenge is to show how discrete replicator populations within a species interact, and how replicator populations of different species interact, each subject to the Couttsian Growth Model. See Evolution - An Exponentialist View for more. The only limits to growth that should be imposed on any such super-model must be the known limits, such as the surface size or mass of the Earth, just as Malthus did. 

Visualise - just as Darwin and Wallace did intuitively - all populations of all species subject to the Couttsian Growth Model (and so, Couttsian Growth or Couttsian Shrinkage) at all times. Nature pits these populations in a struggle for existence, a struggle of endless, powerful exponential forces restrained within limits to growth. Often these forces balance out and may not approach the limits to growth, or rise and fall in dynamic equilibrium, but all replicator populations possess the power to explode given the chance. An algae bloom here, a plague of locusts there. A viral population explosion during the Spanish Flu caused a massive death toll amongst humans, just as the ongoing explosion of human population now causes an era of species extinction on a global scale.

Any Idiot Knows...

Any idiot has heard of exponential growth (W. Michael Kelly, 2002. "The Complete Idiot's Guide To Calculus"):

"Most people have an intuitive understanding of what it means to have exponential growth. Basically, it means that things are increasing in an out-of-control way, like a virus in a horror movie. One infected person spreads the illness to another person, then those two each spread it to another. Two infected people becomes four, four becomes eight, eight becomes sixteen, until it's an epidemic and Jackie Chan has to come in and save the day, possibly with karate kicks."

Elapsed Time
(weeks)
  7 15 27       52
Virus 
(Millions of infected people)
1 2 4 8 9 13 15 16
Elapsed Time
(weeks)
        28 34 42 52
People 
(Millions of uninfected people)
16 15 13 9 8 4 2 1
Total people
(millions)
17 17 17 17 17 17 17 17

Table A. Couttsian Growth Model in action, showing interaction between human population
and non-fatal viral population over one year

As you can see, the assumption is that nobody dies from the virus, and the population is thus stable at 17 million. The viral population is measured by numbers of infected people, or replicator vehicles as Dawkins might call them. The infected population is subject to variable-period population doubling, and the uninfected population is subject to variable-period population halving.

In this scenario, there is no need to worry about the limit to growth problem, because over time the situation depicted in Table A would simply reverse itself with the infected population becoming subject to variable-period population halving, and the uninfected population becoming subject to variable-period population doubling.

Logistic Growth - still a preferred model

Any idiot knows that the Malthusian Growth Model doesn't apply to real-life populations (W. Michael Kelly, 2002. "The Complete Idiot's Guide To Calculus"):

"Truth be told, there are not a lot of natural cases in which exponential growth is exhibited. An exponential growth model assumes that there is an infinite amount of resources from which to draw. In our epidemic example, the rate of increase of the illness cannot go on uninhibited, because eventually, everyone will already be sick. To get around such restrictions, many problems involving exponential growth and decay deal with exciting things like bacterial growth."

Of course, this limitation of the exponential growth model only occurs if we restrict the model to a constant rate of growth. At this point, the Logistic Growth Model (which doesn't actually seem to work very well in practice) is trotted out as the answer to modelling real-life population problems which - any idiot knows - must face limits to growth. In introducing logistic growth Kelly, slightly confusingly, then discusses the virus epidemic again, only this time it is 100% fatal (W. Michael Kelly, 2002. "The Complete Idiot's Guide To Calculus"):

"A more realistic example of growth and decay is logistic growth. In this model, growth begins quickly (it basically looks exponential at first) and then slows as it reaches some limiting factor (as our virus could only spread to so many people before everyone was already dead - isn't that a pleasant thought?)."

Kelly is in good company, as physicist James Trefil (2002), also prefers the logistic growth model as:

"...a better representation of the growth of real populations than the simple exponential".

Elapsed Time
(weeks)
  9 18 27       52 54
Active Virus
(Infected alive people -1000's)
1 32 512 512 256 128 64 32 0
Total Virus
Total of infected and killed  people
(millions)
1 2 4 8 9 13 15 16 17
Elapsed Time
(weeks)
        28 34 42 52 54
People 
(Millions of uninfected people)
16 15 12 7 8 4 2 1 0
Total people
(millions)
17 17 17 17 17 17 17 17 17

Table B. Couttsian Growth Model in action, showing interaction between human population
and 100 % fatal viral population over one year

As Kelly pointed out, the early growth of the viral population is expected to be exponential (at a constant rate), which I've shown in Table B (regular doubling every 9 weeks, followed by a 25 week doubling period). However, although the Couttsian Growth Model easily accommodates growth at a constant rate, in reality I expect variable rates and variable doubling and halving periods to be the norm and not the exception.

Note that the viral population can be viewed as the accumulation of all those infected and killed by the virus. The active, replicating viral population is thus a subset of the total number infected and killed by the virus. I would imagine that the active, replicating viral population would quickly reach its maximum (in my example, a maximum of around 512,000 living infected people). Regardless, as the last person dies, so the viral population must die off (or at least enter a dormant state...).

In this scenario, the limit to growth does not represent a problem to the Couttsian Growth Model. The total viral population grows exponentially (via Couttsian Growth) as the human population shrinks exponentially (via Couttsian Shrinkage). For growing populations, variable positive rates of compound interest result in variable population doubling periods. For shrinking populations, variable negative rates of compound interest result in variable population halving periods. 

In fact, the Couttsian Growth Model applies to all populations of any species at any time, simply by using variable growth rates and allowing those rates to be positive and negative. Why is it that we must immediately leap from one broken model (exponential growth and decay at constant rate) to another (logistic growth and decay), with no discussion whatsoever of the model that actually works in real-life (exponential growth and decay at variable rates - the Couttsian Growth Model refinement of the Malthusian Growth Model)?

In the Table B example, the active viral population - in the process of exterminating the human population - itself grows exponentially for a while before then shrinking exponentially to its own extinction.

Hence, any idiot should know that the Malthusian Growth Model works perfectly well within limits to growth. It is only the insistence that exponential growth requires a constant rate of growth which prevents people from seeing this. As soon as you include variable rates of growth within the Malthusian Growth Model then a scientific law is revealed, through the Couttsian Growth Model. Suddenly you get variable periods of population doubling and variable periods of population halving, just like real-life. 

Bibliography

The Complete Idiot's Guide To Calculus - W. Michael Kelly (2002)

Cassell's Laws Of Nature - James Trefil (2002)

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Copyright 2001 David A. Coutts
Last modified: 15 December, 2011