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?  

Exponential Method

Couttsian Growth Model (available for peer review) - Academic Publishing Wiki

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

e the EXPONENTIAL - the Magic Number of GROWTH - Keith Tognetti, University of Wollongong, NSW, Australia

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

Population Growth Models

(The Search For Holy Chaos)*


The basic population growth models on offer seem to be those with no inherent limit to growth:

  Fixed Rate
(or Constant Rate)
Variable Rate
Simple Interest Linear Growth Model None
Compound Interest Exponential Growth Model
(also known as the Malthusian Growth Model)
Couttsian Growth Model 
( see also Compound Growth)

A very early but hopelessly naive attempt at a population growth model is the Fibonnaci numbers.

Additionally, the Logistic Growth model features an inherent limit to growth, which acts as a feedback mechanism to cause the growth rate to decline as the population increases.

Note that Malthus referred to linear growth as arithmetic growth, and to exponential growth as geometric growth.

As this is the exponentialist web site, I felt it to be worthwhile comparing exponential growth (and Couttsian Growth) against linear growth, compound and logistic growth (see links to these articles - above left). As you will see, it is argued that Couttsian Growth represents a universal law of population growth and is exponential in nature.

Population Dynamics

In the complex world of population dynamics, factors which can influence the population growth rate are said to be endogenous (affecting the population density and dependant on it) or exogenous (affecting the population density, but independent of it), resulting in population dynamics models typically described as stochastic, deterministic or a combination of both (Turchin, 2003). In principle, endogenous and exogenous factors can result in fixed rate or variable rate growth (or shrinkage). However, it is much more likely in practice for the combined effect of all factors to result in variable rates of growth (rather than a fixed rate of growth) or variable rates of shrinkage (rather than a fixed rate of shrinkage).

Generally speaking, to avoid burdening the reasoning with the usual baggage of assumptions, the Exponentialist approach is not to focus primarily on why populations grow and shrink. So, here at least, you can forget terms such as exogenous, endogenous, stochastic and deterministic. They are not used elsewhere in the Exponentialist web site (except the Glossary, and the article - Peter Turchin - An Exponentialist View). The Exponentialist framework used to explore the secondary issue of why populations grow and shrink can be summed up by the familiar theory of Natural Selection (and associated Artificial Selection), which in turn leads to Exponentialist theory of Malthusian Selection (which includes a number of exogenous factors such as seasons).

Instead, the Exponentialist focus is on how populations grow and shrink. It is the per capita rate over time which is the prime focus. Hence, the key questions are not around what drives the per capita rate but what happens to the population if that rate stays constant, and what happens if the rate can vary.

As you will see, this approach fundamentally challenges the usual assumptions built into population growth models around limits to growth and population density feedback mechanisms. This is not to say that limits to growth and population density are not important - they are important. But the first thing is find a single population growth model which applies to all populations of all species for all time. To do that we have to go back to the first principle of population dynamics (Turchin, 2003) - the approximate exponential law first introduced by Malthus in 1798. 

e - The Black Jewel Of The Calculus

Take some graph paper and, for Population A,  plot an exponential curve for a constant rate of growth for 70 years of 1%. Now , for Population B, plot an exponential curve for a constant rate of growth for 70 years of 2%. You will notice, as per the Rule Of 70, that your starting figure has doubled in both cases. A population growing at 1% doubles in 70 years, and a population growing at 2% doubles in 35 years (or doubles twice in 70 years).

Now for Population C plot points for a third line, but this time vary the growth rate for each year between 1% and 2% inclusive for as long as it takes you to double your original figure. Feel free to use the same rate more than once, but never twice in a row (not that it actually matters, but I wouldn't want to confuse people with moments of apparent constant rate exponential growth in this example). Use as many rates as you like.

You will have plotted a wiggly line (not an exponential curve) that runs between your two exponential curves (for 1% and 2%). Your population will double between 35 and 70 years. Population growth plotted in all of the space on your graph between the two exponential curves can be produced by the same exponential growth function as produced the two exponential curves. The only difference is that it requires a constant rate of growth to produce an exponential curve. 

In the scheme of things, as an aid to our understanding of population growth, it is plain silly to insist that the dictionary is right and that we must have a constant rate of growth to produce exponential growth. Given that we so rarely see constant rates of growth in nature, it would be far more useful to define exponential growth as growth based on either variable rate or constant rate compound interest. 

Variable rate exponential growth is what we see in real-world of population growth, and the much less-useful constant rate growth is what is used to beguile generations of mathematicians, economists, evolutionists and demographers with those pretty little exponential curves. Nonetheless, in recognition of the urgent need to differentiate between growth driven by variable rate compound interest, and growth driven by constant rate compound interest, I propose to adopt the following convention:

For a brief description of a mathematical model based purely on Exponential Growth and Exponential Shrinkage, see Malthusian Growth Model. This model is widely regarded as the basis for an approximate law of nature known as the Exponential Law.

For a brief description of a mathematical model based on Couttsian Growth and Couttsian Shrinkage, see Couttsian Growth Model. Note that this model can also accommodate Exponential Growth and Exponential Shrinkage.

Berlinski (1995) compares logarithmic and exponential functions and makes the following rather poetic observation regarding the nature of the transcendental number e:

"The intellectual movement is one of a soapy wave washing forward to define the logarithmic function, and exposing in the back and forth motion the number e, the black jewel of the calculus. Once uncovered, e serves to unify, to amalgamate, all other exponential functions."

Source: http://www.ima.umn.edu/~arnold/calculus/exponential/exponential1/exponential-g.html
(Used with permission)

Population growth at a constant rate of compound interest results in a constant doubling period, and can be neatly expressed via the series 21, 22, 23, 24 ...  Such exponential growth is recognised at best as an approximate law of nature - namely the Exponential Law. Population growth at variable rates of compound interest results in variable doubling periods. This is Couttsian Growth. What is interesting is that such growth can also be neatly (and meaningfully) expressed via the series 21, 22, 23, 24 ... (See article A New Malthusian Scale). What's even more interesting is that such Couttsian Growth (and Shrinkage) is a universal law of nature for all populations of all species for all time. 

It is also worth noting that a population subject to the Couttsian Growth Model can potentially grow and shrink forever without hitting any limit to growth (if it flip-flops between positive and negative rates). Of course, a population which sustains negative rates of growth will reach the zero limit and go extinct, and a population which sustains positive rates of growth will eventually reach its upper limit to growth and be forced to experience zero or negative rates of growth.

The transcendental number e - the base of natural logarithms - lies at the heart of the Rule Of 70, just as it lies at the heart of my own Scales Of 70 (and the more general model the Scales Of e). So, at the heart of the Couttsian Growth Model lies e, the magic number of growth, the black jewel of the calculus. 

Exponential Growth and The Calculus

According to Trefil (2002):

"When the net increase in a population is proportional to the number of individuals, the population will grow exponentially.

...Now, if dN is the net number of individuals added to the population in a time dt, and if there are N individuals all told in a population, then the conditions for exponential growth will be met if

dN = rNdt"

The letter r (the constant of proportionality) in the above differential equation is effectively the rate of compound interest. Trefil continues:

"Ever since Newton invented the Calculus in the 17th century, we have known how to solve this equation for N, the number of people in the population at any given time....The solution is:

N=N0 e rt

where N0 represents the starting population. 

Thinking back to Populations A and B (see e - The Black Jewel Of The Calculus), the former growing at a constant rate of 1% and the latter at 2%, it is clear that for each year of growth that growth can be considered exponential. Also, for both populations, they are growing exponentially across their entire doubling periods (70 and 35 years respectively).

For Population C, each year of growth satisfies Newton's criteria for exponential growth (at a constant rate). Hence, although the growth over the doubling period (somewhere between 35 and 70 years, depending upon the rates you chose) cannot be said to be growing at a constant rate, it is always (somewhat counter intuitively) growing exponentially. Thus variable rate compound interest results in exponential growth as surely as constant rate exponential growth. 

Once again, e - the black jewel of the calculus - lies at the heart of this proof. Oddly Trefil (2002), who only implicitly credits Malthus with the law of exponential growth (see Green Revolution in Trefil's book), seems to miss this point, preferring the logistic growth model as:

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

If simple exponential is fixed rate exponential growth then perhaps it is, but Malthus gave us the variable rate exponential growth model too, if you look closely enough. This is referred to throughout the Exponentialist web site as Couttsian Growth. This, in turn, is a better representation than the failed logistic model.

Malthus, who first attempted to formalise an exponential model of population growth, philosophised in his 1798 population essay about the nature of original thought and genius:

"The finest minds seem to be formed rather by efforts at original thinking, by endeavours to form new combinations, and to discover new truths, than by passively receiving the impressions of other men's ideas."

Ever modest, among those Malthus included in this elite was Newton (but never himself). 

I would add Malthus to that list.


Complex Population Dynamics - Peter Turchin (2003) * Thanks to Professor Turchin for the phrase from Alfredo Ascioti 

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

A Tour Of The Calculus - David Berlinski (1995)

An Essay On The Principle Of Population (1st Edition) - Reverend Thomas Robert Malthus (1798)

If you have any constructive criticism on this article, email me.

Send email to exponentialist@optusnet.com.au with questions or comments about this web site.
Copyright 2001 David A. Coutts
Last modified: 21 September, 2009