Up Exponentialist Evolution - An Exponentialist View Famous Exponentialists Replicators - An Exponentialist View A New Malthusian Scale The Scales Of 70 Population Growth Models Understanding Compound Interest The Exponential Method The Myth Of The Exponential Phase An Exponentialist Glossary What's New EcoFuture - Terms relating to population study

An Exponentialist Glossary

A-life - artificial life, sometimes known as virtual life. These life-forms could be regarded as the first replicators of the virtual world, rising from the primordial silicon within the hard discs of PCs around the world.

A-pop - "alphabet" alternative to pop as used in New Malthusian Scale. Continues through to Z-pop, thus allowing for 26 rows of population doubling

Absolute Increase - with a linear growth model, the absolute increase for a given time period must remain constant. With exponential growth, the absolute increase for a given time period can change. For a population growing at a constant rate of compound interest, the absolute increase itself can be represented via an exponential series.

Actual Population Doubling - historically recorded doubling of a given population.

Actual Population Halving - historically recorded halving of a given population.

Annual Percentage Rate - See my article Calculating the Annual Percentage Yield (APY) For Continuous Compounding. See also Wikipedia article Annual Percentage Rate

Annual Percentage Yield - See my article Calculating the Annual Percentage Yield (APY) For Continuous Compounding. See also Wikipedia article Annual Percentage Yield

Artificial Selection - well known term used to describe manipulation of genetic material through selective breeding of animals and hybridisation of plants. One of the factors which influences differential replication.

Arithmetic Mean - the average of a set of numbers (e.g.. the average of 5, 7, 2 and 6 is [5 + 7 + 2 +6] / 4) = 5. However, when averaging growth rates, the geometric mean should be used.

Birth Rate - term commonly used for replication rate. The term "birth rate" only applies to living entities capable of reproduction.

Cellular Chauvinism - the chauvinistic view that the cell is the smallest unit of life. See Viral Replication - An Exponentialist View for more. For alternative views see Richard Dawkins - An Exponentialist View (for a look at DNA) and Drexler - An Exponentialist View (for a look at Molecular Nanotechnology) for more.

Check - see Population Check.

Compound Growth - growth model which assumes a variable rate of Compound Interest for each period. Professor Don Roper first suggested the term "compound growth" to describe variable rate exponential growth, as the term Exponential Growth is universally (if not very usefully) understood to describe constant rate exponential growth. I have chosen not to take up this suggestion - see article Compound Growth Vs Exponential Growth for more. See also Couttsian Growth Model and Malthusian Growth Model.

Compound Interest - interest paid on principle sum and also on the accumulated interest. This can be at a fixed rate (same as constant rate), or at variable rates. Compound interest is usually expressed as a percentage growth rate for a given (or discrete) period such as an hour, a day, a week, a month or a year (or some multiple of one of these). If the given period is infinitely small then compound interest is said to be continuous. See also Simple Interest. See Wikipedia article Interest.

Compounding Periods - the number of times, for any given period, for which Compound Interest is calculated and applied. See my article Calculating the Annual Percentage Yield (APY) for Continuous Compounding for more.

Constant Rate - a growth rate which never varies, but remains constant for each period. A constant rate of growth is frequently depicted using an exponential curve. See articles Positive Growth - Constant Rate and Negative Growth - Constant Rate for more. See also Variable Rate.

Continuous Compounding - When Compound Interest is calculated and accrued continuously this is known as Continuous Compounding. When calculating continuous compounding the number of compounding periods per growth period is assumed to be infinite. The formula for continuous compounding is as follows:

$a(t)=\left(1+\frac{r}{n}\right)^{n \cdot t}\,$

where

a(t) = amount at time t
t = number of growth periods
r = interest rate
n = number of compounding periods per growth period

Source:
Derived from Wikipedia article Interest

See my article Calculating the Annual Percentage Yield (APY) and Continuous Compounding for more.

Cotes - English mathematician Roger Cotes (1682-1716) who helped Newton edit the second edition of the "Principia Mathematica", stated an early version of Euler's Formula in his own work in 1710, "Harmonia mensurarum" (published posthumously in 1722). Refer Wikipedia Roger Cotes for more.

Couttsian Growth - Given the dogged resistance to the idea that (positive) variable rate compound interest results in a form of exponential growth, and an often blind adherence to the current dictionary definition of exponential growth (which assumes a constant rate), I am coming to the conclusion that it would be better to use a distinct term for growth based on (positive) variable rate compound interest. Amongst the suggestions I've had are "sort-of exponential growth", "exponential-like growth",  "growth" (quite a compliment really, when you consider its universal applicability, but then it causes confusion... after all, have you heard of The "Growth Model"?), and "Compound Growth" (which I quite like, but still find misleading as it also should apply to constant rate exponential growth). I've also read of "faster than exponential growth" in discussions of Moore's Law relating to the ever increasing doubling rate computing power (Moravec, 1998), "superexponential growth" in the same context (Wilson, 2002) and viruses when compared with bacteria (see Viruses - An Exponentialist Explanation), which must therefore allow for the equally useless "slower than exponential growth" model... I also thought of 1) exponentialist growth but an exponentialist is a person with exponentialist views, and 2) Malthusian Growth but this would again imply a constant rate of growth, and 3) I don't like any of the other possibilities, so I've opted for Couttsian Growth. See also Couttsian Shrinkage.

Couttsian Growth Model - a mathematical model describing a scientific law of population growth (and shrinkage) which applies universally to all replicator populations. Named after David A Coutts, the author of this Exponentialist web site. In essence, just like the Malthusian Growth Model, there are only 3 states that a population can be in for any given time period:

• (replication rate + immigration rate) greater than (death rate plus emigration rate) : results in  positive population growth
• (replication rate + immigration rate) less than (death rate plus emigration rate) : results in  negative population growth
• (replication rate + immigration rate) equals (death rate plus emigration rate) : results in  zero population growth

Like the Malthusian Growth Model, the Couttsian Growth Model permits growth at a constant rate of compound growth. The key point to realise is that the Couttsian Growth Model does not rely on a constant rate of growth, but also permits  Couttsian Growth using positive variable rates of compound interest that result in variable population doubling periods. Similarly, Couttsian Shrinkage also uses negative variable rates of compound interest that result in variable population halving periods. Note that both the constant rate and variable rate versions of the model both assume compound interest, and are equally powerful forms of growth.

It is the Exponentialist view that real-world population growth is based on Couttsian Growth and Couttsian Shrinkage. See article The Scales Of 70 for a demonstration of the Couttsian Growth Model, and the underlying significance of the transcendental number e.

Couttsian Shrinkage - Given the dogged resistance to the idea that (positive) variable rate compound interest results in a form of exponential growth, and an often blind adherence to the current dictionary definition of exponential growth (which assumes a constant rate), I am coming to the conclusion that it would be better to use a distinct term for both growth and shrinkage based on variable rate compound interest. For growth based on (negative) variable rate compound interest I've opted for Couttsian Shrinkage. See Couttsian Growth for a fuller explanation of why the term Couttsian is used.

Crude Population Doubling - the rough population doubling period obtained by using the Rule Of 70

 Positive Growth Rate 1% 2% 3% 4% 5% 6% 7% Doubling Period 70 35 23.3 17.5 14 11.6 10

The unit of measurement could be any time period. Typically it is in minutes, hours, days, weeks, months or years.

Crude Population Halving - the rough population halving period obtained by using the Rule Of 70

 Negative Growth Rate 1% 2% 3% 4% 5% 6% 7% Halving Period 70 35 23.3 17.5 14 11.6 10

The unit of measurement could be any time period. Typically it is in minutes, hours, days, weeks, months or years.

Darwin - Charles Robert Darwin (1809-1882), naturalist and co-founder (with Wallace) of the theory of evolution. Author of The Origin Of Species (1859). See article Charles Darwin - An Exponentialist View for more.

Darwinian Wedge - idea introduced by Darwin that individuals of each species strive to increase to increase their numbers, thus pitting populations against one another like wedges in a circle. Force one wedge in and it drives one or more other wedges out. See Darwin's Wedge for more.

Dawkins - Richard Dawkins, British zoologist and evolutionist. See article Richard Dawkins - An Exponentialist View for more (including the original definition of a replicator).

Dawkins' Law - the law that all life evolves by the differential survival replicating entities - Richard Dawkins, The Selfish Gene, 1976, pp191-192. See Dawkins Law section of Richard Dawkins - An Exponentialist View for more.

Death Event - an event which adds to the death rate of a population (the death of a replicator).

Death Rate - universal term for the rate at which members of a population die and thus cease to be living entities. Adding to death rate increases negative population growth or decreases positive population growth

Deterministic - population growth driven by largely non-stochastic factors. See also stochastic.

Differential Reproduction - term commonly used for differential replication. The term "differential reproduction" only applies to living entities capable of reproduction.

Differential Replication - the principle whereby discrete populations of the same or similar replicators rise and fall at different rates in a Malthusian struggle for existence. See Exponentialist article Differential Reproduction for more.

Doubling Series - The standard population doubling series is 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024  - as used in the New Malthusian Scale. See also scientific notation. The doubling series is commonly used in explanations of exponential growth, but exponential growth is not limited to population doubling.

Doubling Period - the length of time it takes a population to double. A doubling period is said to be constant when the length of time remains the same for each subsequent doubling of a population. A doubling period is said to be variable when the length of time varies between each subsequent doubling of a population.

Dynamic Equilibrium - a delicate and natural balance between periods of negative population growth and positive population growth, which over time results in no net gain or loss in population. See also Lotka-Volterra equations and Zero Population Growth.

Drexler - K. Eric Drexler, inventor of the future technology known as molecular nanotechnology. See article K. Eric Drexler - An Exponentialist View for more. Drexler makes good use of the term replicator first defined by Dawkins.

e - irrational and transcendental number, the base of the natural logarithms, which unifies all exponential and logarithmic functions via the calculus. Variously known by mathematicians as Euler's Number and Napier's Constant. Referred to as "the black jewel of the calculus" by David Berlinski (1995), and "the magic number of growth" by Keith Tognetti (1998) - see Exponentialist Population Growth Models page and Wikipedia entry for e (mathematical constant) for more.

Effective Interest Rate - The nominal interest rate adjusted for compound interest (usually calculated as an annual rate). Assuming compound interest, for any given combination of nominal interest rate and compounding periods (e.g.. daily, weekly, monthly, annually) per growth period (e.g.. one year), it is possible to calculate the effective interest rate for that growth period. See my article Calculating the Annual Percentage Yield (APY) for Continuous Compounding for more.

Emigration Rate - the rate at which members of a population leave that population. This is added to the death rate to determine the net rate of negative growth for a population

Envelope Curve - A series of S-curves.

EndogenousDensity dependent factor affecting population growth rate, further divided into first order (immediate effect) and second order (effect after lag period) factors. Typically deterministic in nature.

Euler - Swiss Mathematician (Leonard Euler 1707 - 1783). Refer Wikipedia entry for Leonard Euler for more.

Euler's Formula - $e^{i \theta} = \cos\theta + i\sin\theta \,.$  This formula establishes the deep link between trigonometric functions and the complex exponential function. Source: Wikipedia Euler's Formula. See also Roger Cotes

Euler's Identity - $e^{i\pi} + 1 = 0 \,\!$. This is a special case of Euler's Formula - see Wikipedia Euler's Identity for more. As noted by Eli Maor in his book "e: The Story of a number" (1998)., Euler's Identity links the 5 most important numbers in mathematics , and the 3 most important functions (addition, multiplication and exponentiation) in one golden and mysterious equation. Maor states:

"These five constants symbolize the four major branches of classical mathematics: arithmetic, represented by 0 and 1; algebra, by i; geometry, by π; and analysis by e. No wonder that many people have found in Euler's formula all kinds of mystic meanings."

Also noted by Maor, American mathematician Benjamin Peirce (1809-1880) concluded after proving Euler's Identity in one of his lectures:

"Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth."

Also referred to by American physicist Richard Feynman (1918-1988) as Euler's Jewel, and (The Feynman Lectures on Physics, vol. I, 1977) "the most remarkable formula in mathematics".

Euler's Identity inspired this limerick from Bill Taylor:

(e to the i) to the pi,
And plus one leaves you nought but a sigh.
This fact amazed Euler
That genius toiler,
And still gives US pause, bye the bye.

...and this limerick from Mervyn Cripps:

I used to think math was no fun,
'Cause I couldn't see how it was done.
Now Euler's my hero,
for I now see why 0
i^(pi)= e + 1.

(Limericks sourced from Science Jokes : 1 Mathematics)

Euler's Number: This expression is sometimes used to refer to e, in honour of Leonard Euler. See also Napier's Constant.

Exogenous - Density independent factor affecting population growth rate, typically stochastic in nature.

Exponent - there are 3 main definitions of the word exponent:

• one who explains
• number used in scientific notation to denote how many times a number is to be multiplied by itself

Exponential - mathematical term related to functions, curves, equations, or series which involves one or more numbers raised to an exponent.

Exponential Brownian Motion - Popular term in finance for what is, as far as I have been able to determine thus far, a financial growth model based on variable rate compound interest (which is the Couttsian Growth Model). Pending further investigation I'll stick with the term Couttsian Growth. The phrase Brownian Motion refers to the random (or stochastic) movement of microscopic particles in a fluid, first observed by Robert Brown in 1827.

Possible uses of the Exponential Brownian Motion model include share price fluctuations. However I would suggest that it is normal to name scientific discoveries after their discovers, and Robert Brown did not "discover" this model, nor even suggest its usage in any way. So, though the name is acceptable (and usefully descriptive) in the field of finance, it is not a suitable scientific term. Furthermore, Exponential Brownian  Motion pertains to growth, not to motion.

Read more in my Exponential Brownian Motion article.

Also see Wikipedia article Geometric Brownian Motion.

Exponential curve - graphical representation of exponential growth at a constant rate. Commonly mistaken to represent all forms of exponential growth, the exponential curve is in fact the least useful representation of real-life exponential growth (which is normally the result of variable rates of growth). See Couttsian Growth for more.

Exponential Factor - A way of using e to calculate Couttsian Growth. The Exponential Factor is derived from the Effective Interest Rate. When calculating the Effective Interest Rate for a year of Continuous Compounding, the Exponential Factor equals the Effective Interest Rate plus 1. See my article Calculating the Annual Percentage Yield (APY) for Continuous Compounding for more.

Exponential Function - for example, ax = e x ln(a). Refer Wikipedia article Exponential Function for more.

Exponential growth - term usually given to the result of positive  population growth at a constant rate. Such growth is continuous (lacking discrete values per time period) - see also Geometric Growth. See article Positive Growth - Constant Rate for more. For negative population growth see Exponential Shrinkage below. See Wikipedia article Exponential Growth.

The exponentialist view of exponential growth thus also includes  positive population growth at variable rates of compound interest. However, see also Couttsian Growth and Couttsian Growth Model. The defining feature of exponential growth is compound interest

See article Population Growth Models  for comparisons of alternative population growth models.

Exponential Law - Approximate law of population based on Malthusian Growth Model, discovered by Malthus 1798. Refer Darwin, Dawkins, and Drexler. Refer Exponential Growth law, "Cassell's Laws Of Nature", Trefil 2002. Also, "e: The Story Of A Number" by Eli Maor, 1994 , "What Evolution Is" by Ernst Maor, 2001 , "How Many People Can The Earth Support" by Joel E. Cohen, 1995, "Complex Population Dynamics" by Peter Turchin, 2003, and "The Galilean turn in population ecology" Mark Colyvan and Lev R. Ginzburg, 2003. The Exponential Law is also sometimes referred to as the Malthusian Law - refer "A General Theory of Evolution By Means of Selection by Density Dependant Competitive Interactions" by Lars Witting, 1997 and "Laws Of Population Ecology" by Dr. Paul Haemig, 2005. See also Malthusian Principle.

Exponential Method - Standard mathematical technique used in demography for calculating growth rates over any period. Malthus was familiar with the Exponential Method, and referenced it in Footnote No. 5 to Book II Chapter IX of the 6th edition of his Essay On the Principle of Population.

Refer US Census Bureau - Incorrect Use Of The Exponential Method article for explanation of Exponential Method.

Exponential Rules - Standard mathematical rules dealing with calculations involving exponents.

$\!\, a^0 = 1$
$\!\, a^1 = a$
$\!\, a^{x + y} = a^x a^y$
$\!\, a^{x y} = \left( a^x \right)^y$
$\!\, {1 \over a^x} = \left({1 \over a}\right)^x = a^{-x}$
$\!\, a^x b^x = (a b)^x$

Source: Wikipedia article Exponential Function

Exponential series - any series where the next number in the series is derived by incrementing the  exponent for the starting number by 1. In other words, to get the next number in the series, multiply the previous number in the series by the starting number. The series can be expressed using scientific notation or standard notation. The doubling series is the exponential series most commonly used by exponentialists due its common usage elsewhere and due to the use of the New Malthusian Scale.

Exponential shrinkage - term usually given to the result of negative population growth at a constant rate. See article Negative Growth - Constant Rate for more. The defining feature of exponential shrinkage is compound interest. The exponentialist definition of exponential shrinkage thus also includes  negative population growth at variable rates of compound interest. See also Malthusian Growth Model.

Exponentialist - An exponent of the exponent, with particular reference to the natural processes of replication. See article Exponentialist for more.

Farming - activity which deliberately encourages and harnesses the exponential growth of replicators from one or more other species.

Fibonacci numbers - an sequence of numbers popularised in Europe by Italian mathematician Fibonacci in 1202. The sequence is: 0,1,1, 2, ,3, 5, 8, 13, 21, 34, 55, 89, 144, 233 etc where is each successive number after 1 is derived by adding the last two numbers in the sequence. See Fibonacci numbers section of Rabbits - An Exponentialist View for more.

Fixed Rate - same as constant rate. The term "fixed rate" is used in business, whereas demographers prefer to say "constant rate". Fixed rate compound interest results in exponential growth. Fixed rate simple interest results in linear growth.

Four Riders Of The Apocalypse - the four infamous positive checks on population - War, Famine, Pestilence and Death. See also Moral Restraint.

Fractional Exponential Growth - A percentage growth rate of 1% can also be expressed as a 101/100 growth rate. Thus, all exponential growth can be regarded as fractional exponential growth. Using scientific notation to express fractional exponential growth, a 1% growth rate is represented thus:

 (101/100)0 (101/100)1 (101/100)2 (101/100)3 (101/100)4 etc

Such growth can produce any linear series of whole numbers such as:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Fractional exponential growth can also result in a seemingly random series such as:

8, 9, 10, 11, 12, 15, 14, 12, 9, 10

Most significantly, fractional exponential growth can produce the familiar doubling series.

Generation - moving forwards in time, the word generation is usually taken to mean the members of a population born "around the same time". In this sense it is an approximate term only. However, when assessing your own ancestral tree (and thus going backwards in time), the word generation can be taken to mean that there must be precisely twice as many individuals in each preceding generation (your 2 parents, 4 grandparents, 8 great-grand-parents, 16 great-great-grand-parents etc).

Generational Chauvinism - chauvinistic attitude of most evolutionists, botanists, biologists and zoologists in favouring explanations of exponential growth through a population model based naively on generations rather than the Malthusian growth model. Perhaps the best dismissal of this model comes from Richard Dawkins, himself an evolutionist and zoologist - see article Richard Dawkins - An Exponentialist View for more.

Geometric - see Exponential.

Geometric Growth - term originally applied by Malthus to population growth, usually taken to continuous Exponential Growth today. However Geometric Growth is a form of discrete Exponential Growth, with discrete growth rates per generation or time period.

Geometric Mean: the average of a set of positive growth rates. The geometric mean is always less than or equal to the arithmetic mean. See Wikipedia entry for geometric mean for more.

Gigapop - 1,024 Megapops.

Graphilism - term used to describe the predilection of modern demographers for depicting population growth via graphs, which lead to the false assumption that only growth depicted by an exponential curve can be said to be exponential.

Growth Rate - the rate at which a population grows. Growth can be positive (which results in population doubling) or negative (which results in population halving). The rate is usually expressed as a percentage growth rate that applied to a population for a given time period. A constant rate is said to apply if the growth rate does not vary from one time period to the next, and a variable rate is said to apply when the growth rate does vary from time period to time period.

Halving Period - the length of time it takes a population to halve. A halving period is said to be constant when the length of time remains the same for each subsequent halving of a population. A halving period is said to be variable when the length of time varies between each subsequent halving of a population.

Halving Series - Basically any doubling series in reverse order.

Immigration Rate - the rate at which new members join a population. This is added to the replication rate to determine the net rate of positive growth for a population.

Interest Rate - Refer Wikipedia article Interest

Kardashev - Soviet astrophysicist Nikolai Kardashev who in 1964 proposed three broad levels of advancement for any space-faring civilisation, based upon the natural availability of resources throughout the universe. These are now known as the Kardashev Levels

Kardashev Levels - These levels, and various gradations in-between, are the true limits to growth for any space-faring civilisation:

• K-1 : utilise all the resources and energy available on the home planet
• K-2 : utilise all the resources and energy of the home solar system
• K-3 : utilise all the resources and energy of the home galaxy

kilopop - 1,024 pops.

Limit To Growth - point at which insufficient resources are available to sustain further positive population growth. Once a population hits the limit to growth then it must enter a period of negative population growth (or prove that the assumptions around the perceived limit to growth are false). Although the concept was effectively introduced by Malthus (1798) and is known as the Malthusian Principle, the term itself was popularised in The Limits To Growth by Dennis L. Meadows and "The Club of Rome" (1972) - see Wikipedia article The Limits To Growth, and The Club Of Rome homepage.

Linear growth - population growth which is modelled by adding a fixed amount each time period. The linear growth model thus assumes simple interest at a constant rate

See article Linear Growth Vs Exponential Growth for a comparison of the two growth models.

Example: 1, 8, 15, 22, 29 etc (add 7 to get the next number in this series)

Linear series - any series where each number in the series is derived by adding a fixed amount.

Living Entity - A living entity is the result of a replication event by one or more replicators, and is encoded with the instructions of its own assembly. Note that a living entity need not be a replicator, or could be a replicator but need never cause (or jointly cause) a replication event. Hence, a living entity can either metabolise or replicate, or it can do both.

Logistic Growth - the logistic growth model was developed by Belgian mathematician Pierre Verhulst (1838). It presupposes that the growth rate is dependant on population density and restricted by carrying capacity (limit to growth). Such growth is typically represented by an S-curve, whereby the growth rate declines as the population increases.

See article Logistic Growth Vs Exponential Growth for a comparison of the two growth models.

Lotka-Volterra equations - Sometimes known as predator-prey equations, which result in the dynamic equilibrium of both populations. Refer Wikipedia article Lotka-Volterra equations

Malthus - Reverend Thomas Robert Malthus (1766-1834), an English clergyman recognised for his work in the fields of demography and economics. Also acknowledged by both Darwin and Wallace as a key contributor to their joint discovery of the theory of evolution. See article Reverend Thomas Robert Malthus - An Exponentialist View for more.

Malthusian - pertaining to Malthus and his "Essay on the Principle Of Population" (1798 - 1826, editions one to six), plus "A Summary View" (1830). Any other use of the term Malthusian is outside the exponentialist meaning.

Malthusian Blues - term used by Aldous Huxley in "Brave New World" (1932) for slow soothing music produced by Synthetic Music apparatus (LONDON'S FINEST SCENT & COLOUR ORGAN)  played just after a lively performance by CALVIN STOPES AND HIS SIXTEEN SEXOPHONISTS to an audience high on Soma...."Everybody's happy now". Also used by Marshall T. Savage in his pro-space book "The Millennial Project: Colonizing The Galaxy In Eight Easy Steps." - refer Marshall T. Savage - An Exponentialist View

Malthusian Catastrophe - Population crash and subsequent return to subsistence standards of living. Read more here at Wikipedia, though be warned that this Wikipedia article makes the usual naive assertions regarding  (simple) exponential growth and instead favours logistic growth (which is just as bad a growth model the simple exponential, or Malthusian Growth Model). Refer Logistic Vs Exponential Growth for the Exponentialist view.

Malthusian Drill - term used by Aldous Huxley in "Brave New World" (1932) to describe contraceptive precautions. This view is based on Neo-Malthusianism.

Malthusian Growth Model - a mathematical model which I believe describes a scientific law of population growth (and shrinkage) which applies universally to all replicator populations. Named after Malthus who first introduced the model. In essence, there are only 3 states that a population can be in for any given time period, based on its Malthusian Parameter:

• (replication rate + immigration rate) greater than (death rate plus emigration rate) : results in  positive population growth
• (replication rate + immigration rate) less than (death rate plus emigration rate) : results in  negative population growth
• (replication rate + immigration rate) equals (death rate plus emigration rate) : results in  zero population growth

The main criticisms of the Malthusian Growth Model are the almost universal assumption of a constant rate of growth and the lack of a limit to growth. In recognition of the fact that everyone else thinks of the Malthusian Growth Model in terms of a constant rate of growth, I have opted to put my own name to a population growth model which permits compound growth at a constant rate, or at variable rates. See Couttsian Growth Model.

Malthus was also famous as an economist, and it is no surprise that the Malthusian Growth Model applies equally well to the world of finance and business.

Malthusian Increase - alternative name to an approximate law of nature known as the Exponential Law. However, see also Malthusian Increase section of Lars Witting - An Exponentialist View.

Malthusian Law - alternative name to an approximate law of nature known as the Exponential Law.

Malthusian Parameter - The population growth rate  - usually r - for a population is sometimes referred to in the scientific literature as the Malthusian Parameter of the equation under discussion. This term originates with R. A. Fisher in his "The Genetical Theory Of Natural Selection" (p.25) originally published in 1930. Refer Malthusian Parameter of population increase for more.

Malthusian Principle - the principle that there is an unassailable limit to growth for any population that sustains positive population growth. For most populations of replicators on Earth, the Earth itself represents the ultimate limit to growth. Once a population hits the limit then it must either stop growing or enter a period of negative population growth. Refer Lars Witting - An Exponentialist View for more. Also Malthusian Wall.

Malthusian Relativity - the theory that evolution is directional, and that complex organisms automatically evolve once basic replicators arise. The term refers to the relative values of the Malthusian Parameter of different populations. The term originates with Lars Witting in his "A General Theory of Evolution" (p.9). Refer Lars Witting - An Exponentialist View for more.

Malthusian Selection - exponentialist hypothesis which explains how environment and non-instinctive behaviour affect the rate of growth for a population and thus influences differential replication. Named after Malthus, who first introduced the concepts in his essay on human populations. See also Natural Selection, and Stochastic.

Malthusian Wall - an unassailable limit to growth for any population that sustains positive population growth. Also Malthusian Principle. See Marshall T. Savage - An Exponentialist View for an example of usage.

Malthusian Wheel - a simple alternative to the New Malthusian Scale. The two basic approaches to using the Malthusian Wheel are explained on both the Exponentialist page and New Malthusian Scale page.

Molecular Nanotechnology - a term introduced by K. Eric Drexler in his book "Engines Of Creation" to describe a future technology where replicators and assemblers are capable of exponential growth and the molecular manipulation of otherwise inanimate matter.

Mean - see arithmetic mean.

Megapop - 1,024 kilopops.

Meme - a new term introduced by Dawkins to describe a unit of imitation capable of replication. Good examples are songs, jokes, chain letters and ideas.

Moral Restraint - term introduced by Malthus in the 2nd edition (onwards) of his essay on population for a preventative check on population. The concept includes sexual abstinence followed by a late or no marriage, and voluntary restraint on sex within marriage.

Nanotechnology - any technology which operates at the scale of a billionth of a meter. See also Molecular Nanotechnology.

Napier - Scottish mathematician John Napier (1550 - 1617) who introduced logarithms. Refer Wikipedia entry John Napier for more.

Napier's Constant - This expression is sometimes used to refer to e. See also Euler's Number

Natural Logarithm - The Natural Logarithm of any number (n) is defined as the sum of the accurate positive period required to grow by factor n and the accurate negative period required to shrink by factor 1/n for any given growth rate. Note that this is not the conventional definition - for that see Wikipedia's Natural Logarithm article. However, as demonstrated in my article The Rule of 70 and The Rule of 72 Compared, this is a very good approximation  that is easy to understand and prove.

Natural Selection - relating to the theory of evolution concerned with the origin of species. The theory also popularises the principle of differential reproduction (first introduced via Malthusian Selection), which is typically explained as entirely the result of genetic variation within a species. The exponentialist view is that Natural Selection, Artificial Selection, Malthusian Selection and Unnatural Selection all contribute to the rate of growth for a population and thus each influences differential replication.

Negative Population Growth - the rate of growth for a population for a given time period is negative. Thus, the death rate exceeds the replication rate for the given time period. Sustained negative population growth leads to population halving and exponential shrinkage (for a constant rate) or Couttsian Shrinkage (for variable rates), and ultimately to extinction.

Neo-Malthusian - a supporter of Neo-Malthusianism

Neo-Malthusian Theory - Nothing to do with the original Neo-Malthusians nor Neo-Malthusianism, Neo-Malthusian Theory proposes that evolutionary forces favouring fertile individuals will force the overall population to the same fertility levels, unless kept at or below subsistence levels of existence. See also Malthusian Catastrophe. Read more here at Wikipedia

Neo-Malthusianism - social movement whose central tenet was a belief in contraceptive measures. The first significant such advocate was Francis Place (1771 - 1854). Michael Hart, in his book "The 100  - A Ranking Of The Most Influential Persons In History" (1978,1992), ranks Malthus at number 80. Part of Hart's justification is that Francis Place (1771-1854) was so influenced by Malthus' essay that he wrote the first book advocating contraception in 1822. Hart correctly notes that Malthus himself did not advocate contraception, but preferred to advocate his own suggestion of moral restraint.

New Malthusian Scale - a scale designed by David Coutts for the universal measurement of exponential growth through  population doubling and population halving. Named after Malthus in honour of Malthus' introduction of the Malthusian Growth Model. See article New Malthusian Scale for more.

Nominal Interest Rate - The rate of interest without adjusting for compound interest. Assuming compound interest, for any given combination of nominal interest rate and compounding periods (e.g.. daily, weekly, monthly, annually) per growth period (e.g.. one year), it is possible to calculate the effective interest rate for that growth period. See my article Calculating the Annual Percentage Yield (APY) for Continuous Compounding for more.

Percentage Growth Rate - A rate of growth in hundredths expressed as a percentage (%):

Examples for positive population growth

• 101/100 is expressed as a (positive) 1% growth rate
• 102/100 is expressed as a (positive) 2% growth rate

Examples for negative population growth:

• 99/100 is expressed as a negative 1% growth rate
• 98/100 is expressed as a negative 2% growth rate

Sometimes referred to as the Malthusian Parameter.

Pop - typically an individual member of a population, but a pop can be any starting quantification for a population (usually numeric, or by area, volume or weight). Derived from the nickname "Pop" given to Malthus by members of his family, but also a simple abbreviation of "population".

Population - any collection of individual living entities which can be meaningfully grouped together (often by species). Without replicators a population will experience sustained negative population growth, leading to rapid extinction. Hence, it is reasonable for exponentialists to assume all populations are replicator populations.

Population Check - term derived from Malthus for any check on population growth. See also Preventative Check and Positive Check

Population Doubling - the doubling of a population due to positive population growth. The period can be based on crude population doubling, projected population doubling, or actual population doubling. For examples using a constant rate of growth, see article Positive Growth - Constant Rate for more. For examples using variable rates of growth see The Scales Of 70

Population Halving - the halving of a population due to negative population growth. The period can be based on crude population halving, projected population halving, or actual population halving. For examples using a constant rate of growth, see article Negative Growth - Constant Rate for more. For examples using variable rates of growth see The Scales Of 70. For radioactive isotopes the halving period is described as the decay rate.

Positive Check - term used by Malthus to describe an unpreventable check on population. Collectively, these might be better known to readers as the Four Riders Of The Apocalypse.

Positive Population Growth - the rate of growth for a population for a given time period is positive. Thus, the replication rate exceeds the death rate for the given time period. Sustained positive population growth leads to population doubling and exponential growth (for a constant rate) or Couttsian Growth (for variable rates) and ultimately to a Malthusian Wall

Preventative Check - term used by Malthus to describe a voluntary check on population. Malthus favoured moral restraint over other voluntary checks such as homosexuality, abortion and contraception.

Principle Of Population - Malthus first wrote "An Essay On The Principle Of Population" in 1798. This essay, although it focuses on humanity, reveals the universal nature of exponential growth for all replicator populations. Most importantly, although Malthus made frequent use of constant rates of growth, he realised that populations grow exponentially due to variable rates of growth. This lead to the Malthusian Growth Model, which in turn lead to the Couttsian Growth Model.

Projected Population Doubling - population doubling typically extrapolated from the current rate of population growth which is then projected into the future as a  constant rate of positive population growth. For example, if the rate is 1% for the given time period, then multiply 1.01 by 1 (enter 1, press X, enter 1.01, press =) , and then keep pressing "=" until you get 2. It will take you about 70 presses. For more examples see article Positive Growth - Constant Rate. See also Rule Of 70

Projected Population Halving - population halving typically extrapolated from the current rate of population growth which is then projected into the future as a constant rate of negative population growth. For example, if the rate is 1% for the given time period, then multiply 1.01 by 1 (enter 1, press X, enter 1.01, press =) , and then keep pressing "=" until you get 2. It will take you about 70 presses. For more examples see article Negative Growth - Constant Rate. See also Rule Of 70.

Random Series - any series of numbers for which no mathematical growth model can predict the next number in the series.

Example: 1, 1, 3, 2, 5, 2, 6, 2, 3, 7 etc

Replication - the process of the creation of a copy by a replicator.

Replication Event - an event which adds to replicator population and thus add to the replication rate  - or Malthusian Parameter - of a population.  A birth is a typical replication event, so is cell division, and so is the creation of a virus via the cell taken over by that virus (viral replication).

Replication Rate - universally applicable term for the rate at which replicators are created by other replicators. Adding to the replication rate decreases negative population growth or increases positive population growth.

Replicator - anything in the universe that is individually or jointly capable of causing a replication event. See also Dawkins and Drexler

Replicator Population - Any population which includes a proportion of replicators.

Replacement Rate - the average rate of births per woman at which a population reaches zero population growth. "Replacement Rate" is, in fact, a vague and slightly misleading term for a number of reasons:

• the replacement rate varies with the average age of the mothers

• the replacement rate varies with the infant mortality rate

• a population does not reach zero population growth rate for at least one generation after reaching replacement rate

See article Human Replicators - An Exponentialist View for more.

Reprogenetics - term invented by Lee M. Silver, author of Remaking Eden: Cloning and Beyond in a Brave New World (1998). Silver defines reprogenetics as a new field which will result from the merging of two existing fields: reproductive biology and genetics.

Reproduction - Reproduction for all living entities on Earth is normally categorised as sexual and asexual reproduction. The commonly-held view is that DNA and viruses do not reproduce, hence the introduction of the term replication

Rule Of 70 - This is the well-known rule that you can derive the crude population doubling period by dividing the growth rate for a given period into 70. The population doubling period is then expressed in the same unit of measurement as the given period for the growth rate. Thus, if the growth rate per annum is 1% then the crude population doubling period is 70 years, and for 2% the crude population doubling period is 35 years. For negative growth rates, the crude population halving period is similarly obtained by dividing the growth rate for the period into 70. The Rule of 70 is only useful for growth rates between negative 7% and positive 7% (or, according to Paul Ehrlich in 1990, "The Population Explosion" Chapter 1 Note 6, his limit on the usefulness of the Rule Of 70 is less than 5%).

The Rule Of 70 is typically invoked to extrapolate population growth at a constant rate of positive population growth or, less commonly, negative population growth. In fact, using The Scales Of 70,  it can be used to prove that a mixture of positive population growth and negative population growth at variable rates can also result in population doubling or population halving.

Rule Of 72 is an alternative to The Rule Of 70. See my article The Rule Of 70 and The Rule Of 72 Compared for more.

The S-Curve - common name for sigmoidal curve which is the result of logistic growth. The pattern of growth is a slow start, a middle period of explosive growth, and then a flattening of the curve as growth slows.

The Scales Of 70 - Extrapolation of the Rule Of 70 which aids explanation of population doubling or population halving for a mixture of variable positive growth rates and variable negative growth rates.

• If the totalled positive growth rates exceed ( "outweigh") the totalled negative growth rates by 70 then the population under consideration will double. For each year of negative growth at rate r, reduce the growth  period by r/100s.

• If the totalled negative growth rates exceed ("outweigh") the totalled positive growth rates by 70 then the population under consideration will halve.  Reduce the shrinkage period by 70/100s.

• See my articles See my article The Rule Of 70 and The Rule Of 72 Compared and Rules Of Population for an explanation why negative growth needs this minor adjustment.

See article The Scales Of 70 for more.

The Scales Of e - More general version of the Scales Of 70. For a given factor F:

• If the totalled Natural Logarithms of the positive growth rates exceed ( "outweigh") the totalled Natural Logarithms of the negative growth rates by 100 * Ln(F) then the population under consideration will multiply by F.

• If the totalled Natural Logarithms of the negative growth rates exceed ("outweigh") the totalled Natural Logarithms of the positive growth rates by 100 * Ln(F) then the population under consideration will divide by F.

• The multiplying or dividing period is always the total of the number of periods of growth.

• See my article The Scales Of e for more.

Scientific LawUsually described as a mathematical equation, a scientific law is a law of nature which applies throughout the universe. They are typically a statement of fact which explain a set of behaviours or actions and are generally regarded as being true without requiring formal proof as they can generally be observed to apply.

Scientific Notation - notation used by scientists to denote exponential series. The doubling series would thus be expressed:

 20 21 22 23 24 25 26 27 28 29 210 etc

It should be noted that any number n to the power of zero = 1. Also, number n to the power of 1 = n.

Sigmoidal - graphical representation of "S" shaped growth curve (S-curve) caused by logistic growth

Simple Interest - interest paid on principle sum only which ignores any accumulated interest. Linear Growth assumes Simple Interest. See also Compound Interest.

Species - a discretely encoded population. All species on Earth are genetically discrete.

Standard Notation - common notation used to express exponential series. See doubling series for an example.

Stochastic - From the Greek root stochastikos, relating to guessing or conjecture. This is a word from the field of statistics, used frequently in writing on evolution, population dynamics or even finance. Often used to mean "random". However, the seasons of the year are said to be non-random, and yet still stochastic (Turchin 2003). Basically though, as Ernst Mayr (2001) put it "Much of the differential survival and reproduction in a population are not the result of selection, but rather of chance."

This is close to the Exponentialist hypothesis, as defined by Malthusian Selection. However, the difference is the Exponentialist emphasis on the environment (excluding other populations) and behavioural factors (the latter notably not generally being random). My take is that environmental factors are the main stochastic factor, but that the term stochastic doesn't adequately cover what Malthus described back in 1798. Hence, for the moment, I'm sticking with Malthusian Selection which compliments Natural Selection nicely.

Struggle For Existence - Malthusian term for the competition for resources between discrete populations of the same living entity (originally used by Malthus in 1798 in relation to competing human populations). The exponentialist view is that the Struggle For Existence exists naturally between all populations of replicators. See The Struggle For Existence for more.

Trophic - From the Greek root trophikos, pertaining to food. In population dynamics this term is used to describe the predator-prey, parasite-host, or herbivore-vegetation type of interactions where one population is food for the other.

Universal Law - see Scientific Law

Universal Law Of Nature - See Scientific Law

Unnatural Selection - term sometimes used to describe human interference in the natural process of replication for any population (e.g. genetic engineering, or molecular nanotechnology). This definition excludes Artificial Selection, which has been with us for so long that it seems almost natural. Unnatural Selection is one of the factors which influences differential replication.

Usury - the charging of interest on a loan. Given the fact that the transcendental number e was relatively recently  discovered (compared, for example, with Pi), and was largely discovered through a mathematical examination of compound interest,  the injunctions against usury in most major religions may have delayed the discovery of e, the magic number of growth. Refer Wikipedia entry for Usury for more.

Variable Rate - a growth rate which varies for each period. For compound interest, sustained positive (variable or constant) rates of growth cause exponential growth through population doubling (for positive population growth) or exponential shrinkage through population halving (for negative population growth). See article The Scales Of 70 for more. See also Constant Rate.

Virus - in the natural world the virus is a crude yet powerful replicator, and that is probably why the same term is used for computer viruses. See Viral Replicators - An Exponentialist View for more.

von Neumann Machine - hypothetical replicator proposed by John von Neumann  ("Theory Of Self-Reproducing Automata", 1966) whereby a machine is built which is then capable of replication. See Zyvex page on John von Neumann.

Wallace - Alfred Russel Wallace (1823-1923), English explorer, naturalist and co-founder (with Darwin) of the theory of evolution. See article Alfred Russel Wallace - An Exponentialist View for more.

Zero Population Growth - Watershed between Negative Population Growth and Positive Population Growth. Actual Zero Population Growth is an extreme rarity for most populations. Sometimes confused with the equivalent result (i.e. no growth in population) of dynamic equilibrium.