Malthusian Memes - An Exponentialist View
Viral Replicators - An Exponentialist View
Bacterial Replicators - An Exponentialist View
Cellular Replicators - An Exponentialist View
Rabbit Replicators - An Exponentialist View
Human Replicators - An Exponentialist View
Grey Goo - An Exponentialist View
Death By Replication
Exponential Assembly - An Exponentialist View

Population Doubling Mechanism 

New Malthusian Scale 

The Myth Of The Exponential Phase

External Links:

Natural Selection and Differential Reproduction - from Replicators: Evolutionary Powerhouses

Population and environment - Exponential Growth - from the Australian Academy of Science

Understanding Exponential Growth - Bacteria in a Bottle - From World Population Balance

Growth of Bacterial Populations - Todar's Online Textbook of Bacteriology

Some Wikipedia Links Related to Bacteria:

Antibiotic (substance that kills bacteria)
Bacterial Growth
Bacteriophage (a Virus that infects bacteria)
Cyanobacteria ("blue-green algae", which can cause toxic bloom)
Penicillin (antibiotic derived from fungi )
Staphylococcus aureus ("Golden Staph"), a common cause of food poisoning and staph infection
Syphilis ("the great pox")
Yersina pestis (which caused the Black Death)


Bacterial Replicators - An Exponentialist View

(It is recommended that you also read Cellular Replicators - An Exponentialist View)

"The most common circumstance in which repeated doublings, and therefore exponential growth, occurs is in biological reproduction. Consider first the simple case of a bacterium that reproduces by dividing itself in two. After a while, each of the two daughter bacteria divides as well. As long as there's enough food and no poisons in the environment, the bacterial colony will grow exponentially. Under very favorable circumstances, there can be a doubling every 15 minutes or so. That means 4 doublings an hour and 96 doublings a day. Although a bacterium weighs only about a trillionth of a gram, its descendants, after a day of wild asexual abandon, will collectively weigh as much as a mountain; in a little over a day and a half as much as the Earth; in two days more than the Sun. . . . And before very long, everything in the Universe will be made of bacteria. This is not a very happy prospect, and fortunately it never happens. Why not? Because exponential growth of this sort always bumps into some natural obstacle." (Carl Sagan, 1997)


Bacteria are the oldest and most successful form of life on Earth. They are also known as prokaryote cells, a name which denotes their primacy in the tree of life over the eukaryote cells of animal and plant life-forms. Eukaryotic cells have a nucleus which contain the chromosomes, whereas bacteria have no nucleus. Both types of cells replicate through a process of cell division, and both can form multi-cellular life-forms. Hence a cell division is referred to in my Exponentialist web site as a replication event. Unlike animals and plants, bacteria are not programmed to die and so could go on living forever. However, they are effectively not immortal as they can still be destroyed.

As the above diagram for cell replication shows, a growing population of bacterial cells reaches 16 in just four generations. However, if only the pink cell in the third generation dies then the final population is only 14. Or, if only the yellow cell in the second generation dies, then the final population is 12.

If both the pink cell in the third generation and the yellow cell in the second generation die, then the final population is 10.

Thus, the generational approach to modelling bacterial populations is thus not as straight forward as it seems. See Richard Dawkins - An Exponentialist View for more.

Fortunately the Couttsian Growth Model, coupled with my New Malthusian Scale, provide a much better solution for modelling bacterial population growth. The purpose of this article is to briefly explore key aspects of bacterial life, and to demonstrate the simplicity and universal applicability of the Couttsian Growth Model.

Gene Exchange

One of the problems in studying populations of bacterial replicators is that they can exchange genes and effectively "interbreed". In this sense, they can be regarded as one species. Nonetheless, it is still meaningful to discuss populations of genetically diverse sub-species of bacteria (or geographically discrete populations of the same founding species).

Life is bacteria

Many scientists revere bacteria as the oldest form of life on Earth, and some such as Margulis and Sagan even claim that "life is bacteria" (Margulis, Sagan, 1995, p.69):

"One legitimate answer to the question 'What is life?' is 'bacteria'. Any organism, if not itself a live bacterium, is then a descendant - one way or another - of a bacterium or, more likely, mergers of several kinds of bacteria. Bacteria initially populated the planet and have never relinquished their hold."

Of course, this ignores those life-forms which replicate but do not metabolise, such as the virus (see Viral Replicators - An Exponentialist View for more). Still it is fair to say that cells of one description or another, singularly or in colonies of multi-celled creatures, make up most of the life on Earth. 


Earth's atmosphere is about 20 percent oxygen. Cyanobacteria (sometimes referred to as blue-green algae) are responsible for producing most of that oxygen (Margulis, Sagan,1995, p.72):

"Rapidly reproducing, bacteria properly supplied with food and water double their cells in a half hour or faster. They have been and probably always will be the most important players in maintaining the biosphere. A single photosynthetic blue-green bacterium growing and dividing under ideal conditions could, in theory, produce all the oxygen now in the atmosphere in just a few weeks."

A-pops 1 2 4 8 16 32 64 128 256 512 1024
B-pops 1 2 4 8 16 32 64 128 256 512 1024
C-pops 1 2 4 8 16 32 64 128 256 512 1024
D-pops 1 2 4 8 16 32 64 128 256 512 1024
E-pops 1 2 4 8 16 32 64 128 256 512 1024
F-pops 1 2 4 8 16 32 64 128 256 512 1024
G-pops 1 2 4 8 16 32 64 128 256 512 1024
H-pops 1 2 4 8 16 32 64 128 256 512 1024
I-pops 1 2 4 8 16 32 64 128 256 512 1024
J-pops 1 2 4 8 16 32 64 128 256 512 1024
K-pops 1 2 4 8 16 32 64 128 256 512 1024
L-pops 1 2 4 8 16 32 64 128 256 512 1024
M-pops 1 2 4 8 16 32 64 128 256 512 1024
N-pops 1 2 4 8 16 32 64 128 256 512 1024
O-pops 1 2 4 8 16 32 64 128 256 512 1024
P-pops 1 2 4 8 16 32 64 128 256 512 1024
Q-pops 1 2 4 8 16 32 64 128 256 512 1024
R-pops 1 2 4 8 16 32 64 128 256 512 1024
S-pops 1 2 4 8 16 32 64 128 256 512 1024
T-pops 1 2 4 8 16 32 64 128 256 512 1024
U-pops 1 2 4 8 16 32 64 128 256 512 1024
V-pops 1 2 4 8 16 32 64 128 256 512 1024
W-pops 1 2 4 8 16 32 64 128 256 512 1024
X-pops 1 2 4 8 16 32 64 128 256 512 1024
Y-pops 1 2 4 8 16 32 64 128 256 512 1024
Z-pops 1 2 4 8 16 32 64 128 256 512 1024
AA-pops 1 2 4 8 16 32 64 128 256 512 1024
BB-pops 1 2 4 8 16 32 64 128 256 512 1024
CC-pops 1 2 4 8 16 32 64 128 256 512 1024
DD-pops 1 2 4 8 16 32 64 128 256 512 1024
EE-pops 1 2 4 8 16 32 64 128 256 512 1024
FF-pops 1 2 4 8 16 32 64 128 256 512 1024
GG-pops 1 2 4 8 16 32 64 128 256 512 1024
HH-pops 1 2 4 8 16 32 64 128 256 512 1024

Table A: 256 E-Pops after just 24 hours. 64 HH-Pops after 168 hours of doubling (at a constant doubling rate of every 30 minutes) for a cyanobacteria population, using on the New Malthusian Scale. A single row is 5 hours. A week is 168 hours.

Note:  1A-pop = 1024 pops, 1B-pop = 1024 A-pops, 1 C-pop = 1024 B-pops etc.

Marshall T. Savage explores the possibility of farming blue-green algae in the future (see Savage - An Exponentialist View for more).

I examine Carl Sagan's exploration of explosive bacterial population growth in my article Carl Sagan - Secrets of the Universe


Margulis and Sagan summarise the significance of the breakthrough of photosynthesis (Margulis, Sagan, 1995, pp.78-80):

"The most important metabolic innovation in the history of the planet was the evolution of photosynthesis. By way of  photosynthesis, life freed itself from energy scarcity; from then on life was limited primarily by the scarcity of one material building block or another.

Hence, just like people, bacteria suffer Malthusian population checks (though not "moral restraint" !) on their growth. They run out of building material, they are not always "properly supplied with food and water" either (possibly due to a hugely variable environment) and bacteriophages and other life-forms are also on hand to slow bacterial populations down by eating them. Hence the typical boom and bust nature of population growth for bacteria. This results in variable rates of growth, and variable population doubling times (or variable population halving times for negative variable rates of growth). Hence, Margulis' and Sagan's cyanobacteria example is purely theoretical (based on "ideal conditions") designed to illustrate the power of exponential growth. 

Exponential Phase

Yet bacteria (along with viruses) are routinely trotted out as reasonable examples of exponential growth, which they are not (Kelly, 2002):

"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. ...To get around such restrictions, many problems involving exponential growth and decay deal with exciting things like bacterial growth."

As with Margulis' and Sagan's example, it is then often noted that this is not the normal growth pattern for bacteria. This temporary phase of extraordinary growth is described as the exponential phase (or log phase).

However, because the constant rate version of the exponential growth model (the Malthusian Growth Model) recognised no limits to growth and seems to impel a population towards unrealistic infinite growth, logistic growth is then frequently cited (by Kelly and others) as a more realistic population growth model. Still, as noted in my article, the logistic growth model has problems of its own. See Logistic Growth versus Exponential Growth (and Couttsian Growth) for more. In place of the Logistic Growth Model I propose a universal model of population growth that accommodates variable rates of both positive and negative population growth - the Couttsian Growth Model. See article The Scales Of 70 for an approximation of the model, or The Scales of e for a universal growth model that also lies at the heart the Exponential Method itself.

The Exponential Phase is a term sometimes erroneously applied to human populations - see The Myth Of The Exponential Phase for more.

Boom and Bust

A consequence of a booming population is often one that goes bust. In the terminology of bacterial growth the exponential phase is followed by a temporary static or stationary phase which is then followed by the death phase.

The Couttsian Growth Model accommodates both positive and negative variable rates of growth in the one model, and comfortably deals with boom and bust situations (regardless of the timeframe). A population subject to the Couttsian Growth Model is not necessarily impelled towards a limit to growth, nor to extinction. A population subject to the Couttsian Growth Model can also persist, forever, within upper and lower limits to growth. It's not always a linear progression from boom to bust, resulting in extinction as bust might imply. It can also be a more dynamic boom and bust, boom and bust, boom and bust etc., as circumstances allow.

Hence, rather than a linear progression of Lag Phase ==> Exponential Phase ==> Static Phase ==> Death Phase, what we have is something far more universal, and far more uncertain and dynamic. 

It's a small world, after all

As Bill Bryson rather entertainingly puts it (Bryson, 2003):

"...there is no point trying to hide from your bacteria, for they are on you and around you always, in numbers you can't conceive of. If you are in good health and averagely diligent about your hygiene, you will have a herd of about one trillion bacteria grazing on your fleshy plains - about one hundred thousand of them on every square centimetre of skin.

And those are just the bacteria that inhabit your skin...Every human body consists of about ten quadrillion cells, but it is host to about a hundred quadrillion bacterial cells. They are, in short, a big part of us. From the bacteria's point of view, of course, we are a rather small part of them."

Given that there are now well over 6 billion humans on our little planet, that's a staggering number of bacteria and that's only those making a living out of us!

Bacteria have been discovered living to a depth of 600 metres below  land. Streptococcus bacteria survived in a sealed camera lens on the Moon for 2 years before . In short they're tough, there's plenty of them, and the Earth is their planet (Bryson, 2003):

"This is their planet, and we are only on it because they allow us to be."

Differential Replication

Bacteria are subject to genetic mutations (though not sexual recombination), and hence natural selection based on differential replication applies to them too. In short, depending upon the prevailing circumstances and environment, some bacteria will out-breed their competitors. Nature may well judge them to be winners in the survival of the fittest.

Mutations happen, on average, once in every million divisions. If you try to imagine all the bacteria on Earth, that's an awful lot of mutation going on. Allowing for mutation and gene exchange, it is fair to say that the bacterial pool of life is somewhat prone to evolving. Human control in such circumstances would appear to be an illusion.


Bryson, Bill. A Short History Of Nearly Everything. Doubleday. 2003.

Kelly , W. Michael, The Complete Idiot's Guide To Calculus. Alpha (A Pearson Education Company). 2002.

Margulis, Lynn; Sagan, Dorion, What Is Life? Simon & Schuster. 1995

Sagan, Carl. Billions and Billions: Thoughts on Life and Death at the Brink of the Millennium. Headline Publishing, 1997

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Copyright 2001 David A. Coutts
Last modified: 29 June, 2012