Rabbits - Wolfram Demonstrations Project
Wikipedia Links to Some Related Concepts:
Wikipedia Links To Children's Books About Rabbits:
Rabbit Replicators - An Exponentialist View
Replicator - "In discussions of evolution, a replicator is an entity (such as a gene, a meme, or the contents of a computer memory disk) which can get itself copied, including any changes it may have undergone. In a broader sense, a replicator is a system which can make a copy of itself, not necessarily copying any changes it may have undergone. A rabbit's genes are replicators in the first sense (a change in a gene can be inherited); the rabbit itself is a replicator only in the second sense (a notch made in its ear can't be inherited)." (Drexler, 1986)
Rabbits are famous for self-replication, as per the well known idioms "breed like rabbits" and "multiply like rabbits."
The average rabbit replicator mass is estimated at 2.5 kg, and the average rabbit replication time (here meaning gestation time) is estimated at 2.68 x 106 seconds (Freitas and Merkle, 2004, p.220) - the authors clearly felt that rabbits typify what it means to be a replicator as (many) cute white bunnies appear on the cover of their book about self-replicating machines!
Even as far back as 1202, rabbits were famous for self-replication. For in 1202 the Italian mathematician Fibonacci attempted to model rabbit population growth and discovered the sequence of numbers we now call the Fibonacci numbers:
0,1,1, 2, ,3, 5, 8, 13, 21, 34, 55, 89, 144, 233 etc
More recently, in Australia, European rabbits (Oryctolagus cuniculus) were introduced by the early British colonists and the population has repeatedly grown so fast and so vast that humans have used viral agents against them to reduce the rabbit population. So exactly how fast do rabbits actually breed, and what is the population growth model that best describes their growth?
Apart from answering this question I'll also consider the depiction of those cute little bunnies in children's literature, including a look at an Australian tale about some rather more sinister rabbits whose story nonetheless has something to teach us about human population growth and evolutionary theory.
Note that in this article a rabbit is considered a replicator in the second sense (see Drexler quote above). Hence, if a rabbit gives birth to something that we still call a rabbit then that's good enough as I am only concerned with populations of rabbits and not the evolution of rabbits over time.
The Fibonacci numbers were an early attempt to model rabbit population growth, based on the following problem posed by Fibonacci in his Liber Abaci or The Book of the Abacus (Fibonacci, 1202):
"A pair of rabbits are put in a field and, if rabbits take a month to become mature and then produce a new pair every month after that, how many pairs will there be in twelve months time?"
This problem has become known as the Fibonacci rabbits problem. The answer to the problem is 233 pairs of rabbits, as per the sequence listed above. That's a 233 fold population increase.
Here are Fibonacci's assumptions, many of which are unrealistic:
Although the Fibonacci numbers are a poor population model, they do appear in nature particularly in relation to the Golden Ratio (follow the link to the Wikipedia article, which notes that the sequence in question was also well known to the ancient Indians, long before Fibonacci).
The Exponentialist approach to population modelling is to use the simple yet powerful Malthusian concept of population doubling - to read more on population doubling see The Mechanism of Population Doubling. I have coupled the concept of population doubling with the opposite concept of population halving, via what I call The Scales of 70 (which is an extension of the commonly used heuristic tool an extension of the heuristic tool the Rule Of 70).
Imagine then a set of weighing scales that "weigh" growth rates. You may add any growth rates that you like, constant or variable, positives and negatives, in any order you like. However, the positives go on one side and the negatives go on the other side. The Scales Of 70 predicts that, when the positive side "outweighs" the negative side by 70 then you have a population doubling and when the negative side "outweighs" the positive side by 70 then you have a population halving. It's that simple.
I then extend the Scales Of 70 even further into an accurate model of population growth called The Scales of e, which is based entirely on the use of natural logarithms (the "e" refers to the base of the natural logarithms).
Rabbit Populations In Australia
According to the Australian Commonwealth Scientific and Industrial Research Organisation (CSIRO), although initially brought out with the First Fleet in 1788, rabbit populations did not boom at that time. However a dozen or rabbits were introduced into the wild in 1859 as food for foxes (Williams et al, 1994), which the British enjoyed hunting. This time localised rabbit populations exploded with enormous environmental impacts (for example soil erosion and the extinction or near extinction of native plants) and an estimated billions of dollars in damage over the last 100 years alone (CSIRO).
Early attempts to control rabbit populations included the rabbit-proof fence, poisoning, shooting, and hunting with weasels. These were largely ineffective, compared to the biological warfare waged against rabbits by humans since the middle of the Twentieth Century. When myxomatosis (a disease caused by the myxoma virus) was introduced in 1951 it caused rabbit populations to fall by an estimated 95%, although not immediately. In 1955 and 1956 an estimated 45 million rabbits were caught commercially in Australia each year (Williams et al, 1994). The effectiveness of the myxoma virus was increased by the introduction of the European rabbit flea in 1985 and the Spanish rabbit flea in 1992 (CSIRO). However, populations then increased to roughly 50% of pre-1950 levels. Rabbit Haemorrhagic Disease (caused by the calicivirus), introduced by the CSIRO in 1995, has again heavily reduced rabbit populations. For more on viral replicators, see Viral Replicators - An Exponentialist View for more.
Regarding the fecundity of European rabbits, the CSIRO notes:
"Rabbits can breed from five months of age and mature female rabbits can be continuously pregnant between six to eight months per year if the conditions are right. A single female can produce 30-40 young per year and it is quite common for rabbit populations to increase 8-10 fold in one breeding season."
As can be seen, the CSIRO facts do not support the assumptions made by Fibonacci. Fibonacci calculates a 233 fold increase in rabbits in one year from one breeding pair whereas the CSIRO calculates "only" an 8-10 fold increase for the same period. Yet the CSIRO states that a single female will produce 30-40 young per year whereas even Fibonacci assumes that even his initial breeding pair can only produce a maximum of 24 rabbits (12 pairs) per year. What's going on? How can the CSIRO model assume more offspring per year per breeding pair and end up with a smaller annual increase whereas Fibonacci assumes fewer offspring per year per breeding pair, and yet expects a far greater increase?
Other Australian Government studies confirm the CSIRO's estimated maximum fecundity of the rabbit, but also factor in rabbit mortality (Williams et al, 1994):
"Adult females produce 15–40 young a year, but only 1–10% survive past the first year."
So, a high rabbit mortality rate explains why rabbit population increases are kept as "low" as an 8-10 fold annual increase.
Also, the Fibonacci population model is a model that is built to suit a strange and fascinating number sequence (regardless of the evidence) whereas the CSIRO model is based on empirical evidence. There is no question that the CSIRO model is more realistic. However, the CSIRO uses some rather awkward factors of population increase - somewhere between an 8-10 fold increase per year. The Exponentialist preference is to use population doubling (or population halving for a shrinking population) wherever possible in order to simplify the explanation. How can this be done in the case of a 8-10 fold increase?
Taking the lower estimate, an 8 fold increase in the same as three successive population doublings (2 x 2 x 2 = 8). So the CSIRO lower estimate is the equivalent of saying that the rabbit population can double 3 times per year. Hence with a starting population of just one pair of rabbits, living in isolation just like Fibonacci's rabbits, the minimum CSIRO expectation is that we will have 8 pairs of rabbits (not 233 pairs as suggested by Fibonacci) by the end of a year:
1 ==> 2 ==> 4 ==> 8 pairs
We can also be sure that the rabbit population would double much more than 3 times per year. This is because the maximum CSIRO expectation is for a 10 fold increase per year, and a fourth doubling would give us 16 pairs of rabbits which is a 16 fold increase over the starting pair. Hence, if we say that rabbit populations double 3 times per year that is exactly the lower CSIRO estimate (an 8 fold increase) and almost exactly the higher CSIRO estimate (a 10 fold increase). So, the other point of realism in the CSIRO model is that is does allows a variation of between an 8 and a 10 fold increase per year, depending upon conditions. In other words, the rate of population doubling would vary slightly from year to year.
A better way of describing the annual increase is in terms of the total population of individual rabbits, not rabbit pairs, as a CSIRO "pair" does not always assume one male and one female (though in practice nature has a funny way of balancing the sex ratio). Hence, the CSIRO model is actually:
2 ==> 4 ==> 8 ==> 16 rabbits
This doesn't seem like much, but exponential growth has a way of sneaking up on you. If a rabbit population doubles 3 times a year, then after 3 years you don't have 9 times (3 x 3) the population. Instead, you have 9 times the population doubling, or 29 times the population which is 512 times the population from any starting population of rabbits. In just 15 short years you would have 245 rabbits, or 35,184,372,088,832 rabbits (32 Terapops) . It can be hard to visualise this number of rabbits, or how we get from one rabbit to so many rabbits in less than 17 years of modest growth each year. That's why I developed my New Malthusian Scale, which I've used here to show a rabbit population growing modestly each year for 16 and 2/3 years:
Table A. New Malthusian Scale with projected rabbit population doubling 3 times per year and one row equal to 3 and 1/3 years (as there are 10 population doubling per row)
29 rabbits = 512 Pops or 512 rabbits, and 245 rabbits = 32 Terapops, or 35,184,372,088,832 rabbits (over 35 trillion rabbits).
Note: 1 Kilopop = 1024 pops, 1 Megapop = 1024 Kilopops, 1 Gigapop = 1024 Megapop etc.
Note that a Pop could represent any starting population of rabbits, and 32 Terapops would then represent more than a 35 trillion fold increase of the starting population!
Note that this example assumes a constant population doubling period of a third of a year, just as the Malthusian Growth Model does, which is unrealistic. However, if growth rates vary then so do the population doubling times as demonstrated in my The Scales of 70. The key thing that the Scales of 70 demonstrates is that variable positive growth rates can produce population growth equally powerfully as constant positive growth rates. The Scales of 70 is not so easily dismissed as the Malthusian Growth Model.
Cute Little Bunnies?
In Britain rabbits have sometimes been popularised as cute little bunnies in children's books (and subsequent film adaptations) such as Beatrix Potter's series of books on Peter Rabbit published between 1902 and 1912, and Richard Adams' 1972 book Watership Down.
It is perhaps ironic then that the sort of rabbit numbers now being reported in Britain are an estimated population of 45 million, costing 100 million British Pounds in damages annually according to The Daily Mail (Moore, 2009). The article suggests that the popularisation of the cute little bunnies through books such as Richard Adams' Watership Down may have helped the recent population boom. The article also claims that a French bacteriologist illegally introduced myxomatosis in France in 1952, wiping out European rabbit populations across Europe (including Britain in 1953).
The Rabbits - An Australian Story
In Australia, however, rabbits are the subject of an allegorical children's tale of the colonisation and industrialisation of Australia by the British. Whatever you may think of it, this book belongs firmly to the Australian History Wars, the public debate on the interpretation of the British colonisation of Australia. The book is The Rabbits, written by John Marsden and beautifully illustrated by Shaun Tan. In the book, the rabbits represent the industrial and vastly more numerous British colonists (in their "millions and millions") and a small tree-dwelling marsupial - much like a numbat - represents the indigenous Aboriginal population (Marsden, Tan, 2003). It is intended as a sad tale of colonial imperialism (including the infamous stolen generation incidents) and environmental disaster, and it even includes a mention of non-native animal species introduced by the rabbits. It packs quite a punch for a book of just over 200 words.
I know it's just a children's book, and not a university thesis, but I do have a few small issues with The Rabbits. One of my quibbles is with the depiction of the "British" colonisation of Australia in The Rabbits , and the fact that that ALL of the settlers are described as coming from outside Australia whereas of course many modern Australians of European descent can trace their ancestry back many generations in Australia. Also worth noting is that, although the national flag of the rabbits is clearly intended to emulate the British flag, many of the immigrant rabbits in reality are actually from continental Europe, the Middle East, India, China, and other parts of Asia. One estimate - when the population of Australia in 1996 was 18,310,700 - is that roughly 29% of our rabbits were not, in fact, ethnically British (Jupp, 2001, p.83). But 13, 082,700 Australians (roughly 71% of the total) might still loosely be called "British" (Jupp, 2001, p.83). However, this includes 2, 263,300 Irish (roughly 12% of the total), the majority of whom would not consider themselves of British ethnicity today, although their ancestors may have been part of the British Empire and so might have seemed just as "British" to the Aboriginals. A more appropriate term is Anglo-Celt (Jupp, 2001, p.82). The other Anglo-Celt sub-totals include 8,293,400 English (roughly 45%), 1,997,700 Scottish (roughly 11%), 269,000 Cornish (roughly 1.5%) and 243,400 Welsh (roughly 1.3%). Contrary to overseas perceptions, although Australia was indeed established as a penal colony for the British Empire, most Australians are not descended from convicts. Only 160,000 convicts were ever shipped to Australia, with the last shipment in 1868 when the population already exceeded a million (Jupp, 2001, p.16).
Of course, in 1901 the various Australian colonies federated to form the nation of Australia, and so the national identity gradually became less British and more Australian over time. For example, although the Australian national flag was created for national federation in 1901, it was not given Royal assent or adopted as the definitive Australian flag until 1954 in the Flags Act 1953.Another example is that in a 1986 national census 2.9 million people did not identify with any ancestral ethnicity other than Australian (Jupp, 2001, p80).
Another problem is that many of the cultural aspects of modern Australia that make Australia a great place to live can be traced back to British culture (including language, education, democratic government, law, many sports, and many of the scientific and technological advantages of being a modern civilised nation). A more serious problem is that it is not possible to turn the clock back. Of course this doesn't set things right, and doesn't take away from the fact the immigration of human "rabbits" into Australia is ongoing. As a recent (1994) immigrant "rabbit" from Britain myself, I nonetheless feel that it is time to promote a policy for a sustainable population in Australia.
Of course, the population problem is not just an Australian problem - it is a global problem. In fact in a speech made earlier this year in the Australian Parliament by my own Member of Parliament - Kelvin Thomson - he argued that sustained population doubling is the global problem (Thomson, 2009):
"In the first two million years of human existence, the global human population was only a few million. Up to 1950, it had managed to climb to two billion. In the 50-odd years since, it has trebled to six billion people. And the population is projected to double again.
The consequences of the present population pressure are dramatic. In my belief, it is not plausible that the world’s population could double without the consequences becoming catastrophic. Yet, when it is suggested that the world’s population is a problem, there is zero interest from policy makers. In my view, it is not so much a problem as the problem."
I agree. The aim of my Exponentialist web site is to clarify the nature and challenges of population growth - the world's single most pressing problem - through a re-examination of the work of Malthus. See the summary on the Exponentialist Homepage for more.
Human populations - Differential Replication and Malthusian Selection
In Exponentialist terms The Rabbits depicts the differential replication of separate human populations due to Malthusian Selection. Malthusian Selection is the Exponentialist hypothesis that environmental and human cultural factors can cause one population to be able to sustain a Malthusian Parameter (population growth rate) at a higher rate for longer than another population, and thus out-compete it. See Malthusian Selection section of Reverend Malthus and Evolution for more.
Genetic evidence confirms that Europeans and Aboriginals are almost identical in terms of the genome, so race cannot be a factor. In the case of British colonists and indigenous Aboriginals in Australia, the environment has essentially remain unchanged and is clearly not a factor and so the overriding factor must be human culture. What else can explain how the population of European settlers has grown to a population of over 21 million in 2009 and is still growing, whereas the Aboriginal population numbered only an estimated 200,000 in 1996 (Jupp, 2001, p.83)? Obvious cultural factors include the modern agricultural practices of European settlers, and modern medicine. Another factor is the Industrial Revolution which started in England around the time of Australian colonisation in 1788. Factors such as these would have allowed the colonists to settle in ever increasing numbers right across the continent of Australia as depicted in The Rabbits. In the meantime Aboriginal numbers, which has persisted for tens of thousands of years before British colonisation, declined dramatically until the early Twentieth Century (Commonwealth of Australian, 2008):
Graph 1 - Based on figures taken from 3105.0.65.001 Australian Historical Population Statistics, 2008, Australia Bureau of Statistics (which are sourced from Smith, 1980).
Diseases such as smallpox played a major role in the reduction of the Indigenous population after colonisation. The introduction of smallpox into Australia is commonly attributed to the British (Diamond, 1997). However, at least one comprehensive study has concluded that, contrary to popular myth, the smallpox virus was not introduced by the British (Campbell, 2002 - from the forward by Frank Fenner):
"...the writer concludes that the four or five importations into Aboriginal Australia between the 1780s and 1860s came in with the Macassan traders operating on the northern coast of Australia. She dispels the myths that the outbreaks seen by the colonists in 1789 and 1829-30 were associated with the British settlement in New South Wales."
The eradication of smallpox followed Englishman Edward Jenner's discovery of a vaccine in the late Eighteenth Century, a couple of decades after initial colonisation. People from densely populated Europe had already some acquired some natural immunity to smallpox and other deadly diseases over centuries of intense suffering whereas Aboriginal populations were far more vulnerable to smallpox, the most lethal of the infectious diseases (Campbell, 2002).
Campbell believes that tuberculosis - although not as deadly as smallpox - was in fact the worst disease transmitted to Aboriginals by Europeans (Campbell, 2002, p.228). Ironically, tuberculosis is also sometimes known as the white plague (because sufferers appear paler than usual). However, with mortality rates of up to 60% in Aboriginal communities (Campbell, 2002, p.222), smallpox may well have been a far more deadly invader than the British. It is this fact which makes The Rabbits both ironic and tragic, given the fate of the rabbit populations in Australia, for it is the British who are depicted as rabbits but it was the Aboriginals who were ravaged by infectious disease.
Hence, Darwinian evolutionary theory does partially explain why the Indigenous population collapsed (due to diseases against which they initially lacked both immunity and adequate medical treatment). However, this is not an excuse to ignore the numerous injustices perpetrated against the Indigenous Aboriginal population by the mainly British colonists and subsequent Australians since initial colonisation.
The fate of the Indigenous population - so dramatic when graphed in isolation - appears to be almost a flat line when graphed against the massive and sustained rise of the overall Australian population (Commonwealth of Australian, 2008):
Graph 2 - Based on figures taken from 3105.0.65.001 Australian Historical Population Statistics, 2008, Australia Bureau of Statistics.
According to the modern synthesis of evolutionary theory the Malthusian Parameter for a population is defined as the measure of the fitness of a genome in the context of its environment (Fisher, 1930).The Exponentialist view is this traditional evolutionary view (which includes Natural Selection) fails to explain why the Indigenous population never rose to the same numbers as the non-Indigenous population, before or since colonisation. After all, both populations are living in the same natural environment with pretty much the same genetic heritage. Hence, the Malthusian Parameter for both populations should be the same over time (with the temporary exception caused by the initial lack of disease immunity of the Indigenous population following colonisation).
The Exponential View is that Malthusian Selection is the answer. Human culture is the key differentiating factor in this scenario, with all other factors being roughly equal. Hence, the relative fitness of discrete human populations living in the same environment (at the same or different times) is determined by human culture. Civilised populations are able to sustain higher growth rates for longer than hunter gatherer populations. This is not intended to demean hunter gatherer societies, past or present. They are a vital part of the long human story.
However, facts are facts. For better or worse civilisation - with all its trappings - has empowered human populations to grow faster and sustain larger populations than ever before. This is not a story restricted to Australia, but repeated throughout the history of humanity across the entire globe.
Malthusian Selection is not Social Darwinism
In case you think Malthusian Selection sounds like Social Darwinism I hasten to add that Malthusian Selection is not proposed to further any world view of European (or even British) racial superiority. I do not support racism in any form. Social Darwinism is a theory of social progress (and superiority) based on a corrupted notion of Natural Selection. Social Darwinists included people such as Herbert Spencer (who coined the phrase survival of the fittest) and Sir Francis Galton (Darwin's cousin, who promoted the highly controversial concept of eugenics), but not Darwin himself.
Malthusian Selection, on the other hand, is proposed as a force for selection at the level of populations in addition to Natural Selection. Unlike Social Darwinism, or even the Australian History Wars, Malthusian Selection is a neutral scientific hypothesis and does not seek to favour any one human population over another. Because Natural Selection is unable to explain how the European population was able to grow and sustain a much larger population in Australia than the indigenous Aboriginals, some other selective force must apply. That selective force is what I call Malthusian Selection which, in the case of Australian colonial history, shows that differential replication still occurs even when Natural Selection does not. Hence Malthusian Selection reveals that - from a population perspective - differential replication encompasses Natural Selection and not the other way around. This is why the concept of Malthusian Selection is important.
Malthusian Selection, Memes and The Extended Phenotype
Richard Dawkins introduced the concept of the meme - the unit of cultural evolution - in 1976 (Dawkins, 1976). In 1982 Dawkins then introduced the concept of the extended phenotype. Phenotype are the observable physical attributes of an organism (e.g. a shell, or a leg) whereas a good example of an extended phenotype is the dam built by a beaver (Dawkins, 1982). A much closer (but by no means perfect) analogy for Malthusian Selection is to consider some components of civilisation - roads, farms, cities and so on - as the extended phenotype of humanity. Malthusian Selection is caused by memetic evolution - not genetic evolution - resulting in a massive and complex intertwined set of extended human phenotypes that enhance the Malthusian Parameter for a given population.
However, given the looseness of the analogy (it's a bit of a stretch) and the fact that Malthus first most effectively drew our attention to the differential replication of human populations due to cultural factors I prefer to simply refer to this as Malthusian Selection - a selective force that acts on populations through the population growth rate (Malthus, 1798).
Although the Fibonacci numbers are rightly famous for other reasons, they are not useful for modelling realistic population growth scenarios. As the CSIRO model shows, annual rabbit population increase can appear relativity modest. However, as my use of population doubling clearly and visibly demonstrates, exponential growth can turn a modest annual increase into truly explosive exponential growth if sustained for a short number of years as repeatedly witnessed in the Australian experience. As it turns out, the most effective way to control a population of replicators (rabbits) is with another population of replicators - viral replicators.
Beautifully illustrated in The Rabbits (Marsden, Tan, 2003), we have a new and vivid analogy for human overpopulation as well the differential replication of human populations due to Malthusian Selection. Whereas Natural Selection would explain that discrete human populations with equal genomes will experience roughly the same population growth rates over time in the same environment, Malthusian Selection explains that cultural factors can cause wildly different population growth rates if all other factors are equal. Hence in terms of population the principle of differential replication encompasses both Natural Selection and Malthusian Selection.
Campbell, Judy, Invisible Invaders - smallpox and other diseases in Aboriginal Australia 1780-1880. Melbourne University Press. 2002.
Commonwealth of Australia, 3105.0.65.001 Australian Historical Population Statistics, 2008. Australian Bureau Of Statistics. (website accessed 24th October 2009)
CSIRO, The European Rabbit, Commonwealth of Australia. date unknown (web site accessed 21st September, 2009)
Dawkins, Richard, The Selfish Gene. Oxford University Press. 1976, 1989
Dawkins, Richard, The Extended Phenotype. Oxford University Press. 1982.
Diamond, Jared, Guns, Germs, and Steel: The Fates of Human Societies. W. W. Norton. 1997.
Drexler, K. Eric. Engines Of Creation - The Coming Era of Nanotechnology. Online Glossary (website accessed 23rd September 2009). 1986.
Fisher R. A.. The Genetical Theory of Natural Selection - A Complete Variorum Edition* - Oxford University Press. 1930, 1958, 1999, 2003*.
Freitas, Robert, A., Merkle, Ralph, C., Kinematic Self-Replicating Machines. Landes Bioscience. 2004.
Jupp, James, The Australian People - An Encyclopedia of the Nation, It's People and Their Origins. Cambridge. 2001.
Malthus, Thomas Robert, An Essay on the Principle of Population. J. Johnson. Library of Economics and Liberty. 1798. (1st edition)
Marsden, John & Tan, Shaun, The Rabbits. Lothian. 2003.
Moore, Victoria, With a rabbit population of 45m Britain is under threat from. The Great Bunny Plague. The Daily Mail. 13th April 2009. (Web site accessed 22nd September, 2009)
Smith, L.R., The Aboriginal Population of Australia, Australian National University Press, Canberra. 1980.
Thomson, Kelvin, Global Population SPEECH (Australian Parliament), Monday, 17 August 2009. (Web site accessed 22nd September, 2009)
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