July 2006

The King of Persia and a peasant sat down one day to play a game of chess. The King, with his noble birth, expensive education, and strategic training was confident of an easy victory.

“If you win, I shall give you whatever you ask,” he said to the peasant and began the game. However things didn’t go as the King had anticipated, and he was shocked to be quickly defeated. Expecting the peasant to demand half the kingdom, his daughter, or worst of all his fluffy Persian cat, the King was pleasantly surprised when the peasant asked for nothing but rice.

“How much rice?” asked the King.

“If it pleases your Majesty,” answered the peasant “I would like one grain placed on the first square of this chessboard, two on the second, four on the third, eight on the fourth, and so on, doubling the number of grains with each square.”

The King, greatly relieved and thinking the peasant a worthless fool called his Chamberlain and ordered him to measure out the rice, put it in a bag, and send the scrofulous peasant on his way.

To the King’s horror, his Chamberlain informed him that there was not enough rice in the whole World, let alone Persia to honour his deal with the peasant. In the fashion of tyrants when faced with ruin, the King had the peasant’s head cut off.


A version of this tale is often told to schoolchildren as an illustration of exponential growth. I am sure you’ve heard a version of the phrase “It’s growing exponentially!” which is particularly irritating to the ears of mathematicians as hardly anyone who says it knows what the “exponentially” bit really means, and usually the thing that they think is growing exponentially really isn’t…it’s just growing a lot. For something to be growing exponentially (warning! here comes the maths bit!) its rate of growth needs to be proportional to its current size. In the story, the number of grains on the next square was just double the number on the present one, and doubling-up is very proportional indeed!

The important thing about exponential growth is that numbers can very quickly become massive…VERY massive. As an example, the energy from a nuclear bomb is released exponentially…ka-boom!

I’m sure that by now you’ve calculated how many grains of rice the peasant won, and agree with me that it’s 18,446,744,073,709,551,615, or about 18 quintillion (American quintillion, that is!). If an average grain of rice weighs roughly 10 milligrams this much rice would weigh about 15 times as much as Mt. Everest. Methinks the Chamberlain was right.

Imagine a pregnant puss was washed up on the shores of a hitherto cat-less island, had her kittens, and then watched these kittens grow-up and have kittens, and these have kittens, and so on. If all these kittens survive to have kittens of their own, and with two litters a year being born, and with two females per litter…we end up in the territory of exponential growth. Unlike the doubling of the grains of rice, though, each female cat is making around four other females each year, for however many years she can keep going (not many…poor, exhausted puss!) The cats would exceed the peasant’s grains of rice within 20 years…and each puss weighs a lot more than a grain of rice! The island would sink beneath the waves under the weight of a multi-Everest of cat flesh. So, the conundrum is…as this is clearly not happening, what is happening to all the cats?


Well, it’s not really a difficult answer, and I’m sure you’ve got it already. Until they’re actually born, these cats are just potential cats, a potential that for all but a very, very tiny minority will never be realised. And if a cat were never born, it doesn’t have kittens. Our pregnant puss in the conundrum above may well have seen all her own kittens surviving (assuming there was plenty of food, no fatal kitten diseases, and no predators of kittens about). She may even have seen all her grandchildren, and great grandchildren survive. What we can guarantee though, is that sooner or later (probably sooner…leastways, within a decade) the food resource on the island would become exhausted, and the number of cats that it could support would depend upon how quickly this resource could renew itself each year.

The cats would go on breeding, and the kittens would be abandoned or would starve. At some point an equilibrium would be reached and the number of kittens that survived would roughly balance the number of older cats that died. Good years and bad years would come and go and the cat population would rise or decline accordingly, but the important thing to note is that many, many more kittens would die before reaching breeding age than would survive. Nature is red in tooth and claw, and ten kittens dieing for each one living may well have been judged by evolution to be a fair price to pay for the propagation of cat genes.

But you don’t have to take my word for it...we can do the numbers!



Here you can see examples of models. Far be it for me to miss a chance to put up pictures of semi-clad, beautiful flesh. However, the title of this section refers to a different type of model, one far less exotic, but nonetheless exciting to mathematical geeks like me the World over.

Mathematical models are a common tool used for trying to understand things, from how economies run, how disease epidemics spread, how chemical reactions happen, I could go on for pages, but I shan’t. They’re a very powerful way to get an idea of what is happening, particularly when it’s happening away from human eyes and so isn’t directly measurable. Population growth is a prime candidate for mathematical modelling…I’ve done it myself many times in my career and even (boasting only a little) had my work published. For those of you who are sent to sleep by maths-talk, you could skip now to the Results section below, or drink an espresso and persevere!

The first thing a model requires is a structure. These can be extremely complex or overly simple, but are usually more than adequate when they’re somewhere in between. The model I am going to use to get an idea of a feral cat population is based upon the life-cycle of a cat shown in the above diagram. It may seem complex but really isn’t.

All you have to do is start at the top at “Birth” and follow the arrows. Each step is a year in the life of the cat with all paths eventually ending at “Death”.

The things we want to estimate are the odds of moving from each of the boxes in this model to the others they connect directly to. For example, an un-neutered female of 4 years is found in the “F4” box. When we re-visit her a year later she will either have moved over into the “F5” box (un-neutered female aged 5 years), taken the slope down to the “N5” box (neutered female aged 5 years) or gone up, right and down to “Death”.

What we’re going to find out is if these three possible steps have odds of 40:40:20, or 70:10:20, or 10:10:80 or something else. When you look at all the boxes and arrows, and realise that each arrow needs a likelihood assigned to it, you can appreciate that this is quite a task. To help, we need some rules and some data…and a few assumptions.

The data that we’ll need are the following:

- The number of litters a female cat produces per year,
- The number of kittens per litter,
- The magnitude of the available food and water which the cats compete for,
- The relative resource needs of cats of different gender and age
- The relative ability of cats of different gender and age to find a share of the available food/water
- The numbers of females neutered per year, and
- The physical toll that continual pregnancy exacts upon a female cat, relative to a male or spayed female

Internet searches are very helpful at providing a lot of useful information! A female cat can start her breeding cycle at between 4-7 months depending on her health and the season of the year. It is not uncommon for a female kitten from the first litters of the season in February/March to be pregnant the following September/October. A female cat can have more than one litter in a year, with reported estimates for different populations ranging considerably, including average litters per year varying from 1.4 to 3 from different studies. An average number of kittens per litter is often cited as 3 or 4. Producing kittens in a feral population is exhausting for the cat, and with an average of 5.25 litters in a lifetime, female cats have a life expectancy of about 3 years, compared to the life expectancy of about 7 years for a male. Mortality among kittens is high, with one study estimating 48% dieing in their first three months, and 67% within the first six months, and only about 20-25% surviving to breed.

The assumptions in our model are quite loose. The relative food needs of male/female cats and young/old cats are assumed to be constant and equal, insofar as a grown male is expected to be able to survive on the same amount of food that a grown female can survive on. No correction is made for older/younger adult cats, except for kittens who require less. A kitten is assumed to need half of the food that a fully grown cat needs to survive to reproductive age.

Of more importance is the ability of cats to locate the food they need. A strong, healthy, young male may be more able to locate its “fair share” of the available food at the expense of a weaker cat. In the model, younger cats are considered more able than older cats, and males and neutered females more able than un-neutered females. Young cats (9 months) and nursing mothers are considered among the less able, with older un-neutered females the least able. This pattern can be used to determine the functional description of the relative ability to locate food and water, with parameters chosen to coincide with model results giving life expectancy similar to that found in previous reports (i.e. 3 years for an un-neutered female and 7 years for a male).

Each year, the food needed to be consumed by the existing cats and the new-born kittens is calculated. The ability of the mothers to locate the food is compared to how much is available and the number of kittens that survive or die can then be determined. It is assumed that a male kitten is as likely to survive to six months of age as a female kitten.

The only thing left is to set the amount of food and water that is available before we can run the model. We do this by back-calculation…we make a guess and see what equilibrium position the model reaches, i.e. how many cats it says that amount of food can sustain. If we think this number is too few/many, we increase/decrease the amount of food. Simply put, we decide on what we think the size of the cat population is and then work backwards to the amount of food.


Perhaps the most difficult thing to estimate has been the size of the cat population on Symi. Estimates of 1500 to 3000 have been offered up, and I’ve decided to go with 2500. If you think it’s more or less, then you can scale the results up or down.



From the model, a stable population of 2500 cats consists of 1892 males and 608 females, a ratio of about 3:1 which is reasonably consistent with previous research. The disparity is due to the harder life that females have, succumbing at an average of 3 years of age, as opposed to 7.

Each year, an average of 3831 kittens are born (6.3 per female) of which 3223 die and 608 survive, a mortality rate of 84%

It’s a terrible tale, and it goes on the World over. Where, you may ask, are all these dieing kittens? Why aren’t the streets of Symi littered with tiny corpses? On an island the size of Symi, 3223 is not so huge a number as it may appear. Many are simply abandoned at birth, others are killed by tomcats, many expire in the heat of a long Symi summer. Whatever happens, their bodies are unlikely to be wasted; Symi has too many hungry rats for that. If you find this explanation difficult to accept, then ask yourself where all the dead rats are, and they are surely born and die at a faster rate than the cats.

Symi has had an annual vet visit for some years, so the figure of 3223 is likely to be somewhat reduced. Last year 63 females were neutered. The number of males neutered was 9, but this is completely irrelevant to the population dynamics of the cats on Symi, simply because where one male fails to mate a female, there will always be another to take his place. The male’s job is easily done and quickly over.


In what follows, I am going to look at the impact of three different Trap/Neuter/Return (TNR) programs:

- A single vet visit per year where 60 females are neutered
- A single vet visit per year where 120 females are neutered
- Two vet visits per year, six months apart, where 60 females are neutered in each.


The first thing to note is that by neutering a female cat, not only do you prevent the birth of its kittens, but also of those kittens’ kittens, and so on. With this in mind, we can estimate the impact of the TNR program in a couple of ways:

- by seeing how many kitten-deaths each program prevents compared to no neutering
- by comparing the effect of each program on the size of the cat population in both the short and long term

Here you can see the impact on the cat population of a single vet visit per year neutering 60 females. The model never fully converges but wobbles (I’ll explain why below) about an average population of 1970 cats (255 un-neutered females, 400 females, and 1315 males). As neutered females live longer than un-neutered ones, the ratio of males to females has fallen to about 2:1. The number of kitten deaths per year averages about 1240 which is substantially less than the 3223 which die each year when there is no neutering, about 2000 less, in fact.

You may have noticed in the above graph that the timescale is rather large, 200 years, but don’t worry - this was only chosen to show you clearly how the numbers cycle; the pattern is more or less reached by 30 years and cycles with a period of about 15 years.

Over the first 30 years since the introduction of TNR, we can look to see how each particular policy performs.

With one visit with 120 females neutered, a cyclic pattern predicts 15 year highs and lows in the population. Why this cycling behaviour?

To help explain, have a look at the following graph which shows the number of males, un-neutered females, and neutered females under this particular TNR program for the first thirty years.

The rapid decline in the number of un-neutered females during the first five years is due to the neutering program; most, but not quite all, of the breeding females get neutered.

The problem with a single vet visit is that there are always a number of female kittens who are too young to be neutered when the vet is there. These cats have kittens before the vet comes the following year, and some of these are again too young to be neutered. It is only after several years, when a sufficiently high number of the older males and neutered females have died, thus reducing the competition for the available food resource, that there can be an explosion in the number of kittens, with the birth-rate exceeding the neutering rate. The number of cats begins to grow again and reaches a peak when the available food is no longer enough, competition returns, and kittens once again begin to die in large numbers. The vet is now able to neuter all the older females that he has been unable to neuter in the previous few years and the population once again decreases. This whole cycle gets repeated about every 15 years.

The key to solving this problem of periodic high population is to make the vet visits six-monthly, not annually. Instead of having two weeks in the late summer, we only have one, and those kittens who were too young to be neutered can be neutered the following spring by a second one-week visit. Two visits of one week set six months apart, rather than two weeks close together…and what a difference it makes. All breeding can be stopped with 7 years.

For me, the best way to examine how well these TNR programs do is to look at the number of kitten-deaths that have been avoided over time compared to a policy of no neutering at all.

All three TNR Programs result in very big reductions in the number of kitten-deaths over time. In the first 10 years, the single visit with 60 neuterings has resulted in the cumulative prevention of about 17,000 kitten-deaths. The other two policies are pretty similar at 10 years, with about 27,000 kitten-deaths avoided. At 20 years, the benefit of the 6-monthly visit over the annual one can be seen with 61,000, as opposed to 51,000 kitten-deaths avoided.


So, what are we to make of all of this? I’ll start by stating a few caveats to these results!

Firstly, all I have done is use a mathematical model, and things never quite work out as they do in a mathematician’s workbook. Still, these numbers are instructive. The model has highlighted certain aspects of the population dynamics that we may not have been expected. It may be that the numbers are exaggerated or under-estimated, but the principle is still sound, nonetheless. Large numbers of kitten deaths are happening each year, and some neutering policies are better than others.

Secondly, the TNR policies described here will only work if enough un-neutered females can actually be located and trapped at each vet visit. In practice this may prove difficult especially after several years when un-neutered females become less numerous than neutered ones. I did attempt to include some correction in the model, but realised that any adjustment I made was nothing more than a complete guess. Who’s to say that volunteers out searching for the remaining 30 un-neutered females in a population of 600 won’t be successful? I wouldn’t dare to suggest they wouldn’t be.

Thirdly, I may have got the stable population size when there is no neutering completely wrong. Maybe it is 1000 or 2000 or 3000. For population estimates of 1000 and 2000, the single annual visit will not stop breeding or large numbers of kitten deaths whereas the six-monthly visits will. For 3000, the number of neuterings would have to be increased to 70 per visit to stop breeding with the six-monthly visit.

It may be, of course, that stopping breeding is not the goal. An alternative strategy to maintain a small, manageable, healthy cat population in which kitten-death is all but eradicated, is to only have the spring visit on consecutive years. It should be noted however, that there can be an explosion in the cat population even if a single year goes by with no vet visit at all.

Maintaining the late summer visit is vital. Adding a second visit in the spring is maybe no less important.

Thank you for reading!

Will




After getting his doctorate in Statistics, Will became a Research Fellow at Imperial College, London before deciding that his career needed to be far less interesting. Eighteen soul-rending months as a statistician in the pharmaceutical industry making the tea and with a constant overdraft encouraged him to return to Academia for a pay rise. As Will’s star rose in the statistical firmament, so he gained his reputation as a jobbing statistician who could always be called upon to provide solutions when others wouldn’t and usually couldn’t. A Fellow of the Royal Statistical Society since 1990, Will was made a Chartered Statistician ten years later by his peers, and has struggled ever since to convince people that this is not as boring as it sounds, or at least less boring than being a Chartered Accountant or Actuary. Will is now fortunate enough to have clients beating their path to his e-mail inbox to hire his unparalleled statistical skills (I’m writing this myself…had you guessed?) and is able to do his amazingly wonderful work wherever there’s a plug socket and internet connection. Will spends as much time as his daughter & cats allow him on Symi where he enjoys the culture, the people, the weather, the food and annoying the ex-pats. Will’s favourite author is Sean Damer, who he thinks got it spot-on. With any luck, he hopes to end his days on Symi being old, fat, tanned, wrinkly and bald, but still being served the most excellent nibbles with his ouzaki when playing Tavli at Kantirimi.


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