Referring to this story about how one unspayed cat can spawn 420,000 cats in 5 years, I wanted to see, just how stupid is this? Of course, it seems pretty stupid on a practical level. But that's not enough to say.
There are a few parameters that are relevant in the model. Suppose we start with just one female cat, who gets pregnant on day one. Cats can have from 3-5 litters annually, meaning a 2-4 month gestation period. Typical litters are 3-6 kittens. Cats need 5-9 months to reach maturity. And, of course, only female cats, presumably about half of the cat population, can have kittens. Now, the model is exponential growth, and so I expect it to be pretty sensitive to the input parameters. Just how sensitive?
At t=0 there is just one cat. Call C(t) the number of cats at time t. If our time step is 3 months (average gestation time), then
C(3 months) = C(0) + C(0)*(size of litter) = 1 + 1*(size of litter)
Now, in ANOTHER 3 months the cat may have gotten pregnant again, and will therefore be giving birth to another litter. The cats born at 3 months have not reached the age where they can have any kittens.
C(6 months) = C(3 months) + C(3 months)*(size of litter)*(probability of getting pregnant again)
Now, at 9 months, the kittens born in the first litter could potentially have a litter of their own. The number of eligible pregnant females should be the number of cats who were alive 6 months ago, or two time steps in our scheme, divided by 2.
C(i+1) = C(i) + C(i-2)*(size of litter)*(probability of getting pregnant again)/2
We can now just iterate for twenty time steps to find the total number of cats alive in five years due to one unspayed cat.
The stupidest possible outrageous model would be to say that all female cats get pregnant every cycle, and all have a litter of 6. This is the "Kitties take over the world" model, the blue line. The result is 1.2 MILLION cats in 5 years. Exponential growth.
Now, supposing that the probability was still good of getting pregnant (0.8) but that the average litter is reduced to 4. This is the "Sexy cat" model (black line, in inset), and it still results in a TRULY MASSIVE number of cats, 45000, but note that this is already about a factor of 10 below what the silly people say. A sensible model with probability of pregnancy =0.3 (something like 3 litters a year) and average litter size of 4 results in 638 cats.
This model obviously doesn't take into account cats who die or the dependence of probability on the number of cats, reducing the food supply. And actually, it will be pretty sensitive to the average gestation time, though I believe that is far less variable than the other parameters. But 600 isn't too outrageous, given the number of cats taken in by shelters every year. It's a scary number. Spay and neuter.