# Russell's Blog

## A desirable extinction

Posted by Russell on March 25, 2010 at 3:19 p.m.
Some weeks ago, Buzz (my cat) escaped out my front door while I was carrying my bicycle into the apartment. For ten or twenty minutes, he romped through the ivy and bushes around my apartment while I followed him around rattling a bag of cat treats. Eventually, he let me pick him up and take him back inside. Naturally, he picked up a few fleas. Naturally, they have multiplied.

Oddly, the fleas don't seem to like Neil very much, nor do they like me. It's just poor Buzz that's beset by the nasty little critters.

### Figure 1: A flea.

As it happens, I've been thinking about endogenous metrics for estimating the sampling quality of an environmental shotgun sequencing dataset, and Buzz's little problem presented an opportunity to play with a simplified problem. So, I have decided to make Buzz, or rather his fleas, into a small experiment in ecology. I am going to try to see if I can drive them into extinction.

Now, this is normally what a pet owner does when they discover their pet has contracted some sort of annoying parasite, but I decided to take a more quantitative approach.

### Figure 2: A cat.

It's simple enough to count fleas on a cat, if the cat is willing to cooperate. Buzz loves the flea comb, and will gleefully hop onto the coffee table and wait to be combed if you show it to him. So, in the interest of science, I convinced my roommate to count the number of passes I made with the flea comb and how many fleas I captured (posterity will remember your efforts, Mehdi). Using his tally, I plotted the cumulative number of passes verses the cumulative number of fleas.

### Figure 3: Fleas captured

As expected, it became somewhat more difficult to capture the next flea as more fleas were captured, suggesting a depletion curve. The value of the asymptote should be the actual number of fleas on Buzz at the time, and reaching that number would imply local extinction for the fleas. Of course, there are probably other fleas lurking about that would recolonize Buzz. In principle, if I were to repeat the exercise frequently enough, Buzz would become a sink for fleas, and their migration to his fur would gradually deplete them from the environment.

There are a couple of different ways to model the impact of the combing on the flea population, with various advantages and disadvantages. All we really want to do here is to estimate the value of the asymptote, and so a simple model is probably sufficient. I showed this data to my fried Sharon Shewmake, an economics graduate student. Sharon, after editorializing on the endeavor ("Ew."), suggested this very simple model.

Assume that Buzz is not going to sit still long enough for the fleas to reproduce, for more fleas to migrate to his fur, and that the fleas already on his fur are going to stay put unless captured. Thus, there is a fixed initial population which only changes as a result of capturing fleas. Next, we assume that any given flea is equally likely to be captured on a single pass of the comb. So, the expectation value for number of fleas captured on a single pass is the product of the current population and the probability of capturing a flea.

$N_{i+1}=N_i-pN_i$

where N is the population of fleas and p is the probability of any particular flea being captured on a single pass. One could tart this up a bit by modeling it as a stochastic process and executing a bunch of Monte Carlo trials until the outcomes converge, but that seems like overkill for a simple single variable problem like this. We will put up with the intellectual inconvenience of capturing fractional fleas.

This is a little easier to see if we let N represent the number of fleas remaining on the cat, rather than the number of fleas captured.

$N_{i+1}=qN_i$

If we stretch our credulity far enough to imagine this as a continuous function, we can express it as a differential equation.

$\frac{d N}{d t}=\alpha N$

Sorry if this bothers you. Not only are we extracting fractional fleas, but we are now modeling the combing process as a sort of flea-killing-combine continuously mowing its way through the fur. This is a model, so you shouldn't be surprised to find massless rope and spherical cows. Anyway, it has a nice easy solution.

$N(t)=e^{\alpha t}$

Well, what the heck. This is a decaying function, so let's pluck a minus sign out of the exponential factor, and maybe tack on a scale factor for the initial population.

$N(t)=N_0 e^{-\alpha t}$

While we're at it, why don't we go back to letting the function stand for the number of fleas captured, rather than the fleas on the cat.

$N(t)=N_0(1-e^{-\alpha t})$

This gives us a nice function to use for a linear regression. A little help from scipy, and we find that the initial population is estimated at 39.7 fleas, and the decay factor is 0.011.

### Figure 4: Flea population

I captured 34 fleas, so that means I missed about five or six. In order to be reasonably confident that I'd captured all 39 fleas, I would have had to continued for about 400 passes with the comb, instead of 173. Buzz is a patient cat, but he started to loose interest around 120 passes, and had to be fetched back onto the coffee table a few time times during the last 50 passes. My guess is that 400 passes would require some kind of sedative. On the other hand, he does seem to like Guinness, so there may be something to that.

Science has been served. I'm going to the pet store to buy some flea collars.

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