# Elephant populations (and Dyck words)

Hello,

I'm relatively new to this forum so apologies if I have tagged my question incorrectly.

I have been in contact with a wildlife biologist recently concerning counting elephant populations and I wonder if people could comment on the following approach.

de Bruijn, Knuth and Rice showed that the expected height of a general Catalan tree is $\sqrt{\pi} \sqrt{n}$. In terms of Dyck words this translates to saying that the expected maximum excess of Xs over Ys in a Dyck word of length $2n$ is $\sqrt{\pi} \sqrt{n}$.

Now if we think of X as "an elephant arrives at the watering hole" and Y as the elephant leaving it then should not the largest number seen correspond to the above expression? Thus we could get an estimate of the population. And that is my question, how good an approximation to the population $n$ will this be?

Note that it is not a concern that an elephant revisits the site and is doubly counted, because this will not affect the maximum seen. However, one flaw in the model is that it assumes that elephants arrive independently. Certainly this will not be the case with young calves but there may be cliques who are fellow travellers also. Also, perhaps, the model assumes that the likelihood of an elephant being present at time $t$ is equally likely for all $t$. This may not be realistic either.

Apologies also to those who find all of the above just elephants :-).

Thanks, Patrick healy

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Is this really about elephants for you? That would be cool! – john mangual Oct 24 '12 at 16:27
Of course one has to be careful: what if there are other watering holes, or what if elephants do not drink on a daily basis? Also, maybe there is a day shift and a night shift. And are there some who spend a long time at the water, or not much? As a rough guide to corroborate other measures, it sounds good, but I would be hesitant to use only that measure to derive conclusions based on population estimates. Gerhard "Not A Population Theorist, However" Paseman, 2012.10.24 – Gerhard Paseman Oct 24 '12 at 16:41
@John Mangual: Yes, it genuinely is. I had in mind verifying this at at our uni's gym where people swipe a card to enter but unfortunately there is no "swiping out" so there is no way to know if they left ;-). – healyp Oct 24 '12 at 18:10
@Gerhard Paseman: Your multiple watering hole is relevant, because in this Nat'l Park in Zimbabwe there are 3 possible ones they could visit but my solution to this is, employ a team of students who keep in constant contact :-). I wanted to model teh situation this way to avoid issues of, as you suggest, "some who spend a long time at the water, or not much". Average feeding time would seem to require a different way of modelling it. – healyp Oct 24 '12 at 18:15
The problem is that while sitting at a watering hole you can get a good idea of the distribution of the feeding time, etc., you cannot estimate the population without knowing the distribution of the time spent without drinking. If it is, say, exponential, then arrivals are essentially Poisson but the intensity is the ratio of the total population to the expected time before drinks. Sitting at the hole, you'll observe exactly the same process for completely different populations if the ratio stays fixed. On the other hand, if you start marking elephants, etc., much simpler techniques apply... – fedja Oct 24 '12 at 18:33