Recently I am studying using statistical methods to estimate animal abundance. It seems that the mark recapture method is a widely used statistical method to do it. However, there are some concepts I am a little bit confused about. Here is an example.
Suppose I want to do a population study for wild ducks in a certain area. I caught 360 ducks on the first day, and 19 of them had markers from last year. Then I gave new markers to the rest of the ducks. I caught 189 ducks on the second day, and 8 of them had markers from last year and 92 of them had markers from the day before.
Here is my way to estimate the population.
$n_{1}$ = 360 - 19 = 341 (Sample size from the first day exclude the ducks with markers from last year)
$n_{2}$ = 189 - 8 = 181 (Sample size from the second day exclude the ducks with markers from last year)
$m_{2}$ = 92 (There are 92 ducks that have bands from the day before)
$\hat{N}$ = (341*181)/92 = 670.88
I am not sure if this is correct since I don't know if I need to exclude the ducks that have markers from last year in my calculation. I will be grateful if somebody can help me.
Also, do I need to exclude the ducks that have markers from last year if I want to calculate $x_{11}$ (Number of ducks present on both days), $x_{12}$ (Ducks present on the first day and absent on the second day), and $x_{21}$ (Ducks absent on the first day and present on the second day)?
Without knowing how many ducks were tagged last year, the number of tagged ducks that were captured on the first day, $19$, is not informative. Instead, you must use the second day's capture information.
That said, you should not exclude ducks that were tagged from last year. These are part of the total number of ducks tagged. So for instance, you captured $189$ ducks on the second day, of which $92 + 8 = 100$ were tagged from either the previous year or day. Then, for the number of ducks tagged on the first day, it is the sum of those already tagged from the previous year and the ones tagged on that day, i.e. $360$. So your estimate should be $$\hat N = \frac{(189)(360)}{100} = 680.4.$$
Now, we can also perform this calculation on the previous year's tagged ducks only: in other words, you can pretend that you didn't tag any new ducks. Then you had $19$ ducks already tagged on the first visit, and you captured $189$ ducks on the second visit, of which only $8$ had tags. Then the estimate is $$\hat N = \frac{(189)(19)}{8} = 448.875,$$ which of course is quite a bit less, and is not as reliable because there are so few ducks that had the old tags.
If you had information about the total number of ducks tagged last year, then you could perform a more sophisticated analysis that incorporates more than one capture-recapture visit pair.
Moreover, this particular choice of estimator is not the only one possible; there are Bayesian estimators, for example. Confidence intervals are also important to calculate.