All of the approaches in this section have in common that, whereas previously we were focussed on estimating survival (accounting for detection variation, covariates, etc.) , now interest includes estimating "recruitment", as well as potentially estimating abundance at each sampling occasion if possible. Before going too much further we should be clear on what we mean by "recruitment". Broadly speaking, we are referring to any entry into the population between marking periods of (by definition) unmarked animals. So recruitment can include births from within the local population, but also movement (immigration) of either adults or young from other areas. Depending on the study design sometimes we can separate these components, sometimes not.
The change in focus from survival estimation (which was based only upon recaptures of marked animals) to recruitment and abundance estimation (which by definition includes unmarked animals) has important implications. Number one, obviously we now have to include the unmarked animals explicitly in the estimation model, a departure from CJS. Number two, in order to estimate anything we have to make some assumptions about the capture and survival processes and whether these are the same or different for unmarked and marked animals, or heterogeneous among animals. That is, in addition to CJS assumptions, we are now going to have to assume (most of the time ) that:
Unmarked and previously marked animals have the same probability of recapture, i.e,. capture does not behaviorally affect animals
All individuals in the population have the same probability of capture, i.e., there is not individual capture heterogeneity among animals.
You should recognize these strong assumptions as being essentially the same as the Lincoln-Petersen assumptions, and that is no accident. Essentially, what has happened is that by having multiple capture occasions, we can either
Assume the population is closed, focus on abundance estimation, and robustly model capture heterogeneity or
Allow the population to be open, estimate abundance, survival and recruitment, but make very restrictive assumption about the capture process.
We cannot do both with a the basic, multiple CMR design, because there is no statistical free lunch.
Keep the above in mind as we go through some neat sounding approaches that produce estimates, below.
Jolly-Seber Model
The basic Jolly-Seber (JS) model is, as described in the notes, a straightforward extenstion of CJS that allows estimation of recruitment and abundance (in addition to survival) by explicitly modeling the unknown parameters Ni and Bi (abundance and recruitment). The steps in JS can be thought of informally as:
Estimate Ni at each occasion by captures of previously marked and unmarked animals, in a manner very similar to Lincoln-Petersen
Estimate Phii (survival between i and i +1) from recaptures of marked animals by CJS models
Estimate Bi by difference between next period's N and survivors from this period:
Bi = Ni+1 - Phii (Ni -ni+Ri)
where -(ni -Ri) is included to account for the difference between number of animals caught and those released (so trap loss or sacrifice). For a k - sample CMR study the following parameters will be estimable:
Abundance (N): occasion 2 to k-1
Survival (Phi): occasion 1 to k-2
Recruitment (B): occasion 2 to k-2
Recapture (p) : occasion 2 to k-2
More parameters can be estimable if we make certain assumptions (e..g, Phi constant among occasions) but the above will be estimable under the fully time-specific model. The JS model is incorporated into MARK and RMark, but I have found it to be very unstable, often failing to converge, and thus favor the alternative parameterization of JS provided in POPAN.
Super-population (POPAN) parameterization
The POPAN parameterization models survival and capture probabilities in a similar way to JS, but comes at recruitment and abundance estimation from a different direction. POPAN starts with the idea of a 'superpopulation' parameter Nsuper, which can be thought of as a reservoir of animals that may enter the population during the course of the study. Recruitment is then defined in terms of entry probabities penti , so that Bi = Nsuper*penti . Abundance is then gotten by application of the JS formula in the other direction
Ni+1 = Bi + Phii (Ni -ni+Ri)
N1 = B0
Because of various constraints (e.g., the entry probabilities have to sum to 1) not all the N's or B's are estimable; in fact, the same ones are estimable as under JS (no surprise- this is basically the same model).
I provide an example from Jim Nichols data on meadow voles (Microtus pennsylvanicus) live trapped over 13 occasions in the 1980s at Patuxent. The R script reads the data from a MARK input file and converts it to RMark format, then runs several models with varying assumptions about time dependency in phi, pent, and p. Note that in the data there are several negative frequencies. These refer to animals that were captured and not released on the last occasion, which is needed for JS (but not CJS).
This is a simple example with no grouping variables or covariates; if these were present they obviously could be modeled as well. The script produces model averaged real parameter estimates, as well as derived abundance estimates under the time specific model.
Pradel temporal symmetry approach
Under this approach, the capture histories for each animal are modeled in both forward and reverse direction. In forward direction, this is simple a CJS analysis: animals that are released at i and recaptured at some i+k occasion must have survived k intervals. In reverse direction, we are modeling entry or "seniority": an animal that was in the population at time i and had earlier been captured at i -k, must have entered the population k occasions (or earlier) ago. Putting these approaches together allows estimation of:
Phii - survival
fi - recruitment
pi - recapture
each of which may be modeled as time-specific, group-specific, or in relation to covariates.
The above is the "survival and recruitment" parameterization (Pradrec), but the Pradel model can be cast in terms of population growth rates (lambda) or seniority rates (the original Pradel parameter). There is also a 'recruitment only' version that is appropriate if mortality can be assumed nil.
The R script reads the mouse data, computes alternative Pradel models, conducts model averaging, and produces a table of real parameter estimates for the time specific model. Note that the Pradel approach does not estimate abundance; instead a per-capita recruitment rate f is directly produced. For comparison, the equivalent parameter from POPAN-JS would be
fi = Bi /Ni.
Comments on approaches
Either POPAN-JS or the Pradel approaches provides flexible modeling for CMR data for survival and recruitment estimation. Obviously, POPAN-JS provides additional information in the form of abundance estimates, but it must be kept in mind that the abundance estimates are very subject to biases if capture heterogeneity is an issue. Again, however, because both approaches depend ultimately on inclusion of unmarked animals, there is no avoiding sensitivity to the heterogeneity/ capture effect issue. To cut that knot we need more data: hence, the "Robust Design."