You recently learned how to estimate abundance, survival, and recruitment with a robust-design model, which required each primary occasion within a set of primary occasions to have multiple secondary occasions. Here, we’ll briefly introduce a method that allows you to estimate abundance, survival, and recruitment using live-recaptures data without secondary occasions, i.e., exactly the type of data you analyzed with a CJS model earlier in the semester.
In the live-recaptures modeling that you did earlier in the course, estimation of survival and recapture probabilities depended solely on re-observations of individuals that were marked and released, i.e., there was no attempt to estimate initial capture probabilities of unmarked individuals. To estimate abundance, as you’ve seen in recent weeks, we need to concern ourselves with the number of unmarked individuals on each occasion. We’ll do so using one of the various versions of the Jolly-Seber (JS) model that exist and, like with the robust-design model, be able to estimate abundance, \(\hat{N_i}\), recruitment rate, \(\hat{f_i}\), and population growth rate, \(\hat{\lambda_i}\), as well as apparent survival rate, \(\hat{phi_i}\), and capture probability, \(\hat{p_i}\).
In JS studies, we must model the capture process for unmarked animals. This is what allows us to estimate \(N_i\) and \(f_i\). To do so, we add an assumption to what we used in CJS-type models of live-recaptures data. We assume that \(p_i\) is the same for marked and unmarked animals. If you think about this, it makes good intuitive sense. If you want to estimate survival rate, you can just work with marked animals. But, if you want to know about population size and/or recruitment, you have to learn about the unmarked animals.
The implications are not trivial. You need to record the numbers of unmarked animals in surveys. If you are doing recaptures, this is probably done as a matter of course. In fact, it’s typical in most studies, to mark and release all unmarked animals so this is a non-issue. But, if some unmarked animals are released unmarked, they need to be recorded. In studies relying on resightings, you also need to record the numbers of unmarked animals that are observed during surveys. This is not necessary in CJS studies.
JS models focus on a single age class, typically adults. We cannot estimate abundance of young animals because we cannot estimate a capture probability for them. To do so, you need to use a robust design. Also, the study area must remain a consistent size. If you don’t then the concepts of population size and recruitment become nonsensical. Losses on capture can be accommodated as you’ll see briefly below.
\(\hat{N_i} = \frac{n_i}{\hat{p_i}}\): can estimate for \(i = 2, 3, \ldots, K-1\)
\(\hat{B_i} = \hat{N_{i+1}} - \hat{\phi_i} \cdot (\hat{N_i} - n_i + R_i)\), where \(B_i\) is the number of new individuals that joined the population between samples \(i\) and \(i+1\), \(n_i\) is the total number of individuals caught on occasion \(i\), \(R_i\) is the number released back into the population, and \(n_i + R_i\) accounts for the number of individuals caught but not released back into the population, i.e., allows you to account for losses on capture (mortalities associated with capture events).
As before, you can model parameters as time-varying or constant through time, and you can use covariates in your models. Also, you can fix parameters if you think that is appropriate, e.g. you might fix \(B_i = 0\) for time periods when you know that no recruitment could have occurred. As before, there are multi-state versions available.
Alternative parameterizations of the JS model above now exist. The primary difference among these is how they model the number of unmarked animals that are caught on each occasion. In MARK, there are multiple parameterizations of the JS model.
POPAN - a model developed by Schwarz and Arnason that parameterizes the Jolly-Seber model in terms of a super-population (\(N\)), and the probability of entry (pent) into that super-population at each occasion.
Link-Barker – this is the Schwarz and Arnason model re-parameterized with the entry probabilities replaced by recruitment (\(f\)), where \(f_i\) is interpreted as a per capita recruitment rate, or the net number of new animals entering the population between occasion \(i\) and \(i+1\) per animal alive at occasion \(i\).
Burnham JS – Burnham developed a parameterization of the JS model that provides estimates of the rate of population change (\(\lambda\)) and the population size on the first trapping occasion (\(N_i\)). Convergence is difficult for this likelihood and so it has not seen as much use as the others.
Pradel models - The recruitment parameterization has similarities to the Link-Barker model in that it works with recruitment. It also has similarities to Burnham’s JS model in that it works with \(\lambda\).
This topic is nicely covered in Ch. 12 of CW, which was written by Carl Schwarz and Neil Arnason. Quoting from page 12-11 of that chapter,
First, only certain of the formulations can be used in MARK if losses-on-capture occur in the experiment. Secondly, and more importantly, different formulations give you different types of information and can be used to test different hypotheses. All of the formulations should give the same estimates of survival and catchability, as all formulations estimate these from recaptures of previously marked animals using a CJS likelihood component. Even though all the models give different types of estimates for growth or recruitment or births, it is always possible to transform the estimates from one type to another by simple transformation and the standard errors can be found using the Delta method. The major equivalents are between NET births, recruitment, and population growth parameters.
\[f_i = \frac{B_i}{N_i} = N \frac{b_i}{N_i}\]
\[\lambda_i = \frac {N_{i+1}}{N_i}= \frac{N_i \phi_i + B_i}{N_i}= \phi_i + f_i\] Note that in the above, \(b_i\) represent the probability that a member of the super-population, \(N\), enters the population between occasion \(i\) and \(i+1\). The net number of new individuals (recruits or births) is \(B_i = N \cdot b_i\). Finally, notice that \(f_i\) is a per capita rate, i.e., \(f_i = \frac{B_i}{N_i}\).
Some of the differences in what can be directly obtained from different parameterizations is nicely shown in Table 12.5 on page 12-12 of Ch. 12 of CW, which is copied below.
POPAN – a model developed by Schwarz and Arnason that parameterizes the JS model in terms of a super-population (\(N\)), and the probability of entry into that super-population at each occasion. We’ll work with the POPAN version on a worked example from Ch. 12 of CW. For each group being considered in the analysis, we’ll have 4 PIMs.
\(\phi_i\) - apparent survival (\(t-1\) estimates)
\(p_i\) - capture probability given the animal is alive and on the study area, i.e., available for capture (\(t\) estimates)
\(pent_i\) - probability of entry into the population for the occasion (\(t-1\) estimates for occasions \(2, 3, \ldots, t\)) [The probability of being in the population on the first occasion is: \(pent_0 = 1 - \sum(pent_i)\). Convergence of this model is difficult to achieve unless you use the MLogit link for the pent parameters.]
\(N\) - super-population size (1 estimate), where the super-population consists of all individuals that were available for capture at any time during the study
The members of \(N\) are assumed to enter the sampled population at different times according to the entry probabilities. Recruitment (\(B_i\)) is distributed as a multinomial with parameters \(N, pent_0, \dots, pent_{K-1}\).
All animals present on the 1st sampling period are “new” with respect to sampling such that the following are true.
\[B_0 = N_1\]
\[N = \sum_{i=0}^{K-1}B_i\]
Here, some key aspects of the example presented on pages 12-13 through 12-25 are provided. The example regards data collected on coho salmon (Oncorhynchus kisutch) spawning in the Chase River of British Columbia using electrofishing gear. The experiment took place over a 10-week period in 1989; data from weeks 1 and 2, and from weeks 9 and 10 were pooled and labeled as weeks 1.5 and 9.5. The super-population here is the total number of adult salmon that returned to spawn during the 10 weeks. As with CJS modeling, if you have full, time-varying parameters, some of the parameters are confounded and non-identifiable.