We will begin with a mark-recapture model for an open population, i.e., open to births, deaths, immigration, and emigration. Imagine sampling an open population in a brief window of time, e.g., 1-2 weeks, at yearly intervals. On each sampling occasion, one might be interested in estimating population size. One might also be interested in learning about additions to the population (recruitment of new individuals through processes like birth and immigration) as well as losses due to death or emigration. To begin, we will narrow our focus on survival and use a live-recaptures model that was first developed by Cormack, Jolly, and Seber (see readings for references).

In the mark-recapture (or sometimes termed capture-mark-recapture or capture-recapture) studies we’ll explore in this section, animals are captured or observed on \(K\) relatively short sampling occasions. On each occasion, each unmarked animal is given a unique mark (e.g., a numbered tag), or its unique natural markings are recorded (e.g., spotting pattern). On subsequent occasions, both marked and unmarked animals are caught; identities of marked animals are recorded, and unmarked animals are marked. Animals are released back into the population after each occasion (accidental deaths or losses on capture can be accommodated). For studies of annual survival, intervals would naturally be 1 year long such that the \(K\) occasions are 1 year apart. The intervals might also be shorter (e.g., months or seasons) or longer depending on the nature of the study and the study organism.

Across the \(K\) occasions, each individual’s capture history (or encounter history) is recorded. The capture history for an individual is a row vector of length \(K\). For each of the occasions, the vector contains a 1 if the individual was seen alive on the occasion and a 0 otherwise. In an 5-year study with capture efforts conducted once per year, an animal might have a history like 11001, which indicates that the individual was observed in years 1, 2, and 5 but not seen in years 3 or 4. This type of history shows that it is possible for animals to be alive but not detected in a given year. That is, this model type can accommodate imperfect detection.

Animals could be missed because they have died or because they have permanently emigrated from the study area. Thus, this type of study provides estimates of apparent survival (\(\phi\)) rather than true survival. The probability of detecting animals that are alive is termed \(p\). Because we are focused on survival, we condition on first capture, which means we start working with data for an animal only once it’s captured. Imagine an animal that isn’t observed until year 3 of a 5-year study: we don’t try to determine if it was present in years 1 or 2 or to learn about when it joined the population. We simply try to learn about it’s survival from year 3 to 4 and from 4 to 5.

To get started, we’ll first consider single-age models and the standard Cormack-Jolly-Seber model (CJS) in which \(\phi\) and \(p\) are time-specific.

\(\phi_i\) is the probability that a marked animal in the study population at sampling period \(i\) survives until period \(i+1\) and remains in the population (does not permanently emigrate). Thus, apparent survival in year \(i\) relates to the interval between resighting (or capture) periods \(i\) and \(i+1\).

\(p_i\) is the probability that a marked animal in the study population at sampling period i is captured or observed during period \(i\).

\(\chi_i\) is the probability that an animal alive and in the study population at sampling period \(i\) is not caught or observed again at any sampling period **after** period \(i\). For a study with \(T\) sampling periods, \(\chi_T = 1\), and values for periods with \(i<T\) can be obtained recursively (i.e., working backwards from \(i=T\) to \(i=1\)) as: \(\chi_i = (1-\phi_i) + \phi_i \cdot (1-p_{i+1})\cdot\chi_{i+1}\).

Notice that the equation for \(\chi_i\) expresses the fact that there are 2 ways for an animal to be missed: (1) it can fail to remain alive in the study area \((1-\phi_i)\) or (2) it can survive but not be captured again \(\phi_i \cdot (1-p_{i+1})\cdot\chi_{i+1}\).

The figure below shows the various pathways that an animal can follow for a 3-occasion CJS study and the capture histories that result. Notice that there are multiple ways to have a capture history = 100. Also, notice that there are 4 unique capture histories (all start with a 1 in the 1st year as they all entered the study that year and can then have a 1 or a 0 in year 2 and year 3). We do not observe animals with histories of 000 and so those aren’t included. We can mark new animals in year 2: those will all have histories that begin with 0 and that finish with a 0 or 1 in the 3rd year, so they will have capture histories of either 011 or 010. We might also capture new animals in year 3 but those will not help us learn about survival during the 3-year study.