Recruitment Parameters in Jolly-Seber Models

The Jolly-Seber models estimate the recruitment rate into the population, as well at the apparent survival rate, phi, and the probability of capture given that the animal is available for capture (p). Because the parameterization of the Jolly-Seber model is not unique, multiple parameterizations have been developed and are provided in MARK.

Pradel Models

Pradel's (1996) original paper defined the parameter gamma(i) as the probability that an animal at time i had not entered the population between time i and i-1. In terms of Jolly's original model, gamma(i+1) = 1 - B(i)/N(i+1).

Lambda is the rate of change of the population, so lambda(i) = N(i+1)/N(i).

f is the fecundity rate of the population, so that f(i) is the number of new animals in the population at time i per animal in the population at time i-1, or N(i+1) = N(i) f(i) + N(i) phi(i) .

The following table provides the relationships between these 3 parameters.

gamma(i+1) = phi(i)/[f(i) + phi(i)]

gamma(i+1) = phi(i)/lambda(i)

lambda(i) = phi(i)/gamma(i+1)

lambda(i) = f(i) + phi(i)

f(i) = lambda(i) - phi(i)

f(i) = phi(i) { [ 1 - gamma(i+1) ] / gamma(i+1) }

Because all of the Pradel models are based on the same likelihood, the AIC values of these models are comparable. However, none of the other models in the set of models discussed here, except the Link-Barker data type,have the same likelihood, so AIC values are not comparable.

Individual covariates can be used to model phi and p in the Pradel models. However, the biological meaning of modeling lambda as a function of an individual covariate is not clear. Intuitively, it makes more sense to model f and gamma as functions of individual covariates, even though these parameters can be combined with phi to provide a derived estimate of lambda.

Burnham's Jolly-Seber Model

Burnham developed a parameterization of the Jolly-Seber model that provides estimates of the rate of population change (lambda) and the population size on the first trapping occasion (N(1)). However, the likelihood of this model has been consistently difficult to optimize because of the penalty constraints required to keep the parameters consistent with each other. Thus, its use is not suggested.

POPAN

Schwarz and Arnason (1996) parameterized the Jolly-Seber model in terms of a super population (N), and the probability of entry (pent in MARK, beta in the paper). The POPAN data type implements this model. The MLogit link function provides a constraint that makes the sum of the pent parameters <= 1, with the probability of occurring in the population on the first occasion as 1 - sum(pent(t)).

Link-Barker

Link and Barker (2003) reparameterized the POPAN model from the probability of entry (pent) to the recruitment parameter (f), and is available as the Link-Barker data type. The reason for this reparameterization was to provide a more biologically meaningful interpretation of the parameters of the model, as part of a hierarchical modeling approach. Both the Pradel recruitment and the Link-Barker model have the f parameter. However, these parameters are not exactly equivalent, although they generate identical estimates if there are not losses on capture, i.e., all animals captured are also released. One advantage of the Link-Barker model is that f is not affected by losses on capture, whereas the Pradel parameterization is affected by losses on capture. Thus, if you have losses on capture, then you should be using the Link-Barker model in MARK rather than the Pradel f model.