Joint live-recapture/live resight/tag-recovery model (Barker’s Model)
Joint live-recapture/live resight/tag-recovery model (Barker’s Model)
This model extends Burnham’s (1993) live/dead model to the case where live resightings are reported during the open period between live recapture occasions. The population is not assumed to be closed while resightings are being obtained and recoveries of tags from dead animals (dead resightings) may be reported. Because the resighting interval is open it is possible for an animal to be resighted alive several times within an interval then be reported dead. For these animals, only the last dead sighting is used in the model, the earlier live resightings in that period are ignored. Therefore the status of an animal on resighting (live or dead) is determined on the last occasion on which it was resighted in the open interval.
Although the model is complicated and involves 4 sets of nuisance parameters for the recapture and resighting/recovery process, the additional data from resightings and tag recoveries can lead to substantial gains in precision on survival rate estimates. Currently, the model does not allow estimation of abundance or recruitment. For a detailed description of the model see Barker (1997, 1999). Parameters in the model are:
S(i) = the probability an animal alive at i is alive at i + 1
p(i) = the probability an animal at risk of capture at i is captured at i
r(i) = the probability an animal that dies in i, i + 1 is found dead and the band reported
R(i) = the probability an animal that survives from i to i + 1 is resighted (alive) some time between i and i + 1.
R’(i) = the probability an animal that dies in i, i + 1 without being found dead is resighted alive in i, i + 1 before it died.
F(i) = the probability an animal at risk of capture at i is at risk of capture at i + 1
F’(i) = the probability an animal not at risk of capture at i is at risk of capture at i + 1 (NB – differs from the definition in Barker (1997)
The resighting parameterization used in MARK differs from that described by Barker (1997). An advantage of the parameterization used by MARK is that it enforces certain internal constraints that arise because the joint probability Pr(A and B) should always be less than or equal to the unconditional probabilities Pr(A) and Pr(B). For example, the MARK parameterization ensures that the probability an animal is resighted alive in i, i + 1 and survives from i to i + 1 is less than the probability it is resighted alive in i, i + 1. It also ensures that Pr(resighted alive and dies in i, i + 1 without being reported) < Pr(dies in i, i + 1 without being reported). These internal constraints are not enforced by the other parameterization.
Movement
Between trapping sessions, animals are permitted to leave the study area then return. If an animal is in the study area then it is considered “at risk of capture”. If it leaves the study area it is considered “not at risk of capture”. Animals that are at risk of capture at time i, leave the study area with probability (1 – F(i)). Thus F(i) has the same interpretation as in Burnham’s (1993) live-dead model as the fidelity to the study area.
Animals not at risk of capture are permitted to return to the study area with probability F’(i). In Barker (1997) F’(i) was the probability that an animal out of the study area at i remained out of the study area at i + 1, but the definition has been changed in the interest of having a parameterization in common with the robust design model. Under this parameterization there are 3 types of emigration:
Random F’(i) = F(i)
Permanent F’(i) = 0
Markov no constraint.
A complication is that in the random emigration model the parameters F(i) = F'(i) are confounded with the capture probability p(i + 1). By making the constraint F(i) = F'(i) = 1 in MARK the random emigration model is fitted, but now the interpretation of p(i) is the joint probability that an
animal is at risk of capture and is caught, F(i – 1)p(i).
Under Markov emigration there tends to be serious confounding of movement and capture probabilities. In a model with time-dependent capture probabilities, it is usually necessary to constrain F(i) = F(.) and F’(i) = F’(.) for all i. Even then, the Markov emigration model may perform poorly. In practice the parameters F and F’ are usually estimable only if the movement model is markedly different to the random emigration model, that is, if there is a large difference between F(i) and F’(i).
To illustrate the meaning of the emigration parameters, suppose the animal is captured during the first trapping session, not captured during the second trapping session, and then captured during the third trapping session. One of several encounter histories that would demonstrate this scenario would be:
100010
The probability of observing this encounter history can be broken into 4 factors:
P1 = Pr(animal survives from time 1 to time 3 | released at 1)
P2 = Pr(animal is not resighted between 1 and 3 | released at 1 and survives to 3)
P3 = Pr(animal is not captured at 2 but is captured at 3 | released at 1 and survives from 1 to 3 without being resighted)
P4 = Pr(encounter history after trapping period 3 | events up to trapping period 3)
For describing movement, the relevant factor is P3. An animal captured at time 1 is known to be at risk of capture at time 1. Because it was captured at time 3 we also know it was at risk of capture at time 3. There are two possible histories that underlie this observed history:
1. The animal was at risk of capture at time 2 and was not captured, but was captured at time 3
2. The animal left the study area between time 1 and 2 but then returned and was captured.
Because we do not know which one actually occurred we instead find the probability that it was either of the two, which is:
P3 = {(1 – F1)F’2 + F1(1 – p2)F2}p3
The complicated term in the brackets represents the probability that the animal was not captured during the second trapping session but is at risk of capture at time 3. The first product within the brackets (1 – F1)F’2 is the joint probability that the animal emigrated between the first 2 trapping sessions (with probability 1 – F1) and then immigrated back onto the study area during the interval between the second and third trapping sessions (with probability F’2). However, a second possibility exists for why the animal was not captured — it could have remained on the study area and not been captured. The term F1 represents the probability that it remained on the study area between time 1 and 2 and the term (1 – p2) is the probability that it was not captured at time 2. The final term F2 represents the probability that the animal remained on the study area so that it was available for capture during the third trapping session.
Encounter histories for this model are coded as LDLDLD… .Because animals can be encountered in this model as either alive or dead during the interval between capture occasions, 2 different codes are required in the encounter histories to provide information. A 1 in the D portion of an encounter history means that the animal was reported dead during the interval. A 2 in the D portion of an encounter history means that the animal was reported alive during the interval. A 1 in the L portion of an encounter history means that the animal was alive on the study area during a capture occasion.
The following are valid encounter histories for a 5-occasion example:
1010101002
Animal was captured on the first occasion, and recaptured again on the 2nd, 3rd, and 4th occasions. It was not captured on the 5th occasion, but was seen alive during the last interval.
0000120100
Animal was captured on the 3rd occasion, and seen alive during the 3rd interval. It was reported dead during the 4th interval.
Note that there can be multiple occasions with a 1 in the L columns, and multiple occasions with a 2 in the D columns, but only one D column can have a 1.