A Unified Capture-Recapture Model

Matthew R. Schofield∗†and Richard J. Barker‡

November 22, 2006

∗The author would like to thank the Tertiary Education Commission

for the Bright Futures Ph.D. scholarship that funded this research.

†Department of Mathematics and Statistics, University of Otago, P.O.

Box 56, Dunedin, New Zealand

‡Department of Mathematics and Statistics, University of Otago, P.O.

Box 56, Dunedin, New Zealand

1

1 Summary

A hierarchical framework is developed for capture-recapture data

that separates the capture process from the demographic pro-

cesses of interest, such as birth and survival. This allows users to

parameterize in terms of meaningful demographic parameters.

The model is very flexible with many of the current capture-

recapture models shown to be special cases. The hierarchical

nature of the model allows natural expression of relationships,

both between parameters and between parameters and the re-

alization of random variables, such as population size. Previ-

ously, many of these relationships, such as density dependence

have been unable to be explored using capture-recapture data.

We fit a density dependent model to male Gonodontis bidentata

data and report evidence of negative density dependence in per-

capita birth rates and weak evidence of negative density depen-

dence in survival. Demographic analysis; Density dependence;

Hierarchical analysis; Missing data; Open population estimation

2

2 Introduction

The uses and extensions of open population capture-recapture

modeling have been many and varied since the foundational pa-

pers of Darroch (1959), Cormack (1964), Jolly (1965), and Seber

(1965). The last 20 years in particular have seen a proliferation

of mark-recapture models with more than 100 distinct models

now included in the mark-recapture software MARK (White

and Burnham 1999). The choice of model is governed by fea-

tures of the sampling process and the parameters that are of

interest. Some models are based on more informative study

designs than others, for example the robust design (Pollock

1982). Other models include covariates that allow missing and

possibly uncertain components, for example multi-state models

(Brownie et al. 1993; Schwarz et al. 1993b), multi-event mod-

els (Pradel 2005) and time-varying continuous-covariate models

(Bonner and Schwarz 2006). Further models incorporate addi-

tional information about the parameters in the model, for ex-

ample re-sighting models (Burnham 1993; Barker 1997). Other

models include reparameterizations that are more meaningful

for biological application, for example Pradel (1996) and Link

3

and Barker (2005).

Computer software packages such as MARK, or M-SURGE

(Choquet et al. 2004) for multi-state models, are very good at

allowing classical analyses based on fitting fixed-effects mod-

els using maximum likelihood. These packages also use GLiM-

type structures to allow constraints on parameters. Where these

packages are weak is in allowing users to fit hierarchical mod-

els including those that express stochastic relationships among

parameters in the model.

Mark-recapture models are naturally hierarchical in the sense

that biologists commonly model demographic parameters such

as survival rate, birth rate, population size as population vari-

ables regulated by probability distributions. Moreover, it is nat-

ural to expect relationships among parameters. For example,

the important concept of density dependence implies that pop-

ulation vital rates depend on population abundance (or density).

As noted by Armstrong et al. (2005): “There is evidence for den-

sity dependence in a wide range of species... but most studies

can be challenged on statistical grounds”. Typically methods

are used where the data is used twice; once to estimate abun-

dance and then again to use the abundance estimate in a density

4

dependent relationship. As pointed out by Seber and Schwarz

(2002): “Tools to investigate the whole issue of density depen-

dence and dependence upon the actions of other individuals are

not yet readily available [for capture-recapture data]. Models

that estimate abundance (e.g., Jolly-Seber models) are avail-

able, but the feedback loop between abundance and subsequent

parameters has not yet been complete”.

There has been relatively little development of hierarchical

models, especially those that allow flexibility in the way hier-

archical relationships are expressed. Link and Barker (2005)

proposed a modification to the likelihood of Pradel (1996) that

allowed stochastic dependence between survival and per capita

birth rates. While a step toward flexible hierarchical model-

ing, their likelihood does not allow for relationships to be ex-

pressed in terms of abundance, thus limiting its use for exploring

density-dependent relationships. The lack of a suitable choice

of likelihood has been an impediment to flexible hierarchical

modeling (Barker and White 2004). Parameterizations that are

convenient for likelihood-based estimation are not necessarily

the best to adopt for exploring biological relationships. Also,

some constraints of interest, such as density-dependence, can-

5

not be written in terms of deterministic functions of parameters

that are explicitly expressed in the likelihood.

Mark-recapture models can be naturally thought of as mod-

els for missing data subject to informative censoring. In open

population studies, the time that an animal first entered the

population is often of interest but all that is known is that it

entered the population sometime before the first capture. Sim-

ilarly, the time of death of an individual may be of interest but

all we know is that if it died during the study, it was sometime

after the last time it was caught. The data are also usually

interval-censored as in most designs the population is sampled

at discrete times.

The standard approach to mark-recapture modeling is to

find an observed data likelihood (ODL) expressed in terms of

parameters of interest after first writing a complete data likeli-

hood (CDL). Unobserved terms are then either integrated out

of the model or left in as parameters to be estimated. This step

is needed so that a likelihood function is obtained for parameter

estimation. However, Bayesian inference methods, in particular

McMC allow easy imputing of the unknowns to give the CDL.

Working with the CDL allows modeling to be focused on ob-

6

taining meaningful biological models instead of concentrating

on the intricacies of capture-recapture study design. An addi-

tional advantage of models based on direct use of the CDL is

that it increases the range of parameter constraints that may be

considered, including stochastic constraints.

In this paper we describe a missing data model that exploits

Bayesian multiple imputation to allow any demographic param-

eter to be explicitly incorporated in the model. Our model is

based on an individual-specific factorization that allows rela-

tionships among parameters and also between parameters and

the outcome of random variables, such as population size. A

further advantage of our approach is that it provides a single

unified modeling framework that includes virtually all of the

standard models as special cases.

Our factorization is similar to that of Dupuis (1995), who

imputed the missing data values for a multi-state model con-

ditional on first capture, where death was considered a state.

However, we have separated death from the multi-state model

and extended the model to include birth.

7

3 Model

The observed data from the t-sample capture-recapture study

is the matrix Xobs, where Xobsij = 1 if individual i was caught

in sample j and Xobsij = 0 otherwise. We define the number

of individual ever available for capture as N , like Crosbie and

Manly (1985) and Schwarz and Arnason (1996). Given N , the

capture histories for the unseen individuals, denoted Xmis are

known. This gives the complete matrix of capture-recapture

histories for all individuals in N , denoted X.

To model the demographic changes in the population due to

birth and death we require the interval censored times of birth

and death for each individual. These are expressed through

partially observed birth and death matrices, b (for births) and

d (for deaths). The value bij = 1 means that individual i was

born between sample j and j + 1, with bij = 0 otherwise (note

that bi0 = 1 means that i was born before the study started).

The value dij = 1 means that individual i died between sample

j and j + 1, with dij = 0 otherwise (note that dit = 1 means

that i was still alive at the end of the study). Individuals must

be born before they can die and can only be born and die once,

8

leading to the constraints,

k∑

j=0

bij−

k∑

j=1

dij ≥ 0,∀i, k,∑

j

bij = 1,∀i,∑

j

dij = 1,∀i.

As we are uncertain whether values of Xij = 0 prior to first

capture and after last capture are because individual i was not

able to be caught in sample j, or because i was not alive at

time of sample j the b and d matrices comprise observed values,

denoted bobs or dobs and missing values, denoted bmis or dmis.

We assume no error in the capture histories, so all values of b

after first capture are observed as bij = 0 and all values of d

before the final capture are observed as dij = 0. Consider the

capture history 0110 for a t = 4 period study. As the individual

had to be born before sample 2, the birth matrix b comprises

an observed component, bobs = (bi2 = 0, bi3 = 0) and a missing

component bmis = (bi0, bi1). As the individual could not have

died before sample 3, the death matrix d comprises dobs = (di1 =

0, di2 = 0) and dmis = (di3, di4). The complete b and d matrices

allow us to model the demographic processes of interest directly.

The missing data mechanisms for b and d are modeled through

X.

The complete b and d matrices allow us to obtain demo-

9

graphic summaries of interest, such as the number of individuals

in the population at time of sample j, denoted Nj,

Nj =N

∑

i=1

(

j−1∑

k=0

bik −

j−1∑

k=1

dik

)

.

Note the difference between N and Nj. The parameter N is the

total number of individuals ever available for capture during the

study. It is a nuisance parameter used to specify the model fully

and to include realizations of random variables, such as Nj in

the model.

Other demographic summaries, such as the number of births

between sample j and j + 1, denoted Bj, and the number of

deaths between sample j and j + 1, denoted Dj, can also be

found directly from the b and d matrices.

We introduce the notation bj, dj and Xj to denote the jth

column of the matrices b, d and X respectively, and the notation

b0:j and d1:j to denote the columns 0 through j for the b matrix

and columns 1 through j in the d matrix respectively.

3.1 Modeling the Capture Process

For a multiple recapture study, the complete capture matrix

is assumed to be the outcome from a series of independent

10

Bernoulli trials. While alive, each individual is assumed to be

caught in sample j with probability pj,

[X|b, d, s, p,N, z] ∝N !

u.!(N − u.)!

N∏

i=1

ti2∏

j=ti1

pXij

j (1 − pj)(1−Xij) .

where u. =∑

j uj is the total number of observed individuals,

ti1 is the sample individual i was first available for capture and

ti2 is the last sample that individual i was available for capture.

For example, if the individual was born between sample j and

j + 1 and died between sample k and k + 1 then ti1 = j + 1 and

ti2 = k. The notation [y|φ] is used to denote the probability

distribution or probability density function of y conditional on

φ.

3.2 Modeling the Deaths

Conditional on individual i being alive at time of sample j, death

between samples j and j + 1 is assumed to be the outcome of a

Bernoulli trial. The probability of death is 1 − Sj, where Sj is

defined as the survival probability from period j to j + 1. The

full conditional distribution used in the model is,

[dij|b0:j, d1:j−1, Sj, N ] ∝ S(1−dij)j (1 − Sj)

dij .

11

Note that we assume that an individual cannot die in the

same period that it was born. Incorporating different types of

data or assumptions can relax this assumption, for example,

Crosbie and Manly (1985); Schwarz et al. (1993a); Schwarz and

Arnason (1996) introduce assumptions to allow individuals to

die before they are available for capture.

3.3 Modeling the Births

Many of the current birth parameterizations are “hybrid” in na-

ture, combining aspects of the study with the birth process. For

example, the Jolly-Seber model parameterizes in terms of {Uj},

the total number of unmarked individuals in the population in

sample j, which reflects the intensity with which sampling is

carried out as well as the birth process. The models of Crosbie

and Manly (1985) and Schwarz and Arnason (1996) parameter-

ize birth in terms of {βj}, the probability of being born between

sample j and j + 1 conditional on ever being available for cap-

ture. The βj parameters reflect aspects of the study design as

well as the birth process. Consider a t = 3 study with N = 75

and parameters β0 = 1/3, β1 = 1/3 and β2 = 1/3. Extending

12

the study to a t = 4 periods with N = 100 gives different pa-

rameters β0 = 1/4, β1 = 1/4 and β2 = 1/4. A more natural

parameterization is in terms of per capita birth rates,

ηj = E[Bj|N ]/Nj.

This approach is adopted by Pradel (1996) and Link and Barker

(2005). However, as with the Jolly-Seber model and the formu-

lations of Burnham (1991) and Schwarz and Arnason (1996)

quantities such as Bj and Nj are not explicitly included in the

model. To overcome this, Pradel (1996) and Link and Barker

(2005) replace Nj with E(Nj|N). However, as Nj is explicit in

our model, we can use this to obtain a per-capita birth rate in

terms of Nj.

We assume that the observed birth matrix is the outcome of

a series of individual multinomial trials. Writing this in terms

of a series of binomial trials gives

[bij|b0:j−1, d1:j−1, β,N ] ∝ β′

j

bij(1−β′

j)(1−

∑jk=0

bik), j = 0, . . . , t−2

where β′

j = βj/∏j−1

k=0(1 − β′

k), β′

0 = β0 and βj is the multino-

mial probability of birth used by Crosbie and Manly (1985) and

Schwarz and Arnason (1996). There is no combinatoric term

13

because the arbitrary ordering of the data has already been ac-

counted for in the modeling of the capture histories.

We propose re-parameterizing and modeling in terms of the

parameter β0 and the per-capita birth rate

ηj = E(Bj|N)/Nj, j = 1, . . . , t − 2,

using the transformation ηj = βjN/Nj, where Nj = f(b0:j−1, d1:j−1).

Note that the parameter ηj−1 (or βj−1) is obtained through the

constraint∑

k βk = 1.

3.4 Posterior

For Bayesian inference we require the posterior distribution,

[p, S, η, bmis, dmis, N |Xobs].

This is proportional to the complete data likelihood

[X, b, d|p, S, η,N ],

which can be factored using the rules of conditional probability

to obtain the series of conditional distributions shown in sections

3.1, 3.2 and 3.3. The model can also be represented as a directed

acyclic graph (figure 1).

14

For the model where we assume probability of capture, sur-

vival and per-capita birth rates are all period specific fixed ef-

fects, denoted p(t)S(t)η(t), we are able to choose prior distri-

butions and re-parameterize so that we obtain full conditional

distributions of known form for all parameters except N , for

details see appendix A.

4 Covariates

Fully observed covariates, z, with associated parameters, θz,

that provide information about parameter(s) of interest can be

included in the usual way. To include partially observed covari-

ates, we need to model the covariate z and impute any missing

values in z every iteration, that is, the missing data gets treated

as another unknown updated with the parameters. Examples

of partially observed covariates that can be included in this way

are given below.

15

4.1 Categorical Individual-Specific Time-Varying

Covariates

Categorical individual-specific time-varying covariates are com-

monly collected in capture-recapture studies. For example, one

could assume that the breeding status of an individual affects

its survival probability, however, these covariates can only be

known when the individual is observed and are usually missing

when the individual is not observed. Such data motivated the

multi-state model (Schwarz et al. 1993b) which assumes that the

“state” occupied in sample j only depends on the state occupied

in sample j − 1, that is,

[zij = k|zij−1 = h] = ψhk, j = 1, . . . , t

with the constraint that∑

k ψhk = 1,∀k. We also model the

initial allocation to “state” as

[zij = h|bij−1 = 1] = πjh, j = 1, . . . , t

with the constraint∑

h πjh = 1, where h denotes the “state”.

This model can be extended to allow the “state” occupied

in sample j to depend on the “states” occupied in both sample

j − 1 and j − 2 (Brownie et al. 1993).

16

4.1.1 Multi-Event

A useful recent development is the multi-event framework that

allows for categorical covariates to be uncertain as well as par-

tially observed (Pradel 2005). The framework has a “state”

covariate of interest that we denote z1 and an “event” covariate

z2 that provides information about z1. This can be included

into our framework by having z1 modeled in terms of z2 and θz2.

The covariate z1 together with the parameters θz1then provide

information about the parameter(s) of interest.

4.1.2 Movement

A commonly used categorical covariate is availability for cap-

ture, where individual i in sample j is either available for cap-

ture (zij = 1) or unavailable for capture (zij = 2). This is an

example where one value of the covariate is never observed be-

cause no individual can be caught while unavailable for capture.

In the first sample after birth, we model the value of the co-

variate for individual i as the outcome of a Bernoulli trial with

probability πj = [zij = 1|bij−1 = 1]. For the complementary

allocation zij = 2, the probability is 1 − πj. Three common

17

assumptions about subsequent movement are first order Marko-

vian emigration, random emigration and permanent emigration

(Barker 1997). First order Markovian emigration is when move-

ment between the time of sample j and j + 1 depends only on

the covariate for individual i at time of sample j. The transition

matrix Ψj for Markovian emigration is,

Ψj =

Fj 1 − Fj

F ′

j 1 − F ′

j

,

where

Fj = probability that individual with zij = 1 has zij+1 = 1.

F ′

j = probability that individual with zij = 2 has zij+1 = 1.

Under random emigration the movement probability does

not depend on the previous value of the covariate, that is, F ′

j =

Fj. Under permanent emigration, once an individual becomes

unavailable for capture, it can never be available again, that is

F ′

j = 0.

In the presence of movement, we can model as if there were

no movement under two assumptions: (i) There is permanent

emigration with the times of birth and immigration combined to

give additions to the population and times of death and emigra-

tion combined to give deletions to the population. This results

18

in survival probabilities becoming deletion rates and birth rates

becoming addition rates. (ii) There is random emigration where

the initial allocation rate are the same as subsequent movement

rates, that is, πj = Fj. This results in the probability of cap-

ture becoming joint probabilities of capture and availability for

capture. For more information on movement assumptions under

permanent, Markovian and random emigration see appendix B.

4.2 Continuous Individual-Specific Time-Varying

Covariates

Continuous individual-specific time-varying covariates, for ex-

ample, individual length or weight can be included in the same

way as the partially observed categorical covariate, except the

model for z is continuous (Bonner and Schwarz 2006; Schofield

and Barker 2006). Note that survival and probability of capture

become individual specific due to the effect of the continuous co-

variate.

19

5 Density Dependence

An important feature of the hierarchical framework is the abil-

ity to model relationships between parameters. For example,

one could believe that parameters are drawn from a common

distribution, that is, a random effect. Specifying multivariate

distributions allow parameters to be related to each other, as

in Link and Barker (2005) where survival and per-capita birth

rates are correlated. An important feature of our model is that

parameters can also depend not only on other parameters, but

on the realization of the random variables b and d prior to the

current period. For example, the survival and birth rates for

the next period could be related to the current population size,

that is, density dependence.

5.1 Example: Gonodontis bidentata with Den-

sity Dependence

The data used are of male Gonodontis bidentata, a dataset pre-

viously used by Bishop et al. (1978), Crosbie (1979), Crosbie

and Manly (1985) and Link and Barker (2005). The data is

available from Bishop et al. (1978) and consists of u. = 689

20

unique individuals tagged over 17 periods.

To incorporate density dependence, we assume that

logit(Sj) ∼ N(γ0 + γ1Nj, τS), j = 1, . . . , t − 1

log(ηj) ∼ N(α0 + α1Nj, τη), j = 1, . . . , t − 2

where

Nj ≡ log(Nj) − 5.5

which is centered to reduce the sampling correlation between

parameters. We assume that that the probability of capture is

sample dependent and that either (i) there is no movement, or

(ii) there is permanent emigration, or (iii) there is random em-

igration. All three assumptions require no movement covariate

in the model and have the same algebraic structure. However,

each assumption gives a different interpretation of parameters,

see appendix B.

We use a Gibbs sampler to update all of the unknowns in

the model. We are able to sample all parameters from their full

conditional distributions directly except Sj, β0, ηj and N , which

we do with the Metropolis-Hasting algorithm or extensions of

it, see appendix C.

After an adaptive phase of 20,000 iterations and a burn in of

21

100,000 iterations a posterior sample of size 400,000 was drawn.

We used flat Beta(1, 1) priors for pj, vague N(0, 0.0001) priors

for α0, α1, γ0 and γ1, vague G(0.001, 0.001) priors for τη and τS

and a flat discrete uniform prior for N with a lower bound of u.

and an upper bound of 200, 000. To confirm that the model had

mixed suitably, multiple chains were fitted with over-dispersed

starting values and checked with the Gelman-Rubin converge

diagnostic (Gelman and Rubin 1992). The posteriors of par-

ticular interest are those on α1 and γ1, the density dependent

parameters (figure 2). The 95% central credible interval for α1

is (−1.49,−0.11) which excludes 0 suggesting that per-capita

birth rates are negatively associated with population size. The

95% central credible interval for γ1 is (−1.87, 0.37) which in-

cludes 0, with approximately 82% of the posterior mass below

0. This suggests a negative relationship between survival and

population size, but is far from convincing. These results in-

dicate that the population size is stable, at least in regards to

birth; when the population size becomes large or small, birth

rates adjust so that the population returns to somewhere near

equilibrium.

The analysis of Link and Barker (2005) suggested a positive

22

correlation between the logit of survival and the log of the per-

capita birth rates. Our evidence of density dependence raises

the possibility that the correlation identified by Link and Barker

(2005) was one induced by survival and birth rates both being

negatively density dependent.

6 Discussion

The hierarchical framework we describe offers a unified approach

to modeling capture-recapture data. It allows the investigation

of biologically interesting relationships among parameters, as

well as between parameters and external covariates. The re-

liance of most studies on classical inference, particularly maxi-

mum likelihood estimation, has meant that the machinery avail-

able to allow this sort of analysis has been limited. The devel-

opment of posterior simulation methods for Bayesian inference,

and McMC in particular, has been an important advance in this

regard.

We have shown that it is conceptually easy to incorporate

into our framework standard capture-recapture models as well

as models that incorporate partially observed covariates. How-

23

ever, many different models are easily included in this frame-

work, including models with different study designs and addi-

tional data.

An example of a model with a different design is the robust

design (Pollock 1982), that consists of a series of closed popula-

tion samples within each open population sample. This allows

the probability of capture and sample size to be primarily es-

timated from the closed population samples, with survival and

birth parameters estimated from the open population samples.

To fit the robust design, Sj must be constrained to 1 and ηj

(or βj) must be constrained to 0 between the closed population

samples. Assumptions about the probability of capture within

the closed population periods are expressed through the capture

model specified for the basic case in section 3.1.

Other study designs can introduce errors in the capture his-

tory, for example, genetic tagging using non-invasive methods

(Lukacs and Burnham 2005). The observed capture matrix, de-

noted X2, together with corruption parameters θX provide in-

formation about the true capture matrix X1. At each iteration

values for X1 are imputed and used in the sampling model.

Models that have additional data can also be incorporated.

24

For example, if a census is undertaken that counts the total

population size at time of sample j, denoted Cj, we can use this

to get a better estimate of population size Nj. A reasonable

model could be that the counts are distributed as a negative

binomial with mean Nj and over-dispersion parameter θC , Cj ∼

NB(Nj, θC), j = 1, . . . , t.

Another source of data could be re-sightings of the tags out-

side the study periods (Burnham 1993; Barker 1997). This pro-

vides additional information on survival and probability of cap-

ture, see Schofield and Barker (2006) for a description of fitting

this model in the framework.

Even though including these models is conceptually easy,

the computational implementation of these models is far more

difficult. An attractive feature of the hierarchical framework is

that the models can be thought of as products of conditional

distributions. For example, the p(t)S(t)η(t) model was sepa-

rated into the conditional distributions for capture, birth and

death as shown in sections 3.1, 3.2 and 3.3. More complex mod-

els, for example, density dependence, multi-event or a model

incorporating census data have the p(t)S(t)η(t) core with ad-

ditional distributions that account for the extension, whether

25

covariates or additional data sources, etc. This allows condi-

tional distributions to be specified and pieced together to form

a user-defined model, an important step toward the concept of

the “mother-of-all-models” (Barker and White 2004). However,

many of these conditional distributions are complex and allow-

ing full flexibility in model specification comes as a trade-off

against computational speed. With the ever increasing advances

in computational power and the prospect of biologists being able

to fit models with meaningful parameters and relationships of

interest, we believe that this area is one of great promise.

The use of density dependent modeling illustrates the po-

tential of our analysis and also reminds us of the need to collect

high quality data to achieve reasonable mixing. The quality of

data can be improved either by more intense sampling on each

occasion, or by increasing the number of sampling occasions, or

by using a better design, such as the robust design of Pollock

(1982). Data collected during periods when the population is

closed provide high quality information on pj and Nj, and data

collected between times when the population is open provide in-

formation on the survival and birth parameters, Sj and ηj. The

two components of the density dependence relationship, for ex-

26

ample ηj and Nj, are informed primarily by different subsets of

data. In contrast, with simple multiple recapture studies, all

parameters are estimated from the same information.

Link and Barker (2005) showed that including N is unnec-

essary for the estimation of identifiable parameters that appear

explicitly in the likelihood. However, we include the parame-

ter N in the model so that we can use demographic summaries

such as Nj in the presence of individual specific parameters, even

though there is very little, if any information about N in the

data. This extends Huggins (1989) to open population models,

giving posterior predictions of Nj when there are individual-

specific covariates affecting the parameters. It should be noted

however, that it is possible to include demographic summaries

such as Nj in the model when there are no individual specific

parameters.

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33

7 Figures

Figure 1: Directed acyclic graph of the jth period for the

p(t)S(t)η(t) model. The bold arrows represent deterministic

links and the plain arrows represent stochastic links.

34

−3 −2 −1 0 1 20

0.2

0.4

0.6

0.8

1

γ1

−3 −2 −1 0 1 20

0.5

1

1.5

α1

Figure 2: Posterior density estimates for the density depen-

dent effects on survival (γ1) and per-capita birth rates (α1) for

Gonodontis bidentata.

35

A Appendix: Computation of p(t)S(t)η(t)

Model

The set of parameters {pj} have Beta(αp, γp) priors that yield

Beta(nj +αp, Nj −nj + γp) full conditional distributions, where

nj is the total number of individuals caught at time of sample

j. The set of parameters {Sj} have Beta(αS, γS) priors that

yield Beta(Nj −Dj +αS, Dj +γS) full conditional distributions.

To obtain updates from β0 and {ηj}t−2j=1, we re-parameterize in

terms of β′

j, where

β′

j = βj/

j−1∏

k=0

(1 − β′

k), j = 1, . . . , t − 2

and β′

0 = β0. The set of parameters {β′

j} have Beta(αβ′ , γβ′) pri-

ors that yield Beta(Bj +αβ′ , N−∑j

k=0 Bk +γβ′) full conditional

distributions which are then transformed to {ηj} by taking

β′

j

j−1∏

k=0

(1 − β′

k)N/Nj.

The times of birth/death are obtained by calculating the full

conditional probability of each plausible period of birth/death

for each individual at every iteration. A period of birth/death is

then sampled using these probabilities. The parameter N has a

discrete uniform distribution and is updated using a reversible

36

jump algorithm (Green 1995). A new candidate value for N

is proposed along with associated values of b, d and Xmis and

the group are accepted or rejected together. The chain is run

for a fixed number of iterations to obtain an optimal jumping

distribution.

B Appendix: Discussion on Movement

Assumptions

B.1 Permanent Emigration

Permanent emigration is assumed in most models that include

first captures, (Jolly 1965; Seber 1965; Schwarz and Arnason

1996; Link and Barker 2005). However, when averaging across

the various combinations for b, d and z for each capture his-

tory, the parameters Sj and Fj are confounded and there is not

enough information to separately estimate birth, movement and

survival parameters prior to the first capture.

The standard approach is to consider additions and deletions

instead of births and deaths. Additions combine individuals be-

ing born available for capture with immigrants becoming avail-

37

able for capture. Deletions combine deaths of those available

for capture with emigrants leaving to become unavailable for

capture. The random variables b and d are used to express time

of addition and deletion, instead of birth of death respectively.

This means the covariate z is no longer required because every

individual is available for capture from the sample of addition

until the sample of deletion, when it either leaves or dies before

the next sample. The meaning of the parameters changes so that

ηj becomes the per capita addition rate and 1−Sj becomes the

deletion rate.

B.2 First order Markovian Emigration

Without strong assumptions, first order Markovian emigration

is not identifiable unless more complex study designs are used,

such as the robust design, or models incorporating different

types of re-encounter data. Even with these designs, additional

constraints about the time-specific covariate parameters, Fj and

F ′

j are required. A common constraint is that movement param-

eters are fixed through time, Fj = F and F ′

j = F ′ (Barker et al.

2004). A further problem is that there is not enough informa-

38

tion to separate the per capita birth rates, ηj, and the allocation

probabilities πj. One possible solution is to gather additional

information that can be used to separate ηj and πj. Another is

to assume that all individuals are born unavailable for capture,

that is, specify a distribution for πj with all mass on 0. This as-

sumption can be relaxed by having some mass on πj > 0, allow-

ing some individuals to be born available for capture. Another

potential solution combines individuals being born available for

capture with immigrants becoming available for capture for the

first time, with b parameterizing addition. The advantage of this

is that assumptions are no longer required for the initial alloca-

tions, πj, however, addition rates are being estimated instead of

birth rates, which are of biological interest.

B.3 Random Emigration

Under random emigration Fj is confounded with pj+1. The stan-

dard approach is to consider the identifiable parameter p′j+1 =

Fjpj+1, the joint probability of being available for capture and

caught in sample j + 1. Including the first captures means that

πj is also confounded with pj+1. As with Markovian emigra-

39

tion, one possible solution is to specify a distribution for πj with

all/most mass on 0. Another possible solution is to assume that

initial allocations are the same as subsequent movement proba-

bilities, that is, πj = Fj, (Barker 1997). Under this assumption

the algebraic structure for the model is identical to that of per-

manent emigration with additions and deletions.

C Appendix: Computation of Den-

sity Dependent Model

All parameters updated using Metropolis-Hastings or reversible

jump McMC have an adaptive phase of 20,000 iterations to find

an optimal jumping distribution. We update Sj and β0 using

a Metropolis-Hastings algorithm with a flat beta distribution

for β0 (note that Sj has a hierarchical “prior” distribution). To

update ηj, we change variable for computational reasons and

update βj and use a Metropolis-Hastings algorithm with the

hierarchical “prior” distribution. The parameter N is updated

using a reversible jump algorithm in the same way as in section

A. The parameters {pj} have a Beta(αp, γp) prior that yields a

40

Beta(nj + αp, Nj − nj + γp) full conditional distribution. The

parameter α0 has a N(0, τα0) prior distribution that yields a

N(τη

∑t−2k=1(log(ηk) − α1Nk)/τ, τ) posterior distribution, where

τ = τη(t−2)+τα0. The parameter α1 has a N(0, τα1

) prior distri-

bution that yields a N(τη

∑t−2k=1((N)k(log(ηk)−α0))/τ, τ) poste-

rior distribution, where τ = τη

∑t−2k=1 N2

k +τα1. The parameter γ0

has a N(0, τγ0) prior distribution that yields a N(τS

∑t−1k=1(logit(Sk)−

γ1Nk)/τ, τ) posterior distribution, where τ = τS(t − 1) + τγ0.

The parameter γ1 has a N(0, τγ1) prior distribution that yields

a N(τS

∑t−1k=1((N)k(logit(Sk) − γ0))/τ, τ) posterior distribution,

where τ = τS

∑t−1k=1 N2

k +τγ1. The parameter τS has a G(aτS

, bτS)

prior distribution that yields a G(aτS+(t−1)/2, bτS

+∑t−1

k=1(logit(Sk)−

(γ0 +γ1Nk))2/2) posterior distribution. The parameter τη has a

G(aτη, bτη

) prior distribution that yields a G(aτη+(t−2)/2, bτη

+

∑t−2k=1(log(ηk) − (α0 + α1Nk))

2/2) posterior distribution. The

times of birth/death are obtained by calculating the full con-

ditional probability of each plausible period of birth/death for

each individual at every iteration. A period of birth/death is

then sampled using these probabilities.

41