a finite-state continuous-time approach for inferring regional migration and mortality rates from...

A Finite-State Continuous-Time Approach for Inferring Regional Migration and MortalityRates from Archival Tagging and Conventional Tag-Recovery ExperimentsAuthor(s): Timothy J. Miller and Per K. AndersenSource: Biometrics, Vol. 64, No. 4 (Dec., 2008), pp. 1196-1206Published by: International Biometric SocietyStable URL: http://www.jstor.org/stable/25502202 .

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Biometrics 64, 1196-1206 DOI: 10.1111/j.l541-0420.2008.00996.x December 2008

A Finite-State Continuous-Time Approach for Inferring Regional Migration and Mortality Rates from Archival Tagging

and Conventional Tag-Recovery Experiments

Timothy J. Miller

Large Pelagics Research Center, Zoology Department, University of New Hampshire,

Durham, New Hampshire 03824, U.S.A.

Current address: National Marine Fisheries Service, Northeast Fisheries Science Center, 166 Water Street, Woods Hole, Massachusetts 02543, U.S.A.

email: timothy.j [email protected]

and

Per K. Andersen

Department of Biostatistics, University of Copenhagen, DK-1014 Copenhagen, Denmark

Summary. Spatially structured population dynamics models are important management tools for har

vested, highly mobile species and although conventional tag recovery experiments remain useful for esti

mation of various demographic parameters of these models, archival tagging experiments are becoming an

important data source for analyzing migratory behavior of mobile marine species. We provide a likelihood

based approach for estimating the regional migration and mortality rate parameters intrinsic to these mod

els that may use information obtained from conventional tag recovery and archival tagging experiments.

Specifically, we assume that the regional location and survival of animals through time is a finite-state

continuous-time stochastic process. The stochastic process is the basis of probability models for observa

tions provided by the different types of tags. Results from application to simulated tagging experiments for western Atlantic bluefin tuna show that maximum likelihood estimators based on archival tagging ob

servations and corresponding confidence intervals perform similar to conventional tagging observations for

a given number of tag releases and releasing tags in each region can improve the behavior of maximum

likelihood estimators regardless of tag type. We provide an example application with Atlantic bluefin tuna

released with conventional tags in 1990-1992.

Key words: Finite-state continuous-time process; Implanted archival tag; Markov process; Pop-up satellite

archival tag; Spatially structured population; Tag-recovery.

1. Introduction

Tagging experiments have proven to be powerful tools to gain

demographic information from animal populations (Pollock,

1991; Schwarz and Seber, 1999; Seber and Schwarz, 2002). Statistical models are available to estimate a variety of param

eters including abundance (Darroch, 1958), mortality (Hearn,

Sundland, and Hampton, 1987), size-specific harvest (Taylor,

Walters, and Martell, 2005), and growth rates (Laslett, Eve

son, and Polacheck, 2002). For populations of migratory an

imals that may be subjected to different levels of mortality in various regions, migration rates among adjacent regions

have also been estimated through tagging experiments (e.g.,

Hilborn, 1990; Hestbeck, Nichols, and Malecki, 1991; Brownie

et al., 1993; Schwarz, Schweigert, and Arnason, 1993), but the

models usually require assumptions that all tagged animals

of a particular release group are released at the same time,

recaptures are made in discrete time, and migration occurs

only once and instantly between intervals. The assumption of

migration time in discrete time models is known to provide bi

ased estimation of migration rates when mortality rates vary

across regions (Hestbeck, 1995). Furthermore, many popula

tion dynamics models used to manage fisheries are parame

terized with instantaneous mortality rates (e.g., Quinn and

Deriso, 1999) and it would appear natural to consider migra

tion in the same continuous time framework.

Tag-recovery experiments continue to be used widely to in

fer rates of movement and mortality of fish populations, but

the relatively new pop-up satellite and implanted archival tags

provide extensive observations of movements of pelagic mi

gratory species (e.g., Block et al., 2001; Bonfil et al., 2005)

and allow inference of location continuously in time (e.g.,

Sibert, Musyl, and Brill, 2003; Sibert et al., 2006). Because

many species that are the subject of archival tagging studies

may have been or continue to be the subject of conventional

1196 ? 2008, The International Biometric Society



Finite-State Continuous-Time Tagging 1197

tag-recovery studies as well, a common estimation framework

that can combine the information provided by these studies

would be useful for their management.

We provide likelihoods that (1) provide a basis for esti

mating instantaneous migration and mortality rates, (2) al

low conventional and archival tagging studies to be combined,

and (3) are unencumbered by the unrealistic assumptions of

separation of migration and survival processes in time and

of tagged animals being released in groups at the same time

that are required for most existing models to estimate migra

tion from tag-recovery experiments. To assess the behavior of

maximum likelihood estimators (MLEs), we provide simula

tion results for experiments using different types of tags in a

population of animals that migrates between two regions and

experiences different levels of natural and harvest mortality in either region. We also provide an example application to

Atlantic bluefin tuna (Thunnus thynnus) conventional tagging

studies.

2. Methods

2.1 Finite-State Continuous-Time Markov Processes

Stochastic processes that describe the transition among a fi

nite number of states in continuous time are referred to by

various names including multistate processes or finite-state

continuous-time (FSCT) processes (Karlin and Taylor, 1975;

Taylor and Karlin, 1984; Andersen et al, 1993). We use FSCT

to emphasize the continuous-time aspect of these models and

distinguish them from the discrete-time multistate processes

more commonly used to model mark-recapture experiments

(e.g., Nichols and Kendall, 1995; Lebreton and Pradel, 2002). The fundamental characteristics of FSCT processes are the

instantaneous intensities of transition between states h and

i at time t,ah,i(t). The intensities are formally defined as

ahii(t) =

\im6t->QPh,i(t, t + 8t)/8t where 0 < ahil(t) < oo and

Ph,i(si *) =

p(Y(t) = * I Y(s)

= h) is the probability that the

process is in state Y(-) = i at time t > s given it is in state

Y(-) = h at time s (Andersen and Keiding, 2002). In general,

these transition intensities may change through time (non

homogeneous) and depend on the previous transition history

of the process. We consider here the FSCT Markov process

that is a class of FSCT processes where the instantaneous

transition rates dh,i(s) depend only on the state at time s

(Markovian) and are constant through time (homogeneous) or are piecewise constant through time.

The H x H infinitesimal matrix of homogeneous transi

tion intensities, A = {a^}, has diagonal elements, a,h,h

?

~Yli=i i^hCLh>i' ^ne vames7 ~ah,h-> are often termed hazards

or forces of transition out of the corresponding states (Hoem,

1971). These requirements imply that each row of A sums to

zero. For homogeneous FSCT Markov processes, the probabil

ity transition matrix P(s, t) =

{phji(s, t)} is the exponential of the infinitesimal matrix

P(s, t) = exp{A(t -

?)} = V {Mt ~

S)r , (1) z?' XI x=()

where t > s (e.g., Cox and Miller, 1965, pp. 178-186). The

infinitesimal matrix can be diagonalized with its eigenvalues and eigenvectors so that the probability transition matrix can

be calculated as P(s, t) = Vexp{D(?

- s)}V-1 where D

is a diagonal matrix of the eigenvalues and V is the matrix

of eigenvectors (Kalbfleisch and Lawless, 1985; Commenges,

2002).

2.2 Defining States for Animal Populations In general, an animal may at any instant survive or die from

one of several causes, but these outcomes may be further de

tailed. For example, in many studies we discretize the space

where a migratory animal may be found so that the resulting

regions correspond to habitats that the animals use for dif

ferent purposes (e.g., breeding and feeding grounds) or areas

used to manage harvest of the species (e.g., area-specific har

vest limits). Within a discrete-space system, an animal may,

at any instant, remain alive in a region, die from one of sev

eral causes in that region, or move to any of the adjacent

regions. If H = {1,..., H} is the set of states that an animal

may exhibit, let R, T, and M represent the subsets of states

where the animal is alive and in one of R regions, dead due to

harvest in one of the R regions, and dead due to other causes

(i.e., natural mortality) in one of the R regions, respectively.

We will use these subsets of states in formulating observation

models for the three different types of tagging experiments

below.

The FSCT framework we consider is a generalization of

simpler FSCT Markov processes assumed for many animal

populations. For example, it is common in many population

dynamics models for fish populations, where migration is not

considered, to assume animals either survive or die from nat

ural or harvest mortality sources during a time interval r.

This equates to a process under our framework where there

is a single region and two sources of mortality in that region.

Letting F and M represent the instantaneous rates of transi

tion (within t) into the harvest and natural mortality states,

respectively, the infinitesimal matrix for this process in the

defined time interval is

(-(F + M) F MX

\ ? 0 0/ where 0 is a 2 x 1 column vector. The probability transition

matrix for the interval that provides the probabilities of being in the living and two different mortality states at the end of

the interval is

P = exp(Ar)

=

(-(F+M)t F fi -(F+M)t^ M r, -(F+M)tA

F + MX f F + M1 s \ 0

0 0 1 /

(2) When N0 animals are alive at the start of the interval, the

expected numbers in each state (product of Nq and the prob abilities in the first row of equation (2)) provide results widely known as the Baranov catch equations (Chapman, 1961;

Ricker, 1975; Seber, 1982, p. 329). In fact, equation (2) is the stochastic analog of the Baranov equations and these proba bilities are general results for competing risks models with ex

ponentially distributed life spans (Dupont, 1983; Kalbfleisch and Prentice, 2002, pp. 247-248). Similarly, for a population

occupying two regions with migration rates fii2 and p2\ from



1198 Biometrics, December 2008

regions 1 to 2 and 2 to 1, respectively, and no mortality during the interval r, the infinitesimal matrix,

A(r)=(-"u "12) V M21 -A*2l/

yields the probability transition matrix,

P = exp(Ar)

=

/ fl21 + g12e-^^)r m c_(u?+u?)T]

\

I M12 + M21 M12 + M21 |

I /X21 u

_ e-(^^n)r}

Ml2+M2ie-^+^)- I '

\/Xl2 + /i21 ^12+^21 /

where the diagonal provides the probabilities of an animal

being present in each region at the end of the interval given

its presence in that region at the beginning of the interval.

These probabilities were also deduced using different means

by Whitehead (2001, equation (5)) for a model to estimate abundance and migration rates for sperm whales. Analytic so

lutions for P = Vexp{Dr}V_1

are readily obtained when the

set of states are small or when there are no transient states,

but obtaining these solutions analytically becomes too cum

bersome for more complex scenarios and numerical methods

may be more appropriate.

2.3 Probability Models for Tagging Experiments The probability models we present below for observations de

rived from conventional tags, pop-up satellite archival tags

(PSATs), and implanted archival tags (IATs) account for the data provided by the corresponding observations as func

tions of the instantaneous migration and mortality parame

ters. These probability models can also be combined to make

likelihood-based estimates of the instantaneous parameters

for studies that use multiple types of tags. 2.3.1 Pop-up satellite archival tags. When PSATs are de

ployed on animals, they collect measurements of various en

vironmental characteristics including light intensity until the

tag releases from the animal and floats to the surface (pops

up) where it relays information via satellite to the researcher.

The tagged animal may theoretically be harvested prior to

the pop-up time or the tag may pop-up when a tagged an

imal stays at a constant depth for specified amount of time

because the animal is dead due to natural mortality sources.

The relayed measurements of environmental characteristics

can in turn be used to estimate geographic location of the

tagged animal at a given time during the interval that the

PS AT was attached to the animal (e.g., Sibert et al., 2003).

When the ocean is compartmentalized into regions, the lo

cations at time translate into regional occupation at time and

times of migration between regions. Although the locations

are estimated rather than known, it may be safe to ignore the

resulting uncertainty in times of migration between regions

when the regions are large relative to the uncertainty in the

locations. Times of entry into states corresponding to harvest

mortality are directly observed and not subject to measure

ment error. Likewise, when entry into natural mortality states

are observed by pop-off of the tag due to constant depth, the

times of transition into these states may be assumed known.

Here we consider the migration times as known so that the

data on a tagged animal k observed from the time the animal

is tagged ?0,fc to the pop-up time ta,k are the times of tran

sition, ti?fc =

?1)fc,..., tijk,..., tnk,k '

to,k <

ti,k <

ta,k, and the

states, Yi,fc =

Y(thk),..., Y(tz>fc),..., Y(tnktk) : Y(tlik) G H,

the animal takes on during the period the tag is operating,

Tk = ta,k

? ^o,fc- As these are data for a continuous-time ob

servation of a FSCT process (Andersen et al., 1993; Andersen

and Keiding, 2002; Commenges, 2002), the probability func tion for the set of observations from a PSAT is

Pi,k(Yi,k,t\,k | to,k,Y(tojk))

i=o (Jtik )

x exp I J aY{tnktk)iY{tnktk)(t)dt \ . (3)

Note in equation (3) that the first line represents the prod

uct of probabilities of remaining in states between transition

events and the transition to the next state. The second line

of equation (3) represents the probability of remaining in the

final observed state after the final observed transition. We as

sume that the release time (ta,k) is fixed or at least indepen

dent of the process Y(-) in equation (3) (i.e., an independent

censoring time; Andersen et al., 1993, Ch. III). When the transition intensities are constant within the in

terval rk, equation (3) reduces to

H

Vhk(Yi,k,ti,k \to,k,Y(t0,k)) =

Y[ea^T^ l\anh^k, (4) h=l i^h

where raMjfc =

XXV* I(Y(tijk) = h, Y(tl+hk)

= i) is the

number of transitions from state h to state i and Th,k =

{?r=5>~l7(y(*',*) =

>0(*i,fc -

*i+i,*)} + i(Y(tnk,k) =

h)(ta[k -

tnk,k) is the total amount of time spent in state h. Determin

ing the MLE for the transition rate ah^ is straightforward

(Albert, 1962) and asymptotic variance estimators have

also been derived (Andersen and Keiding, 2002). When a

piecewise homogeneous model is used where transition rates

are constant between time points T\,..., T3 ,,..., T/v, some or

all of the instantaneous rates may be interval specific.

2.3.2 Conventional tags. In conventional tag-recovery ex

periments, an animal is released with a unique tag and people

harvesting the species may recover the tagged animal during

the course of their efforts, so given the state at time of release

(Y(to,fc)) we observe the time of recovery (?r,fc) and states at

the instant prior to recovery (Y(tr,k?) G 1Z) and at recovery

(Y(?r>fc) G T) for harvested animals. For unharvested animals,

we know that the animal is either still alive or dead from nat

ural sources at the time of analysis (Y(ta^k) G {IZ, M}). Note

that Y(tr,k) =

Y(tUfk) for recovered animals. Thus, the data

are

({Y(tr,k-),Y(tr,k),tr,k} ifY(ta,k)eF {

u,k,tii,k}-^Y^^e{nM^ otherwise




The probability function for the set of observations from a

conventional tag is

^VfcCYii^, tnffc | t0,fc, ta,k, Y(tQ,k))

rD \i(Y(ta,k)er) = VY(t0,k),Y(tr,k-)aY(tr,k-),Y(tr.k)}

{\ HY(tatk)e{R,M})

^2 PY(t0,k),Y(ta,k) > , (5)

Y(ta,k)e{TZM} )

where I(Y(ta,k) T) and I(Y(ta,k) <E {11, M}) are indicators of whether the animal is in harvested and nonharvested states,

respectively, at the time of analysis. The first line in equa

tion (5) is the product of the probability of being alive in the

region of recovery just prior to harvest at time tr,k ?

given

Y(t0jk) and the instantaneous rate of harvest in the region

where recovery occurred. The probability Py(t0,fc),y(trifc-) is

the (y(t0,fc), Y(tr,k?)) element of the probability transition matrix P(*0,fc, tr,k-) and

ay(.rifc-),y(t0ffc) is the (Y(tr,k~),

Y(ta,k)) element of the infinitesimal matrix. The second line in equation (5) is the probability of being in any of the non

harvested states at the time of analysis given Y(tQ^k), which is

the sum of the elements of the probability transition matrix

P(to,fc, *<_,*) in row Y(*0,fc) where Y(ta,k) ? {11, M}. All tags are assumed to be reported when animals are har

vested, but incomplete tag reporting could be incorporated

into the model by expanding and redefining the possible states

(see Section 4 below for an example). In practice, we de

termine the maximum likelihood estimates using numerical

methods similar to Kalbfleisch and Lawless (1985) where we

determine the instantaneous rates and the probability transi

tion matrices using the eigendecomposition of the infinitesi

mal matrix and equation (1) such that the required elements

maximize equation (5). When a model is used where transition rates are piecewise

constant between time points T\,..., 7},..., TN and j0,fc =

min{j : T3 > ?0,j_} and jr>fc

= max{j

: 7} < ?r,fc}, the prob

ability transition matrices required are the product of the

interval-specific matrices,

P(?o,fc,?r,fc) =

P(tQ,k,TJok)P(TJok,Tjo+])

due to the Markov assumption. The probabilities necessary for an unrecovered tag are obtained by the product of the

interval-specific matrices where ta,k is used instead of trjk. 2.3.3 Implanted archival tags. Like conventional tags, IATs

will remain in tagged animals for the remainder of their lives and we only obtain postrelease information from animals that

are harvested. However, like PSATS, we can observe times of

transition and states before the time of tag battery expiration

(te,k) for a recovered animal (ti.fc =

*i>fc,..., tLk,..., tnk,k :

to,* <

U,k <

te,k, YI>fc =

Y(t_,fc), , Y(t.,fc), , Y(tnk,k)). If

the animal is recovered after the battery expires, we also have

an observation analogous to that of a conventional tag where

the state at the time of battery expiration (te,k) and states

just prior to and at the time of recovery (tr,k) are known.

Finally, we also have animals that are tagged and not recov

ered at the time of analysis (?a,fc) as in conventional tagging

experiments. Thus, the data for an I AT are

{Yni,fc, tin,fe}

{Yilfc,ti,fc if Y(te,k)ef

Y,lfc,ti>fc,Y(tc,fc), if Y(te,k)eU

Y(tr,k-),Y(tr,k),tr,k and Y(ta,k) G T '

[Y(ta,k) G {II, M}} otherwise

The probability function for the set of observations from

an IAT is

/^>iii,fc(Yn])/e, tin,*; | t0,k, ta,k, Y(t{)^k))

= ?u(Yi,*,ti,fc \tQ,k, Y(to,fc))/(y(t-fc)e^

rD i/(y(*e,fc)eW,y(tO|fc)GJ0 x vy(te../c)^(tr.fe-)ay(tr,fc-),y(tr,fe))

{^ \ HY{tatk)e{K,M})

Yl py^.khY(ta,k) \ > (6)

y(ta.fc)G{7e,A^} J

where 7\fc(Yi,fc, ti)fc | ?o,fc> Y(?o,fc)) is the component represent

ing continuous-time observation between times to,k and te,k

(i.e., equation (3)), the second line represents the observation

of recovery after battery expiration (analogous to the first line

in equation (5)), and the final line represents the observation

of an animal that remains unrecovered at the time of analysis

(analogous to the second line in equation (5)). As with the release time for PSATs, we assume that the battery life (tBjk) of an IAT is fixed or at least independent of the process Y(-) in equation (6).

3. Behavior of MLEs for Conventional

and Archival Tagging Experiments

Through simulation, we explored the behavior of MLEs for the

regional migration and mortality rate parameters based on the

three likelihoods presented above for conventional, PS AT, and

IAT observations. We intend the process observed through the

tagging experiments to reflect the life history of adult west

ern Atlantic bluefin tuna. The northern Atlantic Ocean is split into western and eastern regions at the 45?W meridian for in

ternational harvest management (e.g., ICCAT, 2007) and fish

may migrate between western and eastern regions of the At

lantic Ocean and die from harvest or natural causes in either

region.

We assumed instantaneous west-to-east (pw,e) and east

to-west (pe,w) migration rates, region-specific instantaneous

harvest (Fw and FE), and natural mortality (Mw and ME) rates are constant throughout the time of analysis for each

animal to reduce the complexity of the results and better ob

serve the behavior of the MLEs for the six instantaneous rate

parameters. Thus, the infinitesimal matrix we use throughout simulations is

(ai,i Vw,e Fw 0 Mw 0 \

Ve,w a2,2 0 FE 0 ME , (7) 0 0 0 0 0 0/

where aM =

-(pw,E + Fw + Mw),a2j2 =

~(pE,w + FE +

ME), and 0 is a 4 x 1 column vector. Note that states 1 and




2 (alive in the western and eastern Atlantic Ocean) comprise the subset IZ, states 3 and 4 (harvested in the western and

eastern Atlantic Ocean) comprise T, and states 5 and 6 (nat ural mortality in the western and eastern Atlantic Ocean)

comprise M. The values we assumed for harvest and natu

ral mortality (Fw =

0.18, FE =

0.35, Mw =

0.1, and ME =

0.15) are similar to those stock assessment scientists believe

to operate on adult Atlantic bluefin tuna in the western and

eastern Atlantic Ocean based on population dynamics mod

els (ICCAT, 2007). The western-to-eastern and eastern-to

western Atlantic migration rates we assumed (pw,E = 0.073

and Pe,w =

0.066, respectively) were estimated from prob abilities of western Atlantic bluefin tuna being in different

regions of the Atlantic at times subsequent to PSAT experi ments reported by Block et al. (2005). All rates are specified on the yearly time scale.

The general algorithm for simulating the transition history is the same for all tag types. The animal is released in a given

region (Y(tQjk) G IV) and the waiting time until the animal either leaves the region or dies in the region is an exponen

tial random variable with rate parameter ?

a^/i- Given that

the animal either leaves or dies, the (H ?

1) x 1 indicator

vector for the next state is a multinomial random variable

with probabilities ?

a/lji/a/l)/l, i / h (e.g., Taylor and Karlin,

1984, pp. 253-255). This procedure is repeated until the anal

ysis time is reached or the animal dies. As described above, each tag type yields observations with different information

on the process. We simulated 1000 experiments for each ex

periment type. Experiment type is defined by the type of tag used (PSAT, IAT, or conventional tag), the total number of

tags released, and the proportion released in each of the two

regions. We generated data and calculated likelihoods for each

experiment using C programs compiled for R (R Development Core Team, 2006).

For each simulated experiment we maximized the corre

sponding likelihood and determined whether parameters were

estimable. For parameters that were estimable, we calculated

MLEs, coefficients of variation (CVs), and 95% confidence in

tervals so that we could compare bias, precision, and 95%

confidence interval coverage, ^(0.95), of different experiment

types. As the possibility of making inferences is as important as (or more important than) the behavior of inference pro

cedures when it is possible to use them, we also calculated

the proportion of experiments where each parameter was es

timable ((p) for each experiment type.

We maximized the likelihoods using the Nelder-Mead algo rithm of the optim function in R (R Development Core Team,

2006). The maximizations were carried out with respect to the

log-transformed instantaneous rates (A^ =

log(aij:/)) to avoid

boundary problems and variance estimates were derived from

inverting numerically derived Hessian matrices in the corre

sponding space. We calculated asymmetric confidence inter

vals as exp{Ai?j ?

ZQ,975SE(Xij)}. There is a high incidence of release of PSATs prior to the

scheduled time from causes other than mortality (e.g., Sibert

et al., 2006), which we assume is independent of the FSCT

process. In our simulation of PSAT experiments, we modeled

the release time of the tag (?0,j_) using a delta-beta distri

bution: indication of tag release prior to the scheduled time

is a Bernoulli random variable with probability p and given

t I

5> \ ?5 " > _

\

q _ -n^^

p _ I_I_I_I_I_I_|_I_1 I >] S-1-1-1-1-H

0.0 0.2 0.4 0.6 0.8 1.0

Proportion of maximum tag life

Figure 1. Distribution of the scaled release times of pre

maturely released pop-up satellite archival tags scheduled to

release more than 250 days after deployment (n =

192). The curve represents a beta distribution with parameters (a

=

0.679,6 = 1.508) determined by fitting the scaled release

times. The probability of premature release is p = 0.255.

the tag releases early, the time of release scaled by the sched

uled release time is a beta random variable with parameters a and b. We used values for delta-beta parameters that we

estimated from release times of pop-up tags that have been

deployed by Lutcavage and colleagues (Figure 1). We assumed

that the scheduled time of release for tags is 0.83 years (10 months), which is similar to the intended release time of cur

rent PSATs. For IATs and conventional tags, we assumed the

time of analysis is 5 years after the time of release and the bat

tery life of IATs is similar to that of batteries currently used

(2 years).

3.1 Simulation Results

3.1.1 Effect of tag release size on MLE behavior. The prob

ability that assumed instantaneous rates are estimable con

verges to one for all tag types, as the total number of released

tags increases (Figure 2, row 1). The rate of convergence of

(j) is generally higher for harvest rate MLEs than migration and natural mortality rate MLEs. Bias of the MLEs dimin

ishes with total number of released tags for all tag types, but

the rate of convergence to unbiasedness is greatest for har

vest rate MLEs (Figure 2, row 2). The rate of convergence to

unbiasedness of migration rate MLEs is generally lowest for

PSATs. The precision of MLEs for all migration and mortal

ity parameters increases with total number of released tags

and is similar for all types of tags when estimation is possi

ble (Figure 2, row 3). For a given total number of released

tags, the CVs of estimators for migration and harvest mortal

ity parameters are lower for IATs and conventional tags than

PSATs and the CV of the natural mortality rate estimator in

the western region is lowest when PSATs are used. The cover

age of 95% confidence intervals converges to the appropriate




\^w,e ^e,w Fw FE Mw ME ? .-. .-~_-:-. .-. .-, .-. .-. . _ qoo-?>o-o - Q0'?9m..m?.-

~ o - ooooo ? o-o - ooooo ? o-o - q?~~?-?

~ o8?

? ?-?

- if f^ 1 I ?* ?!

~H ? - -

i -I d-

-J

- - -I

^ _ ? _ o 0 -i.-j-j-j-j-_j -i_-__-(-!-j-_j "Xj-j-j-j-j-_j -i_-j-|-!-]-.J -l.-j-j-j-(-_J "1-,-j-,-(-,-_J

i?TTT I -n\ I tn I fl I tl I 11 I >s- U -

5\ -i - -

k - s

I-?o\ -

c\ -

v, -i - v- - i

(So- lK.-:-S^^rr,8 - ?

o:^^-:? - S-8^--_Ji___B - ^8^8=^__g . \?--^^g

- "^^^z^.%

1 t-1-1-1-1-r t-1-1-1-1-r t-1-1-1-1-r h-1-1-1-1-r h-1-1-1-1-r t-1-1-1-1-r

oo . _ ; 6 . O ~ o. o :

~ o".

|j>

o _ _ _ _ ~~.0 _ ? -L--j-j-j-j--pi "__-j-,-j-,--J -tj-j-,-j-j--J -L.-,-(-(-(--J T-,-(-(-(-(--J "l--(-(-(-f--J

O_ ,_ _;_ _, _ ,_,_, ? _j : : I _j n _i : i j _j I _j

^-f?. -f^^ -|s>?. ?**? .^r^, .j_-_?.

ts:i : : :' T :? O ? 8 00 _

? n-1-1-1.i?r h-r?i-1-1-r n-1?-~i-1-1-r n-1-1-1-1-r n-1-1-1-1-r n-1-1-1-1-r 0 200 600 1000 0 200 600 1000 0 200 600 1000 0 200 600 1000 0 200 600 1000 0 200 600 1000

Number of tags released

Figure 2. Probability of estimation ((f)), percent relative bias, coefficient of variation (CV), and probability of approximate

95% confidence interval containing true parameter value (^(0.95)) for maximum likelihood estimators of the migration and

mortality rates (columns) when experiments use pop-up satellite archival (solid line), implanted archival (dashed line), or

conventional (dotted line) tags released in equal proportions in the western (W) and eastern (E) regions. Bias, CV, and

^(0.95) results are based on the subsets of 1000 simulations of tagging experiments for each tag type and sample size in which

parameters are estimable.

probability for all types of tagging experiments as the total

number of released tags increases (Figure 2, row 4). The rate

of convergence is highest for MLEs of harvest rates, but low

est for the MLE of the natural mortality rate in the western

region (Mw) using IATs and conventional tags in particular.

Because the relative bias, CV, and confidence interval cover

age for a given parameter are all based only on simulations

where the parameter is estimable, the values observed when

4> is low may be less informative than when <j> is high.

3.1.2 Effect of regional allocation of tag releases on MLE

behavior. For PS AT experiments, behavior of a given MLE

generally improved with the proportion of tags released in

the region corresponding to the state prior to transition

(Figure 3). For example, pw.E^Wi and Fw are more likely to be estimable and exhibit less bias, greater precision, and

better confidence interval coverage when more PSATs are re

leased in the western region. For studies using conventional

tags or IATs, the relationship of a region where all releases

occur to the instantaneous rate parameter is less predictable.

For example, the probabilities of Pw.e^e.w^w^ and ME

being estimable for IATs are low whether all tags are released

in the western or eastern region. When some tags are released

in both regions, behavior of MLEs is generally similar across

tag types with the exception that MLEs for migration rates

can be slightly more likely to be estimable and have slightly

less bias with IAT and conventional tag studies. As with the

results above on the effect of tag release size on MLE behav

ior, the values observed when <j> is low may be less informative

than when cf) is high.

3.1.3 Effect of release size and regional allocation on corre

lation of MLEs. The correlation of migration rate MLEs with

each other and other MLEs is negligible for all tag types when

tags are released in either region (Figure 4). For IAT and con

ventional tag studies, harvest mortality rate and natural mor

tality rate MLEs for a given region are positively correlated

but are negligibly correlated between regions. The correlation

of MLEs in conventional tag and IAT experiments can cause

confounding of parameters and the poor behavior of MLEs

we observe at low tag release sizes and releasing all tags in a

single region.




\*>w,e \ê,w Fw FE Mw ME - 0 - - - O - ? - @ -i-^? O - O ??ZTg?-

-* O -. - _ 0 ~0 - Qz-=^*~0 .o??o -o o o o.O -

_,q.^-^Q -o" - O O 0=^^-0.o 00 - / j/ \ - >' \

N - >'/ - V - i':? \ _ / V o ; ox \ / \ \ / / \\ /./ \ / \v ; j \ O/ O S O/ I \ \ /. / ^ f \ \

^_? / - \\ - / - \ -": - \? 0/ '

\ ? / ? / \

7 - \

-? - -/ ? o ~S-1-1-1-1-H ~h-1-1-1-1-r ~h-1-1-1-i-r ~h-1-1-1-1-r ~h-1-1-1-1-r ~H-1-1-1-1-r

^sn 1 f1 n n 11 n kî i i n 1 . v .-" -;

A--'

"

/

" v

" I,'

^^_ - - - - _8" 1 t-1-1-1-1-r n-1-1-1-1-r n-1-i-1-1-r n-1-1-1-1-r n-1-r~?i-1-r i-1-1-1-1-r

CJ "L,-,-!-(-!-jJ -L,-,-j?-,-1-jJ -1^-j-,-!-!-,J "1,-(-,-,-,-^ "t,-,-,-,-,-,J "t,-,-,-,-,-,J o ,_, ,_, ,_, ,_;_, ,_, ,_, O -I I -I 51 -I I -I -fft

,J O ^ ?' ?. ^00 o ** ft -: - - a ̂ J-~~' 8_. 8 ?t- Q *- - O . . O O ?_n n _Q- ? '?. ._? -' '"' ? ̂ ̂ a_a-" ? s ' n

^ _ ,8 ?-^8=^- _0-?f

-8^- _ . ??or-0 -8--ê-^r=?-,,A,.,@ _ y o-o ^0. .o^-o-ê..,

,.<..-o

71...H-../,,..H. ... .\ o n-1-1-1-1-r t-1-1-1-r-r V?i-1-1-1-r n-1-1-1-1-r n?h-1-1-1-r n-1-1-1-1-ir

0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80

% tags released in western region

Figure 3. Probability of estimation (</>), percent relative bias, coefficient of variation (CV), and probability of approximate

95% confidence interval containing true parameter value (^(0.95)) for maximum likelihood estimators of the migration and

mortality rates (columns) when experiments use 200 pop-up satellite archival (solid line), implanted archival (dashed line), or conventional (dotted line) tags released in varying proportions in the western (W) and eastern (E) regions. Bias, CV, and

ij)(Q.95) results are based on the subsets of 1000 simulations of tagging experiments for each tag type and sample size in which

parameters are estimable.

4. An Example: Atlantic Bluefin Tuna

Conventional Tagging Data

Conventional tagging experiments for Atlantic bluefin tuna

have been conducted throughout the northern Atlantic Ocean

for several decades and the resulting data have been archived

by the International Commission for the Conservation of

Atlantic Tunas (ICCAT). We considered a subset of the archived data where tagged fish were released in 1990-1992

and recaptures were made prior to the beginning of 1995

(assumed end of study). We used the longitudes of tagged fish at release and recovery to determine the regional locations

at those times. The dates of release and recovery are used

to determine the times on study (ta^k ?

?o,& !__ 5 years). Of

the 2022 tagged fish released in the western region of the

Atlantic Ocean, 65 were recaptured in the same region

and 6 were recaptured in the eastern region. Of the 4288

tagged fish released in the eastern region, 72 were recap

tured in the same region and 6 were recaptured in the western

region.

For this example, we have the same instantaneous rates

as the simulation study above, but the probability of report

ing harvested tags is thought to be substantially less than 1.

Because there is no auxiliary information to estimate report

ing probability and it is completely confounded with natural

mortality rates, we assume a value for natural mortality rate

(Mw =

ME ?

0.15), similar to values believed by Atlantic

bluefin assessment scientists (ICCAT, 2007), and estimate re

porting probability (0 < p < 1) along with migration and harvest mortality rates. The resulting infinitesimal matrix is

(ai,i Pw,e PwFw 0 (1 -

pw)Fw + Mw 0 \

Pe,w a2i2 0 p^Fb 0 (1 -

pE)FE + ME\ , (8) 0 0 0 0 0 0 /




% tags released in western region 0 20 60 100 0 20 60 100 0 20 60 100 0 20 60 100 0 20 60 100

J_I_I_I_L_L_ J_1_I_I_1_L_ J_I_I_I_I_L_ J_I_I_I_I_L_ J_I_I_1_I_L p

\lWE -- ; ?^-- ^:- iî^îrrr_

^^^??^

.

Â_~~-^- L^?^-]-?

?

T p_ I.. ,i_1_i_i., ,i i,, ., i_i i i i J_I ?,i, . , I_i_l_ I. I_i i i I q

g-^^. - ---

\*>E,W -y77^^-.-_ -^-^?^- ^?-**- ^--?y,:?

? "

? ~

4.-r?T------,-pi - - - -1--'

p _,_,_,_ _:____ J_I_I_I_1_L J_I I_1_I_L_ J_I_I_1_I_i__ p

? ^ -t,-,-,-._,-pj -_.-,-,-,-,-_j - -

-h^ ?_ O ,_?_. ,_ ,_._. J_1_I_I_I_L J_I_I_I_I_L O

s -1 i r ~

U q-t-_- -1^^.~~-^--~^_?_?: -u^^^^~^-^ J7j-, =^^~~~^zîL._ ill_LL_.2_i___._q

? p --, -_.-___?j-j-,-pJ -4--,-,-,-,-pJ -L.-j-,-(-,--J

-- --.

p _._,_,_ _ J_I_I_I_I_L O

q _\z^?Lzjt^z^^=^ J^?^?___-__-______- -?c^-.--.--..-.->^--.. A/1\rr isl*^^.^^^^-^

? ~| ? --, H( t ,-f-(-!--J -4--___?|-]-!-pJ -__-(?__--j-(--J -Lj-j-j-j-j-pJ

- -p

O _,_,_ _,_:_ _;_ _ _

? ---

H_-,-!-,-,--J -4--,-,-,-,-jJ 4_-(-,-(?_--pJ -4--(-(-(-(--J -L.-,-,-(-pJ 0 2000 4000 0 2000 4000 0 2000 4000 0 2000 4000 0 2000 4000

Number of tags released

Figure 4. Correlation of maximum likelihood estimators of the migration (pw,E and /jLe,w) and harvest (Fw and Fe) and

natural mortality (A_V and ME) rates when experiments use 200 pop-up satellite archival (solid line), implanted archival

(dashed line), or conventional (dotted line) tags released in varying proportions in the western (W) and eastern (E) regions

(upper right) and released in equal proportions in the western (W) and eastern (E) regions (lower left). Results are based on

the subsets of 1000 simulations of tagging experiments for each tag type and sample size in which parameters are estimable.

where aM =

-(pw,E + Fw + Mw),a2j2 ?

-(ps,w + FE +

ME), and 0 is a 4 x 1 column vector. We fit a suite of nested

models where all rates are the same for each region (pw,E ?

(J>e,w,Ew ?

FE, and pw =

Pe\ mo)^ only harvest mortality rates are region specific (mi), harvest and migration rates

are region specific (m2), migration rates are region specific

and harvest rates are both region and year specific (m3), and

reporting probability is also region specific (m?). For each model, we maximized equation (5) with respect

to the migration and harvest mortality rates and reporting

probabilities. The probabilities necessary in equation (5) are

calculated using equation (1) and eigendecomposition of equa

tion (8) (year specific for models m% and m4). For the best

model, we estimated variances and calculated asymmetric

confidence intervals for the migration and mortality rates us

ing the same methods as the simulation study above, but a

logit transformation (rather than log transformation) of re

porting probability is estimated due to its constraints (i.e, 0 < p< 1).

We calculated x2 goodness-of-fit test statistics (x2gof)

f?r

each fitted model by comparing the observed numbers of tags

reported in each region each year for each region of release

and the observed numbers of tags never reported with the

corresponding expected values under each model. As there

are five years and two regions of reporting and one category

of tags never reported for each region of release, the degrees

of freedom for the goodness-of-fit test is dfcoF ? 20. We also

compared nested models sequentially using likelihood ratio

tests with degrees of freedom equal to the difference in the

number of parameters between models.

The goodness-of-fit tests show that models mQ,m\, and m^

are not plausible mechanisms for generating the pooled num

bers of reported tags in each region and year (Table 1). Both

models m3 and ra4 could not be rejected using the goodness-of fit test, but the model with region- and year-specific harvest

mortality rates and region-specific migration rates (m%) pro

vides a better fit to the Atlantic bluefin tuna data using the likelihood ratio test. Based on model m3, there is evidence of

a very low tag-reporting probability from harvested individu

als (~0.122) and estimates of harvest mortality are greater for

the western region than the eastern region within a given year,

but the estimates tend to decrease over time for a given region

(Table 2). The migration rate from the western region to east

ern region of the Atlantic Ocean is greater than the opposite




Table 1

Goodness-of-fit tests and sequential likelihood ratio tests of nested models fit to Atlantic bluefin tuna conventional tagging data. The fitted models have all migration rates, fishing mortality rates, and reporting

rates constant over years and regions (mo), harvest mortality is region specific (mi), migration rates are also

region specific (m2), harvest mortality rates are also year specific (ra3), and reporting rates are also region

specific (ra4). The goodness-of-fit statistic (Xgof) and corresponding p-values (Pgof), log likelihood at the maximum (log(L)), number of parameters (np), (log)likelihood ratio statistics (xIrt) and corresponding

degrees of freedom (dfrnr) and p-values (Plrt) are provided.

Model XcoFidfcoF =

20) PG0F log(?) np xIrt dfLRT PLRT

mo 90.57 <0.0001 -1786.65 3 mi 76.99 <0.0001 -1780.38 4 12.54 1 0.0004 ma 65.62 <0.0001 -1776.26 5 8.24 1 0.0041

m3 9.34 0.9786 -1752.03 13 48.46 8 <0.0001 m4 8.48 0.9881 -1750.81 14 2.44 1 0.1183

Table 2 Maximum likelihood estimates for migration rates (pw,E and

He,w), region- and year-specific harvest mortality rates (F) and reporting probability (p), and corresponding asymmetric

95% confidence intervals provided by model m3 from fitting Atlantic bluefin tuna conventional tagging data

Parameter Estimate SE CI

pw,E 0.1832 0.0750 (0.0821-0.4089) pE,w 0.0374 0.0158 (0.0164-0.0855)

^,1990 0.5168 0.2491 (0.2009-1.3292) Fe'mm 0.1773 0.0816 (0.0719-0.4369) Fvy,i99i 0.3946 0.1545 (0.1832-0.8498)

FE',i<m 0.1360 0.0538 (0.0627-0.2953)

FW1992 0.1255 0.0580 (0.0507-0.3107) FE',im 0.0370 0.0167 (0.0153-0.0895) F^i993 0.1256 0.0584 (0.0505-0.3123) F^'1993

0.0641 0.0279 (0.0273-0.1505)

Fw,m4 0.0910 0.0523 (0.0295-0.2809) FE]vm 0.0300 0.0157 (0.0107-0.0838)

p '

0.1222 0.0063 (0.0630-0.2238)

migration rate, which would result in a greater probability of

living individuals occurring in the eastern region over time.

5. Discussion

The results we present here demonstrate the potential utility

of the FSCT approach for combining information from the

different tagging studies to estimate migration and mortality rates. Although the simulation results assume instantaneous

rate parameters presumed to be operating within western At

lantic bluefin tuna populations, the general result of increased

precision, reduced bias, and accurate confidence interval cov

erage with increased numbers of released tags in each region

of interest should apply to any scenario.

There are other attributes of a tagging study we did not

consider that can affect the behavior of MLEs. The battery

life of archival tags, rates of migration and harvest mortality,

and duration of the study will also affect the behavior of the

MLEs for the various migration and mortality rates. In the

extreme case of zero harvesting in any region, none of the pa

rameters will be estimable with any number of conventional

tags or IATs released because no tags will be recovered. For

both types of archival tags, the longer the battery life, the

more continuous-time observations are made, which in turn

provides more information. When the expected duration in

a particular state is long with respect to a given tag battery

life, duration of study, and/or number of deployed tags, the

expected number of transitions to and probabilities of being in other states will be small and vice versa. Also, the longer the duration of the study given a set of migration and mor

tality rates, tag battery life, and number of deployed tags, the more transitions would be expected. The form of the

asymptotic variance estimator for MLE with PSAT experi ments (Andersen and Keiding, 2002) shows that the precision

increases with the number of transitions. That the same like

lihood is a component of the IAT experiments implies that

increased precision with more transitions would also, to per

haps a lesser extent, occur for those experiments. However, the interval censoring that occurs for IATs and conventional

tags makes the relationship less clear for these experiments.

In practice the rates of migration and mortality are unknown,

which would further argue for release of tags in all regions. Realities such as less than total tag reporting and tag shed

ding not considered here will also diminish the information

provided from all types of tagging experiments. In more realistic population models, the rates of migration

and mortality may change through time and the life history of the animal. For example, migration rates from the region of

origin may increase with age for Atlantic bluefin tuna or har

vest mortality rates may be zero outside of or may change dra

matically within harvesting seasons for many species. In these

cases, several sequential infinitesimal matrices with rates con

stant over shorter intervals may be considered. With a given

number of releases, duration of analysis, and tag battery life

(for IATs), the instantaneous rates for these more complex

life-history processes would also be more poorly estimated

than those driving the simpler stochastic process assumed in

the simulations. As such, more releases throughout each year

would provide better information for estimating these param

eters.

Although we partition space and parameterize the FSCT

process for fisheries management, the partitioning and

parametrization can also address behavioral or ecological

characteristics of the population of interest. With Atlantic

bluefin tuna, for example, we might further partition the space




to represent spawning and feeding grounds within each re

gion because adults are thought to migrate between these

areas during certain periods each year. We might also con

strain migration rates to the spawning and feeding grounds to be higher at the appropriate periods to reflect these life

history attributes of the population. The regional migration

and mortality model is also directly applicable to manage

ment regimes for many species that employ protected areas

where no harvest is allowed (e.g., Roberts et al., 2001; Field

et al., 2006). Harvest rates would be set equal to zero in the

protected areas and migration rates among the protected and

unprotected areas could be estimated from tagged animals.

Acknowledgements

We thank Daniel Commenges, Andy Cooper, Andy Rosen

berg, and John Sibert for productive discussions on ideas

involved in this work and Victor Restrepo and Papa Kebe

at ICCAT for help with conventional tagging data. We also thank Carl Schwarz for providing helpful comments

on an early draft and a referee and associate editor who

also provided comments that improved this manuscript. This

work was generously funded by contract NA04NMF4550391 awarded to M. Lutcavage by NOAA National Marine Fish

eries Service.

References

Albert, A. (1962). Estimating the infinitesimal generator of a

continuous time, finite state Markov process. Annals of

Mathematical Statistics 33, 727-753.

Andersen, P. K. and Keiding, N. (2002). Multi-state models

for event history analysis. Statistical Methods in Medical

Research 11, 91-115.

Andersen, P. K., Borgen, 0., Gill, R. D., and Keiding, N.

(1993). Statistical Models Based on Counting Processes.

New York: Springer-Verlag.

Block, B. A., Dewar, H., Blackwell, S. B., Williams, T. D.,

Prince, E. D., Farwell, C. J., Boustany, A., Teo, S. L. H.,

Seitz, A., and Walli, A. (2001). Migratory movements,

depth preference, and thermal biology of Atlantic bluefin

tuna. Science 293, 1310-1314.

Block, B. A., Teo, S. L. H., Walli, A., Boustany, A., Stokes

bury, M. J. W., Farwell, C. J., Weng, K. C, Dewar, H., and Williams, T. D. (2005). Electronic tagging and pop

ulation structure of Atlantic bluefin tuna. Nature 434, 1121-1127.

Bonfil, R., Meyer, M., Scholl, M. C, Johnson, R., O'Brien,

S., Oosthuizen, H., Swanson, S., Kotze, D., and Paterson, M. (2005). Transoceanic migration, spatial dynamics and

population linkages of white sharks. Science 310, 100

103.

Brownie, C, Hines, J. E., Nichols, J. D., Pollock, K. H., and Hestbeck, J. B. (1993). Capture-recapture studies

for multiple strata including non-Markovian transitions.

Biometrics 49, 1173-1187.

Chapman, D. G. (1961). Statistical problems in the dynam ics of exploited fisheries populations. Proceedings of the

Berkeley Symposium on Mathematical Statistics and Prob

ability 4, 153-168.

Commenges, D. (2002). Inference for multi-state models from

interval-censored data. Statistical Methods in Medical Re

search 11, 167-182.

Cox, D. R. and Miller, H. D. (1965). The Theory of Stochastic

Processes. New York: Wiley.

Darroch, J. N. (1958). Estimation of a closed population.

Biometrika 45, 343-359.

Dupont, W. D. (1983). A stochastic catch-effort method for

estimating animal abundance. Biometrics 39, 1021-1033.

Field, J. C, Punt, A. E., Methot, R. D., and Thompson, C. J. (2006). Does MPA mean "Major Problem for As

sessments?" Considering the consequences of place-based

management systems. Fish and Fisheries 7, 284-302.

Hearn, W. S., Sundland, R. L., and Hampton, J. (1987). Ro

bust estimation of the natural mortality rate in a com

pleted tagging experiment with variable fishing intensity.

Journal Du Conseil International Pour L'exploration De

La Mer 43, 107-117.

Hestbeck, J. B. (1995). Bias in transition-specific survival

and movement probabilities estimated using capture

recapture data. Journal of Applied Statistics 65, 429

433.

Hestbeck, J. B., Nichols, J. D., and Malecki, R. A. (1991). Es

timates of movement and site fidelity using mark-resight

data of wintering Canada geese. Ecology 72, 523-533.

Hilborn, R. (1990). Determination offish movement patterns

from tag-recoveries using maximum likelihood estima

tors. Canadian Journal of Fisheries and Aquatic Sciences

47, 635-643.

Hoem, J. M. (1971). Point estimation of forces of transition

in demographic models. Journal of the Royal Statistical

Society, Series B 33, 275-289.

ICCAT. (2007). Report of the 2006 Atlantic bluefin tuna stock assessment session. Collective Volume of Scientific Papers

ICCAT 60, 652-880.

Kalbfleisch, J. D. and Lawless, J. F. (1985). The analysis of

panel data under a Markov assumption. Journal of the

American Statisical Association 80, 863-871.

Kalbfleisch, J. D. and Prentice, R. L. (2002). The Statistical

Analysis of Failure Time Data. New York: Wiley.

Karlin, S. and Taylor, H. M. (1975). A First Course in Stochas

tic Processes. New York: Academic Press.

Laslett, G. M., Eveson, J. P., and Polacheck, T. (2002). A

flexible maximum likelihood approach for fitting growth curves to tag-recapture data. Canadian Journal of Fish

eries and Aquatic Sciences 59, 976-986.

Lebreton, J. D. and Pradel, R. (2002). Multistate recapture models: Modelling incomplete individual histories. Jour

nal of Applied Statistics 29, 353-369.

Nichols, J. D. and Kendall W. L. (1995). The use of multi

state capture-recapture models to address questions in

evolutionary ecology. Journal of Applied Statistics 22, 835-846.

Pollock, K. H. (1991). Modeling capture, recapture, and re

moval statistics for estimation of demographic parame

ters for fish and wildlife populations: Past, present, and

future. Journal of the American Statistical Association

86, 225-238.

Quinn, T. J. and Deriso, R. B. (1999). Quantitative Fish Dy namics. New York: Oxford University Press.




R Development Core Team. (2006). R: A Language and Envi

ronment for Statistical Computing. Vienna: R Foundation

for Statistical Computing.

Ricker, W. E. (1975). Computation and the Interpretation of

Biological Statistics of Fish Populations. Bulletin of the Fisheries Research Board of Canada, Number 191, Ot

tawa, Canada.

Roberts, C. M., Bohnsack, J. A., Gell, F., Hawkins, J. P.,

and Goodridge, R. (2001). Effects of marine reserves on

adjacent fisheries. Science 294, 1920-1923.

Schwarz, C. J. and Seber, G. A. F. (1999). Estimating animal

abundance: Review III. Statistical Science 14, 427-456.

Schwarz, C. J., Schweigert, J. F., and Arnason, A. N. (1993).

Estimating migration rates using tag-recovery data. Bio

metrics 49, 177-193.

Seber, G. A. F. (1982). The Estimation of Animal Abundance and Related Parameters. London: C. Griffin & Co., Ltd.

Seber, G. A. F. and Schwarz, C. J. (2002). Capture-recapture:

Before and after EURING 2000. Journal of Applied Statistics 29, 5-18.

Sibert, J. R., Musyl, M. K., and Brill, R. W. (2003). Hori

zontal movement of bigeye tuna (Thunnus obesus) near

Hawaii determined by Kalman filter analysis of archival

tagging data. Fisheries Oceanography 12, 141-151.

Sibert, J. R., Lutcavage, M. E., Nielsen, A., Brill, R. W., and

Wilson, S. G. (2006). Interannual variation in large scale

movement of Atlantic bluefin tuna (Thunnus, thynnus) determined from pop-up satellite archival tags. Cana

dian Journal of Fisheries and Aquatic Sciences 63, 2154

2166.

Taylor, H. M. and Karlin, S. (1984). An Introduction to

Stochastic Modeling. Orlando, Florida: Academic Press.

Taylor, N. G., Walters, C. J., and Martell, S. J. D. (2005). A

new likelihood for simultaneously estimating von Berta

lanffy growth parameters, gear selectivity and natural

and fishing mortality. Canadian Journal of Fisheries and

Aquatic Sciences 62, 215-223.

Whitehead, H. (2001). Analysis of animal movement using opportunistic individual identifications: Application to

sperm whales. Ecology 82, 1417-1432.

Received April 2007. Revised November 2007.

Accepted November 2007.



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