© 2004 by ICES/CIEM International Council for the Exploration of the Sea/Conseil International pour l'Exploration de la Mer
Estimating migration rates from two tag–release/one recovery experiments
a Ocean Research Institute, the University of Tokyo 15-1 Minamidai, Nakano-ku, Tokyo 164-8639, Japan
b Tokyo University of Fisheries 4-5-7 Konan, Minato-ku, Tokyo 108-8477, Japan.
*Correspondence to K. Shirakihara: tel: +81 3 5351 6493; fax: +81 3 5351 6492. e-mail: shirak{at}ori.u-tokyo.ac.jp.
We propose a method for estimating migration (movement) rates from two tag–release/one recovery experiments, regardless of tag-shedding or incomplete tag-reporting which are major problems when tag–recovery techniques are applied to commercially exploitable populations. The entire survey area is divided into multiple strata in advance. The first release is limited to one stratum; then, the second release occurs in every stratum. Recoveries from both releases occur in every stratum. The migration rate between the time of the first and second release is estimated together with its variance. To check the applicability of our method, we applied it to tag–recovery data for skipjack tuna from various places at different times and unequal time intervals. The precision of the estimates was low and the coefficient of variation was 22.6–42.1% because of the small number of recoveries. The experimental design necessary to improve precision is discussed.
Keywords: migration rate, movement, skipjack tuna, tagging experiment
Received 31 July 2003; accepted 5 January 2004.
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1 Introduction |
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 Top 1 Introduction 2 Methods3 DiscussionAppendixReferences |
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Migration rates are an essential parameter in studies of fish population dynamics, as many fish populations migrate. Their estimation is important for managing migratory populations (e.g. Phiri and Shirakihara, 1999) and evaluating the effect of closed areas (e.g. Polacheck, 1990).
Mark–recapture or tag–recovery techniques may be useful for determining the migration rates. However, when tag-shedding or incomplete tag-reporting occurs, as is often the case for commercially exploitable populations, the estimate will be biased. Evaluating the tag-shedding or tag-reporting rate, which is necessary to obtain unbiased estimates, is never a straightforward task (e.g. Hearn et al., 1991, 1999). This paper presents a method for estimating migration rates from tagging experiments regardless of the presence of tag-shedding or incomplete tag-reporting. Tagging experiments have been also used to determine migration routes rather than to estimate migration rates. The proposed method has some applicability to the data from such experiments that have been carried out in various places, at different times, and at unequal time intervals.
Various methods of estimating migration rates using tag–recovery techniques have been reported. Darroch (1961) extended the basic Petersen method to a geographically stratified population and proposed maximum likelihood estimators using two samples (one tagging and one recovery). Arnason (1972) extended the theory to a case with three samples. Schwarz et al. (1993) proposed a more comprehensive method, extending the multiple-sampling model of Brownie et al. (1985) to a stratified population. In addition to these studies that use only tag–recovery data, Tanaka (1976) extended the Jolly and Seber method to a stratified population. Hilborn (1990) considered the simulation approach used by Ishii (1979) and proposed maximum likelihood estimators that make it possible to use available fisheries data such as fishing effort. Extensions and applications of this method have appeared in several subsequent studies (e.g. Deriso et al., 1991; Anganuzzi et al., 1994; Xiao, 2000).
This study follows Darroch's initial approach. Quinn and Deriso (1999) stated that it is often more straightforward to use Hilborn's approach. Darroch's approach is, however, also meaningful in that it provides information that is independent of fisheries statistics. Darroch (1961), who dealt with simultaneous tag–releases in each of s strata (areas) and simultaneous recoveries in each of u strata, pointed out that migration rates are not identifiable when s<u. Schwarz et al. (1993) considered the case of s=u and multiple tag–recovery experiments in which releases take place simultaneously in all strata and both releases and recoveries continue at equal time intervals. They proposed a method to estimate survival/migration rates that are not separated into components and not affected by tag-reporting rates. However, implementation of their method, which requires sampling at equal time intervals, is likely to be difficult when recoveries depend on reporting from commercial fishing boats. By contrast, our method allows s<u as well as unequal time intervals, and focuses on determining migration rates by means of two tag–release/one recovery experiments that are somewhat more specific: the first release is in one stratum, the second release is in all strata, and recoveries are from all strata.
The term "migration" refers to a consistent and direcional movement of some component of a population, whereas "movement" is a more general term that refers to any change in the location of individuals in a population (Quinn and Deriso, 1999). We do not distinguish these terms and use "migration" for both cases. We begin by developing three models to estimate migration rates. Then, we apply the models to tag–recovery data for skipjack tuna (Katsuwanus pelamis) and discuss their applicability.
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2 Methods |
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Top1 Introduction2 Methods3 DiscussionAppendixReferences |
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2.1 Design of two tag–release/one recovery experiments
First, the entire study area is divided into s strata. The first tag–release is carried out in one stratum at time t1. Later, a second tag–release is carried out in every stratum simultaneously at t2, or at t2i in stratum i. Finally, recovery surveys for both releases are conducted in every stratum simultaneously at t3, or at t3i. The tag with a record of the place and time of release is used, but individual recognition is not necessary in our experiments. The recovered fish are not returned to the sea.
2.2 Models
We propose three models for parameter estimation, based on the absence (Basic Model, Figure 1) or presence (Extension Model I, Figure 2) of migration between t2 and t3, or on the time difference of the second releases (Extension Model II, Figure 2). The assumptions common to all three models are as follows:
- The rate of tag loss at the time of tagging, owing to immediate tag-shedding and deaths caused by tagging, is the same for all fish, regardless of when they are tagged, because the same protocol is used for all tagging.
- The rates of chronic tag-shedding, long-term death due to tagging, and natural mortality are constant for all fish. Emigration from the survey area is included as natural mortality.
- The probability of capture is the same for fish present in stratum i at the time of the second release in stratum i. This is because tag finders do not discriminate between tags from the two tagging events and because fish belonging to each event have mixed in each stratum by the time that recoveries occur.
- Survival rates between t1 and t2 are the same for fish belonging to the first event regardless of which stratum individual fish move to. This assumption is required to distinguish stratum-dependent migration rates and stratum-independent survival rates for this period.
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Note that the survival rates between t2 and t3 are allowed to differ among strata, due to spatial heterogeneity in fishing mortality rates or to the effort involved in obtaining recovered fish. The reporting rates are also allowed to differ among strata, so that we can make use of recovery data obtained by fishers.
2.2.1 Basic Model
We consider the case in which the second release is carried out simultaneously at t2 in all strata and add the following assumption.
- Migration between t2 and t3 can be ignored.
Let M fish with tags be released at time t1 and 
be the mortality rate between t1 and t2. The number of individuals surviving until t2 from the first release is
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| (1) |
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| (2) |
i is the migration rate from the stratum in which the first release was made to stratum i and n = 
i ni. Let Yi fish with tags be released newly in stratum i. Individuals from both releases stay in this stratum until t3, under assumption (v). Since death and tag-shedding are indistinguishable, the number of individuals from the first release that survive with tags at t2 is not ni but is ni
, where is the tag-retention rate. Let mi and yi be the numbers of individuals recovered in stratum i from the first and second releases, respectively. Because the following proportionality is expected: ni:mi=Yi:yi (Figure 1, upper right), an estimate of ni is
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| (3) |
. Substituting Equation (3) into this equation, an estimate of is given by
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| (4) |
into each component in our experiments, but it is possible to estimate i. From Equations (1) and (2), we obtain the relationship 
. Substituting Equation (3) into this equation, we obtain an estimate of the migration rate to stratum i:
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| (5) |
and E[yi]=piYi, respectively, where pi is the product of the tag-retention rate between t2 and t3, the recovery rate in stratum i, and the tag-reporting rate in this stratum. As shown in Equation (5), no estimate of pi is required to obtain 
, but the ratio of the two observed recoveries, mi/yi, is necessary. Estimates of the variances of 
, 
, and 
are derived in the Appendix.
Let us consider the effect of using t3i instead of t3 on parameter estimation. As shown above, all the estimates are based on the ratio of the two observed recoveries. As long as these two recoveries in stratum i are obtained simultaneously, such a difference has no effect on the estimates. Next, we examine the situation in which multiple recoveries take place in stratum i over a period, i.e. the 
th recoveries are mi and xi, where it is acceptable that some of them are zero. The ratio is estimated as mi / yi, using the ratio estimator (Cochran, 1977). In other words, mi and yi can be regarded as mi and yi, respectively. Hereafter, we do not use t3i.
2.2.2 Extension Model I
Assumption (v) of the Basic Model is likely to be appropriate when all the recoveries are obtained only from the release stratum. We now consider the case where this assumption is violated and the second release is carried out simultaneously in all strata (Figure 2, left). In this model, mi, yi, and pi are replaced by mij, yij, and pij, respectively, where the left and right subscripts are the strata in which individuals are present at t2 and t3, respectively.
The numbers of recoveries, mij, are not observable, but their sum over the release strata, m·j, is observable. The expected values of m·j and yij are
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| (6) |
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| (7) |
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| (8) |
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| (9) |
cannot be expressed as an explicit function of the observed statistics, but is obtained by solving s linear equations (Equation 9). Since i = n i / n i ; , 
is obtained by
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| (10) |
, 
, and 
are given in the Appendix.
2.2.3 Extension Model II
We now consider the case in which assumption (v) of the Basic Model is violated and the second releases are carried out at different times in each stratum: t2i i =...,s (Figure 2, right). To estimate migration rates, we make an additional assumption.
- During the period of the second releases, the ratio nit/nt remains constant, where nit is the number of individuals from the first release that survive and remain in stratum i at an arbitrary time t and nt = i nit.
This assumption is an approximation rather than a reflection of the truth. For example, if the fish tend to move northward, the ratio in the northern stratum will steadily increase with time. Here, however, we assume that some mean ratio is maintained approximately during this period. We define t2 as the earliest time of the second release, i.e. the minimum of {t2i} and define n*i as the number of individuals of the first release that survive with tags and remain in stratum i at t2.
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| (11) |
The number n*i, which corresponds to ni in Extension Model I, is estimated by solving the following equations in the same manner as Equation (9).
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| (12) |
, from Equation (11) we obtain 
. Replacing and n*i with their estimates, we obtain
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| (13) |
, 
is found by solving Equation (13). Using 
and 
, ni is estimated as:
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| (14) |
. Estimates of the variances of 
, 
, and 
are derived in the Appendix.
2.3 Example
The Fisheries Agency of Japan conducted tag–recovery experiments for skipjack tuna in the Pacific. Typically, tag–release experiments, which consist of capturing, tagging, and releasing the fish, were done for several days at different places from research vessels equipped with fishing gear of pole and line. The number of released fish was recorded with the release date, and latitude/longitude of the release site. Recovery data, such as number of fish, date, and latitude/longitude, were reported mainly from fishing boats.
We applied our models to these data. Since we were interested in examining the applicability of the models rather than improving our understanding of fish migration, we subdivided the survey area without referring to fish biology studies. The results of the estimation should be considered preliminary.
Two vessels carried out five tagging experiments from October 1988 to January 1989 (Report on tagging surveys of skipjack tuna in 1988–1989, Tohoku National Fisheries Research Institute, Fisheries Agency of Japan, 1989). Larger numbers of fish were released and recovered in three experiments (Table 1). Recoveries were usually reported within 200 days from the date of release. The survey area was divided into two strata: stratum 1 began at 10°N and stratum 2 was at lower latitudes. Experiment A was regarded as the first release in stratum 1, B as the second release in stratum 1, and C as the second release in stratum 2. Since the periods of the second releases (B and C) overlapped and were less than one week apart, both releases were regarded as occurring on the same day, 25 January 1989. The date of the first release A was also regarded as the first day of the experiment. Nine individuals from release B were found in a stratum different from the release stratum, which suggests that we should use Extension Model I. For comparative purposes, the other two models were also used by modifying the raw data (Table 2). To apply the Basic Model, the data for y12 and y21 were neglected. To apply Extension Model II, the date of t22 was arbitrarily changed.
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The estimates are shown in Table 3. Extension Model I suggests that more than half of the fish remained in the release stratum between 25 November 1988 and 25 January 1989. The other models did not give substantially different estimates. The precision of the estimates was not high and the coefficient of variation was 22.6–42.1%. An experimental design to yield higher precision will be discussed later.
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3 Discussion |
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Top1 Introduction2 Methods3 DiscussionAppendixReferences |
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Our method was originally developed to estimate migration rates using data from tag–recovery experiments that have been carried out in various places at different times. Such experiments are commonly used not to estimate migration rates but to determine migration routes, as illustrated in Figure 3. When recoveries occur at unequal time intervals, it is difficult to tabulate the period and stratum of the recoveries in matrix form and to estimate parameters simultaneously using the general model of Schwarz et al. (1993). By contrast, there may be cases in which it is suitable to apply our models. For example, experiment A in Figure 3 can be regarded as the first release, and C and D as the second releases in each stratum. Using the recovery data from the three experiments, the migration rates for the period between A and C would be estimated. Next, B is regarded as the first release, and C and D as the second release or C as the first release, and E and F as the second release. Then, the estimation procedures are iterated. To obtain proper estimates, the survey area should be divided into strata with nonzero recoveries from the second releases. If there are some strata without recoveries, estimates cannot be obtained.
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Among the assumptions to determine migration rates, the critical one is constant fishing mortality between t1 and t2 in all strata. If this assumption is violated, it is impossible to discriminate between the migration rates and mortality rates, as Schwarz et al. (1993) pointed out. As our method does not use the recovery data before t2, the assumption of constant fishing mortality is unavoidable. However, if there is evidence that suggests heterogeneity in fishing mortality between t1 and t2 for individuals from the first release, then any estimates of migration rates are preliminary.
Although our method uses a simple procedure to obtain point estimates of migration rates, it generally requires a numerical evaluation of their variances, except for the simplest case of the Basic Model when s=2. Therefore, it is difficult to investigate analytically an experimental design for obtaining reliable estimates with sufficient levels of precision. Here, we only discuss the design for the simplest case, where the coefficient of variation (CV) for 
is estimated as follows from Equation (A5) in the Appendix:
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and each CV is estimated as 20.3%. Stochastic variation in each recovery will surely give a poorer result. To achieve a CV below 20%, each release should number a few thousand. For cases other than the Basic Model with s=2, numerical evaluation of the CV is recommended for each combination of numbers of releases and recoveries in advance of the experiments. Here, we show another example using the Basic Model with s=3: M=Y1=Y2=Y3=1200 and m1=m2=m3=y1=y2=y3=18 (keeping i mi and i yi the same as in the first hypothetical example). Then 
and each CV is estimated as 27.1%. These two hypothetical examples suggest that larger numbers of strata should result in less precision. This is because the number of parameters to be estimated increases and the number of recoveries per stratum decreases. Therefore, a design focusing on the number of strata is required to obtain reliable estimates of migration rates. Our method assumes that recoveries follow binomial or multinomial distributions, as do most studies that estimate migration rates. This assumption is valid when recoveries result from simple random sampling, but their variances will be larger if fish is clustered. In such a situation, our method underestimates the precision of the estimates. Introducing over-dispersion (e.g. Kitada et al., 1994) to our models, which needs replicates of recoveries, is left for a future study.
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Appendix |
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Top1 Introduction2 Methods3 DiscussionAppendixReferences |
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A.1 Variances of the parameter estimates
In each of the three models,
and 
are obtained as follows, given 
.
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| (A1) |
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| (A2) |
can be expressed approximately as follows from Equation (A1).
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| (A3) |
can be directly formulated as follows from Equation (A2).
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| (A4) |
and 
can be found if 
and 
are given, where 
is evaluated from Equation (A1). We will derive 
and 
for each model later in the three following subsections.
It is generally difficult to show 
and 
explicitly as a simple function of observed statistics, but it is easy to obtain them numerically if we use mathematical software, such as Mathematica, to calculate the partial derivatives. An exception is the simplest case of the Basic Model with s=2. In this case,
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| (A5) |
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| (A6) |
A.1.1 
and 
for the Basic Model
As shown in Equation (3), ni is estimated as:
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| (A7) |
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| (A8) |
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| (A9) |
and 
, the following are necessary: 
, 
, 
, 
, and 
. Provided that recoveries are obtained by simple random sampling, {mi} and yi follow multinomial and binomial distributions, respectively. Therefore, 
, 
, and 
, where i
j. Provided that recoveries are obtained independently, 
and 
. Furthermore, 
, whose proof is shown later. Substituting these variances and covariances into Equations (A8) or (A9), we obtain
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| (A10) |
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| (A11) |
A.1.2 
and 
for Extension Model I
Unlike the Basic Model, 
should be obtained by solving the following equations.
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| (A12) |
as a function of the random variables {m·j} and {xjk},
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| (A13) |
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| (A14) |
and 
are evaluated from Equation (A12) and variances and covariances are: |
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A.1.3 
and 
for Extension Model II
The estimation from this model is more complicated than for Extension Model I because 
cannot be solved directly. Instead 
can be solved from the following equations of Equation (12).
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and 
are obtained in the same way as in the previous subsection, except that 
is replaced with 
in Equations (A13) and (A14). From Equation (14), |
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and 
can be also expressed as follows:
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| (A15) |
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| (A16) |
is a constant, Equations (A15) and (A16) give linear equations with respect to the unknowns 
and 
. If the inverse exists, they can be solved.
A.2 Proof of 
Under all the assumptions of the Basic Model, the probabilities of obtaining mi from M and of obtaining yi from Yi are ipi and pi, respectively. The likelihood of {mi} is
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i mi. The likelihood of yi is |
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, , {i}, and {pi} as 2s parameters of {
i} and {pi}, where |
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i} and {pi} are obtained by solving 
and 
. Here |
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and 
is equivalent to solving 
and 
, which means that we can estimate {i} from M and {mi}, and estimate pi "independently" from yi and Yi. Therefore, we can consider 
. A similar situation is seen in Darroch (1961), who considered the maximum likelihood estimation of migration rates for the case of one release and one recovery in multiple strata. |
Acknowledgements |
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We thank M. Ogura, Y. Uozumi, and J. Suzuki, of the National Research Institute of Far Sea Fisheries, Fisheries Agency of Japan, for allowing us to use the tag–recovery data of skipjack tuna and for making useful comments on our manuscript. W.S. Hearn of CSIRO, and H. Hilborn of the University of Washington provided valuable reviews of our manuscript. H. Matsuda, K. Tatsukawa, T. Katsukawa, and all the members of the Fish Population Dynamics Laboratory of ORI gave us encouragement, for which we are grateful. Finally, B.A. Megrey, the editor, greatly helped us to improve our manuscript.
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References |
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Top1 Introduction2 Methods3 DiscussionAppendixReferences |
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