estimation of juvenile survival, adult survival, and age-specific pupping probabilities for the...

Estimation of juvenile survival, adult survival, andage-specific pupping probabilities for the femalegrey seal (Halichoerus gryprus) on Sable Islandfrom capture–recapture data

Carl J. Schwarz and Wayne T. Stobo

Abstract: We use a longitudinal capture–recapture study from resightings of grey seals (Halichoerus gryprus) brandedas young on Sable Island to estimate (i) the juvenile survival rate from the time of branding to age 4, (ii) the yearlyadult survival rate from age 4 to age 9, and (iii) the age-specific pupping probabilities, i.e., the probability that a sealwill first give birth at each age. The estimated juvenile survival rate from branding (just after weaning) to age 4 rangedfrom 70 to 80%; however, the lower values are known to be biased low because the study was terminated early. Theestimated yearly adult survival rates for ages 4–9 ranged from 0.88 to 0.92·year–1. The estimated probabilities of firstgiving birth to a young seal (pupping) at ages 4–9 are 0.28, 0.41, 0.18, 0.06, 0.05, and 0.02, respectively, and the esti-mated average age of first pupping is 5.2 years.

Résumé : Nous avons réalisé une étude longitudinale de capture-recapture fondée sur les réobservations de phoquesgris (Halichoerus gryprus) marqués en bas âge sur l’île de Sable pour estimer (i) le taux de survie des juvéniles depuisle marquage jusqu’à l’âge 4; (ii) le taux annuel de survie des adultes de l’âge 4 à l’âge 9; et (iii) les probabilités demise bas selon l’âge, c’est-à-dire la probabilité d’une première mise bas à chaque âge. Le taux de survie estimé des ju-véniles depuis le marquage (juste après le sevrage) jusqu’à l’âge 4 était de 70 à 80 %, les valeurs les plus basses étantdes sous-évaluations parce que l’étude n’a pas duré suffisamment longtemps. Les taux de survie annuels estimés desadultes depuis l’âge 4 jusqu’à l’âge 9 variaient entre 0,88 et 0,92. Les probabilités estimées d’une première mise basentre les âges 4 et 9 sont respectivement de 0,28, 0,41, 0,18, 0,06, 0,05 et 0,02, et l’âge moyen estimé de la premièremise bas est de 5,2 ans.

[Traduit par la Rédaction] Schwarz and Stobo 253

Introduction

The key parameters in studies of population growth aresurvival, reproductive rates, and ages of first birth. In thispaper, we use a longitudinal capture–recapture study basedon resightings of grey seals (Halichoerus gryprus) brandedas young on Sable Island to estimate (i) the juvenile survivalfrom the time of branding to age 4, (ii) the yearly adult sur-vival rates from age 4 to 9; and (iii) the age-specific puppingprobabilities, i.e., the probability that a specific-aged sealwill give birth for the first time.

Previous studies (e.g., Mansfield and Beck 1977; Har-wood and Prime 1978; Hammill and Gosselin 1995) have es-timated these rates based on cross-sectional studies. Asample of seals was sacrificed and aged, and an internal ex-amination of the reproductive tracts revealed if the seal had

ever bred in previous years. Life table methods were thenused to estimate the adult yearly survival rates assuming thatthe population had a stable age distribution. A simple differ-ence of the proportion of ever-bred seals at each age esti-mates the individual age-specific pupping proportions.Finally, these two quantities and the rate of increase in thepopulation were used to derive the juvenile survival rates.

Clobert et al. (1994) presented a method to estimate theseparameters from a longitudinal capture–recapture study ofanimals marked as young. In this study design, successivecohorts of animals are marked as young at a known age.Prior to age k at which the youngest individual breeds, ani-mals cannot be observed. Once an animal starts to breed, itmay be recaptured or resighted. They fit a Cormack–Jolly–Seber (Cormack 1964; Jolly 1965; Seber 1965) capture–recapture model to the sighting histories. This gives directestimates of the juvenile survival and adult survival rates,and from changes in the capture probabilities during the ac-cession to breeding status, they estimate the age-specificbreeding proportions. Recently, Schwarz and Arnason (2000)developed a method to estimate these parameters from thisstudy design using a new parameterization of an ordinaryJolly–Seber model. Furthermore, they showed that theirmethod avoids many of the problems encountered by Clobertet al. (1994). Because seals have an approximate 12-monthgestation period, mating and breeding occurs approximately

Can. J. Fish. Aquat. Sci. 57: 247–253 (2000) © 2000 NRC Canada

247

Received June 9, 1999. Accepted October 27, 1999.J15179

C.J. Schwarz.1 Department of Statistics and Mathematics,Simon Fraser University, Burnaby, BC V5A 1S6, Canada.W.T. Stobo. Department of Fisheries and Oceans, BedfordInstitute of Oceanography, P.O. Box 1006, Dartmouth,NS B2Y 4A2, Canada.

1Author to whom all correspondence should be addressed.e-mail: [email protected]

J:\cjfas\cjfas57\cjfas-01\F99-259.vpFriday, January 14, 2000 9:27:54 AM

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1 year prior to giving birth to a pup, but the methods appliedto estimate first breeding times can be used with minor mod-ifications to estimate the first pupping times.

In this paper, we use the method of Schwarz and Arnason(2000) to estimate the survival and pupping rates. We com-pare our estimates with previously published estimates ob-tained from cross-sectional studies. Finally, we outline theadvantages and disadvantages of the two study designs anddiscuss some design considerations for the longitudinal study.

Methods

Sable Island is an island off the coast of Nova Scotia, Canada,and is the pupping site of one of the largest breeding colonies ofthe grey seal in the Northwest Atlantic (Mansfield and Beck 1977).In 1969, 1970, 1973, 1974, 1985, 1986, 1987, and 1989, samplesof pups born on the island were sexed and branded with uniquebrand numbers.

Seals return to Sable Island over a 6-week period starting inmid-December of each year to mate and to give birth. Departuresof females from the breeding colony occur over several weeks untilabout mid-February; the departing females will not rejoin thebreeding group until the next year. In each year, several surveyswere taken of the island searching for branded seals. The surveysare conducted by a team of four to six observers using small all-terrain vehicles to move among the seals. Each observer is as-signed an area of the breeding colony. All adult seals are scruti-nized and brand numbers read at distances ranging from 2 to 5 m.The individual numbers are at least 10 cm high on adults and usu-ally are easily discerned at these sighting distances, but errors canoccur (Schwarz and Stobo 1999). The first survey each year oc-curred after some animals had arrived, but few were believed tohave departed. Subsequent surveys were approximately 1 weekapart. We collapsed the multiple weekly surveys within the firstmonth into a single sighting value for that year. If a seal wassighted in at least one of the weekly surveys, it was treated as seenin that year.

The yearly sighting records define a history vector for each sealindicating when it was born, branded, and subsequently spotted.For example, a history vector 1000011010 for a seal indicates thatit was branded in 1985 (the first value that is a 1) and was seen onthe breeding grounds in 1990, 1991, and 1993 (presumably to givebirth to a pup) but was not seen in 1986, 1987, 1988, 1989, 1992,or 1994. However, just because the seal was “first” sighted on thebreeding grounds in 1990, this does not imply that she first gavebirth in 1990 (at age 5). She, for example, could have had a firstpup at age 4 (in 1989) but because of the timing of the survey, con-ditions during the survey, other logistical problems, or just bychance was not sighted. The Jolly–Seber model allows for the pos-sibility of these types of unknown events and based on the com-plete set of capture histories (or the sufficient statistics deducedfrom the histories) estimates the recapture rates, the interperiodsurvival rates, and the number of new entrants into the population,in this case, the number of seals giving birth to their first pup. Fe-male seals return both to mate and to give birth, but observers sel-dom saw a branded female seal without a pup beside her.Consequently, we believe the estimates for pupping probabilities tobe largely unbiased by the seals returning without a pup.

As in Clobert et al. (1994), we made the following assumptionsin addition to those commonly made in Cormack–Jolly–Sebermodels. (i) All female seals have a zero probability of resightingprior to the year of their first pup. (ii) All seals giving birth topups, whether for the first time or not, have the same probability ofresighting. (iii) All seals, regardless of whether they are a first-timeor a repeat mother, have the same probability of survival in a year.We make no assumptions about the survival rate of seals who have

not yet had their first pup until we estimate the juvenile survivalrate, at which time, it is assumed that the survival rates prior tofirst pup are the same as for other seals. (iv) All seals in each co-hort have pupped by the end of the study. Otherwise, the age-specific estimates will be conditional upon seals having their firstpup before the latest age in the study.

The basic principle of the Schwarz and Arnason (2000) methodis to ignore the initial capture at birth, treat the first resighting asan initial mark, and treat the second and subsequent resightings (inlater years) as recaptures after the initial mark. In this way, thepopulation of seals who have given birth to their first pup is treatedas an open population in the Jolly–Seber framework. Seals that pupfor the first time are treated as new entrants (“births”) into thepupping population. Schwarz and Arnason (1996) parameterizedtime-specific births in the Jolly–Seber model by the proportion ofthe total entrants over the course of the study following Crosbieand Manly (1985). This now corresponds directly to the age-specific proportions of seals giving birth for the first time. Theadult survival rates are an intrinsic parameter of the Jolly–Sebermodel and are estimated directly. Lastly, the ratio of the total ani-mals that returned to give birth to the number branded estimatesthe juvenile survival rate.

Figure 1 illustrates the parameters (recapture, survival, and age-specific pupping proportions) in a Jolly–Seber model of the sight-ing histories as they would apply in our study for seals branded atbirth in 1985–1987 and resighted in 1986 onwards with one addi-tional cohort representing seals from earlier branded cohorts.

Unfortunately, the most general model shown in Fig. 1 is notvery useful, as each cohort has its own set of parameters and thereis confounding of parameters at the start and end of the study, aslisted in Schwarz et al. (1993). This implies that not all parametersare estimable unless further assumptions are made. For example, inFig. 1, some of the confounded terms are the products b85,4 p85,89and b86,4p86,90, which imply that b85,4 and b86,4 cannot be estimated,or the products φ85,93p85,94 and φ86,94 p86,94, which imply that theparameters b85,9 and b86,8 cannot be estimated (see Table 1 for no-tation).

The easiest way to resolve the confounding at the end of thestudy is to ensure that the last capture occasion is sufficiently lateso that all cohorts have had their first pup. Unfortunately, this isnot possible for this study. However, if the yearly survival rate isvery high, then the product φ85,93 p85,94 essentially measures the fi-nal capture rate and the final age-specific pupping proportion canbe approximately resolved. As well, models where survival ratesare neither time dependent or cohort dependent, e.g., φ85,93 = φ85,92,also can resolve the confounding. (Of course, these models must beconsistent with the data at hand; otherwise, other biases are intro-duced into the estimates.)

Resolving the confounding at the start of the study can be doneby assuming that certain parameters are equal across cohorts or bymodeling the capture probabilities as functions of covariates. Forexample, one may be willing to assume that the capture rates arecohort independent, i.e., that p85,j = p86,j = p87,j. This would implythat all seals giving birth, regardless of cohort, have the same prob-ability of being recaptured in year j. This would allow the con-founding at the start of the study to be resolved. However, theconfounding for the first cohort would still be a problem becausethe first cohort’s first capture probability is not shared with anyother cohort. However, because we have a separate cohort of sealsthat have given birth previously that were marked prior to the startof the first recaptures of the new cohorts, this additional cohortcould be used to estimate the early capture probabilities (under theassumption of independence of recapture probabilities among co-horts) and this would fully resolve the confounding in all cohorts.

Alternatively, the capture probabilities could be modeled asfunctions of covariates (as was done in Clobert et al. (1994)), or

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248 Can. J. Fish. Aquat. Sci. Vol. 57, 2000



possibly, it may be tenable to consider models where the captureprobabilities are constant over time.

There could be a large number of possible models that will haveto be considered in reference to the data at hand. For example, thecapture rates could be cohort independent; survival rates year inde-pendent, etc. Some of the more interesting submodels are thosewhere the last age of first pupping is successively reduced to at-tempt to identify the age by which all animals have given birth totheir first pup. Another interesting set of models is where the age-specific pupping proportions are the same for all cohorts.

Burnham and Anderson (1998) suggested that model selectionbe done using both formal likelihood ratio tests and considerationof the Akaike information criterion (AIC) for the various models.The latter attempts to measure both the fit of the model and thenumber of parameters needed to describe the model, and modelswith similar AIC values should be considered as fitting equallywell. The various Jolly–Seber models described below were fit tothe data using the methods outlined in Schwarz and Arnason (1996)and the computer package POPAN (Arnason et al. 1998) based onthe usual summary statistics for capture–recapture data.

Following the model fit, the average age of first pupping was es-timated as

$ $

,A jbij

m

i j==∑

1

and the estimated standard error was found using a Taylor-seriesexpansion and the estimated variance–covariance matrix of the age-specific estimates.

The juvenile survival rate is estimated by extrapolating the num-ber of seals returning to give first birth at each age back to age 4assuming that the survival rates from age 4 onwards are applicableto these seals, i.e., for cohort i:

$

[ $ $ $ $ ($ $ ) ... ],

, , , , , ,θθ θ θ

ii i i i i iB B B

0 44 5 4 6 4 5

− =+ + +� �

Ni

Here, $

,φi a refers to the yearly survival rate for a seal of age a fromcohort i, which is found directly from year-specific survival rates(φ$ ),i j . The estimated standard error is again found by a Taylor-series expansion.

Results

The usual Jolly–Seber summary statistics for the 1985–1987 branded cohorts and a fourth cohort consisting of allbranded seals from previous cohorts are shown in Table 2.

Female seals were first observed returning to the breedinggrounds to give birth at age 4. Because no females were ob-served giving birth before age 4, some of the rows of thesummary statistics are zero. This is strictly an artifact of theway the data are presented for modeling — it is necessary tohave the same sampling occasions available for all cohorts —but it is a simple matter to constrain the sighting rates to bezero in these cases.

Because no seals were observed giving birth prior to age 4,the age-specific pupping rates are assumed to be zero forages 1–3. Unfortunately, this study was terminated in 1994(at age 9 for the 1985 cohort). Consequently, all estimatesare conditional upon seals first giving birth at or before age 9.Fortunately, Hammill and Gosselin (1995) found no evi-dence of first pupping after age 9 based on their cross-sectional studies, so we expect any biases resulting from thisconditioning to be slight.

© 2000 NRC Canada

Schwarz and Stobo 249

Fig. 1. Representation of the parameters for the four cohorts of grey seals in the study: pij is the probability that a seal in cohort i isseen in year j conditional upon it pupping in year j, φij is the probability that a seal alive in cohort i and in the population of havinggiven birth to a pup in year j is alive and in the pupping population in year j + 1, and bi,a is the probability that a seal from cohort iwill give birth for the first time at age a. Note that the last cohort consists of a combination of many previous cohorts, and hence, thebi,a should be interpreted as the proportion of animals from previous cohorts who give birth for the first time in calendar year a.



An initial model was fit allowing for each of the four co-horts to have their separate resighting, survival, and age-specific pupping proportions (log-likelihood value –1674.1with 57 parameters and AIC 3462.2). Each cohort was alsosubjected to the four goodness-of-fit tests described byPollock et al. (1990) and Pradel (1993). There was no evi-dence of lack of fit except for some evidence that animalsseen for the first time in any year (i.e., first time breeders)are somewhat less likely to be seen in subsequent yearscompared with animals first seen in previous years. A plot ofthe estimated recapture rates (Fig. 2) showed that these ratesare similar, but there is some evidence of differential sight-ing rates. For the 1985 and 1986 cohorts, there was some ad-ditional effort expended outside the weekly sighting trips tolocate animals of these cohorts for other studies, but the cur-rent data records and field notes do not indicate if this addi-tional effort was included in the sighting records.Unfortunately, this model does not allow estimation of allthe age-specific pupping proportions, and so, some restric-tions need to be applied to the resighting rates to allowestimability. A second model was fit where resighting rateswere independent of cohort (log-likelihood value –1683.5with 48 parameters and AIC 3463.0). A likelihood ratio testbetween these two models shows some evidence that thissimpler model may not be tenable (χ2 = 18.8, 9 df, p =0.02), but the AIC values are virtually the same so that ei-ther model may be acceptable. The significance level mayreflect the large power in this study with the large samplesizes present. Models with a constant cohort-specific annualadult survival rate were also tenable (log-likelihood value–1687.7 with 39 parameters and AIC 3453.4).

The estimated age-specific pupping proportions and aver-age age of first giving birth for this last model are shown inTable 3. Note that because of the termination of the study in1994, the 1986 and 1987 cohorts did not reach age 9, and sothe estimates are conditional upon first pupping at or beforethe last sightable age.

The average annual yearly survival rate is very close to0.90·year–1. Approximately 70–80% of seals survive fromthe time of branding to age 4. However, because the studywas terminated in 1994, the juvenile survival estimates forthe 1985 and 1987 cohorts are biased downwards becausesome seals in this cohort have not yet returned to give birthto their first pup and would not have been included.

A fourth model in which the age-specific pupping propor-tions were assumed to be equal in the 1985–1987 cohortswas fit (log-likelihood value –1695.05 with 39 parametersand AIC 3468.1) giving the estimates shown in Table 2.Note that because the latter two cohorts were not observeduntil age 9, they contributed information only on the age offirst pupping for the earlier ages. When compared with thesecond model, the likelihood ratio test (χ2 = 23.06, 9 df, p =0.006) indicated that this third model did not fit well com-pared with the second model and the change in AIC was rea-sonably large. Nevertheless, this provides a reasonablesummary of the age-specific pupping proportions and the av-erage age of first birth from the three cohorts.

Discussion

Hammill and Gosselin (1995) reported mean ages of firstbirth ranging from 3.61 (SE 0.76) to 5.06 (SE 0.12) years, as

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Parameters

bia Probability that an animal in cohort i that survives until it gives birth for the first time will give birth at age a; i = 1,..., G;

j = 1,..., m; biaa

m

=∑ =

1

1

Bia Number of animals in cohort i that survive until they give birth for the first time at age a; i = 1,..., G; j = 1,..., mφij Probability that an animal in cohort i that is alive in year j and has given birth at least once will be alive in year j + 1;

i = 1,..., G; j = i,..., T – 1φi, 0 4− Probability that an animal in cohort i that was branded will survive to age 4. This is the juvenile survival rate

pij Probability that an animal in cohort i that has previously given birth or that will give birth in year j will be recaptured inyear j; animals that have not previously given birth or are not giving birth have a zero probability of being recaptured;i = 1,..., G; j = i,..., T

Ai Average age of first pupping for cohort i; A jbi i jj

m

==∑ ,

1

StatisticsNi Number of animals branded as young in cohort inij Number of animals in cohort i, marked at age 0, that are recaptured in calendar year j; i = 1,..., G; j = i,..., Tmij Number of animals in cohort i, marked at age 0, that are captured in year j and were previously recaptured prior to year j;

i = 1,..., G; j = i,..., TRij Number of animals in cohort i, marked at age 0, recaptured in year j and released in year j; Rij may differ from nij because

of losses on capture or injections; i = 1,..., G; j = i,..., Trij Number of animals in cohort i, marked at age 0, from Rij that are subsequently recaptured after release; i = 1,..., G; j = i,..., Tzij Number of animals in cohort i, marked at age 0, recaptured before year j, not recaptured in year j, and recaptured after year

j; i = 1,..., G; j = i,..., T

Note: For convenience, we assume that animals give birth for the first time at age 1, that all animals have given birth for the first time by age m, thatsuccessive capture occasions are 1 year apart, and that observations on the population start in calendar year 1, the first year when the first cohort givesbirth for the first time. Observations continue until calendar year T with T ≥ m.

Table 1. Notation used in the paper. In this study, there are G cohorts of animals marked as young (for simplicity, at age 0).



determined by examining the corpora lutea in the wall of theuterus, and from 5.03 (SE 0.22) to 6.08 (SE 0.32) years, asdetermined by examining for the presence of a fetus. Our re-sults are comparable. Their estimated age-specific first preg-nancy proportions were 0.01, 0.44, 0.45, and 0.10 for femaleseals aged 3, 4, 5, and 6+ years, respectively, as determinedby examining the corpora lutea, and 0.18, 0.68, 0.02, and0.12 for female seals aged 4, 5, 6, and 7+ years, respec-tively, as determined from the presence of a fetus. Harwoodand Prime (1978) estimated these proportions as 0.16, 0.45,and 0.39 for ages 5, 6, and 7+, respectively. Mansfield andBeck (1977) obtained values of 0.16, 0.55, 0.18, and 0.11for ages 4, 5, 6, and 7+, respectively. Our results are compa-rable with these.

The greatest value of our analysis is that it provides rates basedon successful pregnancies. This could be particularly important foryounger animals for which ovulation but no fertilization occurs orfor which spontaneous abortion is prevalent.

The life table methods of Mansfield and Beck (1977) andHarwood and Prime (1978) gave estimated annual adult survivalrates of 0.87 and 0.94·year–1, respectively. Zwanenburg andBowen (1990) estimated that the survival rate was 0.96·year–1 inorder to obtain a 12.6% annual increase in pup production. Ourestimates are again similar.

The life table approaches cannot unambiguously estimatejuvenile survival rates, as these are determined by the re-quirement that the population growth from the derivedLeslie matrix match the observed population growth. Har-wood and Prime (1978) found that the juvenile survival ratecould range from 0.41 to 0.61 in their model and arbitrarilychose the value 0.49. Zwanenburg and Bowen (1990) used avalue of 0.79 in their model. On the surface, our estimatesappear to be much higher; however, our estimates excludemortality from birth to the time of branding, which cannotbe estimated from this study. Consequently, our estimatescannot be used directly in a Leslie matrix without this fur-ther information. The estimates are consistent with the lowerbound derived from the minimum number of seals alive(MNA = Σ(ni – mi)), which range from 0.60 to 0.72. Theminimum number alive is known to be an underestimate ofthe total number alive, as it does not account for the sealsalive but not seen and consequently provides a lower boundto the juvenile survival rate.

Similar methods could be used to estimate the populationparameters for the males. However, “first breeding” must becarefully defined for a male, as during the intervening periodbetween sexual and social maturity, these males are forcedto the periphery of the breeding colony, and infrequently ob-served, and only occasionally succeed in copulating with fe-males (Hewer 1961; Godsell 1991). As such, first breedingcould be defined as first attaining social dominance. Unfor-tunately, the current study was terminated before sufficientdata could be accumulated on the males.

Our proposed method has a number of advantages overthe longitudinal methods used by Clobert et al. (1994), asdiscussed by Schwarz and Arnason (2000). In particular, the

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Resighting year j nij mij Rij rij zij

Seals branded in i = 1985; total branded 4001987 0 0 0 0 01988 0 0 0 0 01989 35 0 35 32 01990 114 17 114 108 151991 180 95 180 152 281992 174 137 174 132 431993 155 131 155 92 441994 146 136 146 0 0



Seals branded in i = all previous years1987 514 0 514 366 01988 252 234 252 232 1321989 252 229 252 221 1351990 274 258 274 245 981991 295 276 295 254 671992 240 237 240 208 841993 241 231 241 178 611994 248 239 248 0 0

Table 2. Summary statistics for the 1985–1987 branded cohortsof female grey seals and a fourth cohort consisting of all previ-ously branded seals.

Fig. 2. Plot of the estimated resighting probabilities from the fullmodel allowing grey seal each cohort to have its own parame-ters. Note that in the full model, the resighting probabilities areconfounded for the first and last year of sightings and cannot beindividually estimated.



method is much simpler conceptually and does not require acomplex model that tries to account for the heterogeneity insighting rates caused by pooling both breeders and non-breeders. Pradel (1996), Pradel et al. (1997), and Pradel andLebreton (1999) used a slightly different method to estimatethe age-specific breeding proportions based on reading thecapture histories “backwards” in time. However, as shownin Schwarz and Arnason (2000), Pradel’s age-specific breed-ing proportions are conditional upon animals surviving untilthe age at which all animals have become breeders. Forlong-lived animals, his estimates of the age-specific breed-ing proportions should be very similar to our estimates;however, for short-lived animals, his method will tend tooverestimate the proportion in the older age-classes and un-derestimate the proportion in the younger age-classes, whichwill lead to a positive bias in the estimate of the average ageof breeding.

There are a number of advantages and disadvantages tocross-sectional versus longitudinal designs. In a cross-sectional study, it is only necessary to sample in one year, soresults can be obtained relatively quickly. However, a key as-sumption is that the seals of each age-class are a randomsample of all seals alive at that age. If nonbreeders are es-sentially uncatchable, then it may be difficult to obtain sucha sample. For example, if sampling were restricted to SableIsland in January and February, then only seals giving birthwould be sampled and a cross-sectional study would not bevalid. In a longitudinal study, animals that have a very longbreeding entry curve will require a study that spans manyyears. This may be too costly. Furthermore, either anotherset of known breeders must be additionally marked or someadditional assumptions about the equality of parametersacross cohorts must be made to ensure that parameters areidentifiable. In a cross-sectional study, the seal must be sac-rificed to determine if she has previously given birth and toage the animal; this may not be acceptable for endangered

populations or may not be wise from a public relations view-point. In the longitudinal study design, the marked animalonly need to be individually observable and breeders distin-guishable from nonbreeders. In this study, we are makingthe explicit assumption that a seal first returns to Sable Is-land only when ready to give birth, and hence, any seal ob-served will be pupping. This assumption is reasonable forSable Island grey seals because these mothers stay with theirpups continuously throughout the lactation period (Bowen etal. 1992). It is not necessary to assume site fidelity. If siteswitching (e.g., between the Gulf and Sable islands) occurs,then as long as the temporary emigration is random (i.e.,does not depend on what occurred in previous years), thisjust reduces the effective sighting probabilities but does notaffect survival or reproductive estimates (Burnham 1993).Of course, if no animals have been previously tagged, then alongitudinal study is impossible and a cross-sectional studymay be the only feasible study. While advances in DNAtechnology or self-marking schemes (e.g., individuals identi-fiable from photographs) may make it feasible to conduct alongitudinal study, achieving the necessary sample sizewould be problematic.

It is also difficult to envision how a cross-sectional studycould be used to investigate the age of first social dominancefor males. In theory, this could be attempted using our meth-ods in a longitudinal study.

Lastly, in all longitudinal studies of this type, there areseveral issues that must be carefully considered during thedesign phase. First, the study must be long enough that allanimals in at least one cohort enter breeding status beforethe end of the study. Otherwise, the estimates of the age-specific breeding proportions will be conditional upon enter-ing breeding status by the end of the study. Our study suffersfrom this flaw, but we do not believe that it causes seriousbias because previous work has shown that most seals havegiven birth to their first pup by age 9. Second, great care

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Model allowing for separate age-specific pupping proportions for each cohort Model with commonproportions for all cohorts1985 cohort 1986 cohort 1987 cohort

Estimate Estimated SE Estimate Estimated SE Estimate Estimated SE Estimate Estimated SE

Juvenile survival(φi,0–4)

0.83 0.04 0.77a 0.03 0.70a 0.02

Adult survival (φi) 0.92 0.014 0.91 0.017 0.88 0.03

Age of first pup4 0.19 0.032 0.39 0.041 0.26 0.033 0.28 0.0215 0.40 0.047 0.39 0.048 0.47 0.047 0.41 0.0276 0.23 0.044 0.14 0.038 0.19 0.045 0.18 0.0237 0.09 0.031 0.04 0.026 0.08 0.033 0.06 0.0158 0.07 0.025 0.04 0.020 Unknownb — 0.05 0.0139 0.02 0.018 Unknownb — Unknownb — 0.02 0.011

Average age 5.5 0.09 4.9 0.082 5.1 0.076 5.2 0.061

Log-likelihood value –1687.7 –1695.05aBecause the study was terminated early, these values will be biased downwards. Refer to text for more details.bBecause the study was truncated before this age-class recruited, it is impossible to estimate the proportion for this age-class and the estimates in its

column are conditional upon first pupping at or before the last observable age.

Table 3. Estimates of the age-specific grey seal pupping proportions from the 1985–1987 cohorts when a model is fit with separateproportions for each cohort and when a common model is fit to all cohorts.



© 2000 NRC Canada


must be taken to properly define the study population and toensure that this definition does not change over time. For ex-ample, if the area searched for seals giving birth changesover time, then the estimates of the age-specific puppingproportion will be confounded with changes in the studyarea because recruitment is now confounded with populationchanges in the different-sized study area. In our case, thiswas not a problem because Sable Island is relatively smalland all breeding areas, both old and new, were easily dis-cerned by both aerial and ground preliminary surveys. Al-though the population on Sable Island has been increasing ata rate of about 12%·year–1, with both expansion of the geo-graphic area occupied by existing breeding concentrationand development of new areas (see Stobo and Zwanenburg1990), the survey design was expanded with the populationincrease to ensure that all breeding areas were surveyed withequal intensity throughout the study period. Third, becausethese estimates are directly related to estimates of “births” inthe ordinary Jolly–Seber method, all of the problems associ-ated with heterogeneity in capture probabilities and bias inthe estimate of birth” will also be important here. Becausenumerous trips were made within each study year and thesame trained observers were used, we do not feel that hetero-geneity of sighting probabilities is a serious problem. Fourth,a sufficiently large number of animals must be marked asyoung to ensure that adequate sample sizes will be observedfrom the subsequent breeders. In this study, we were fortunatethat many seals were branded as young; a simulation studycould be used to examine if a proposed program leads to esti-mates that are sufficiently precise to be useful.

Acknowledgments

This work was supported by Natural Science and Engi-neering Research Council of Canada research grants.Funding for the development of POPAN was provided byManitoba Hydro.

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