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Ecology, 83(12), 2002, pp. 3257–3265 ᭧ 2002 by the Ecological Society of America ESTIMATING POPULATION PROJECTION MATRICES FROM MULTI-STAGE Biology Department MS 34, Woods Hole Oceanographic Institution, Woods Hole, Masachusetts 02543 USA Multi-stage mark–recapture (MSMR) statistics provide the best method for estimating the transition probabilities in matrix population models when individual capturehistory data are available. In this paper, we improve the method in four major ways. Weuse a Markov chain formulation of the life cycle to express the likelihood functions inmatrix form, which makes numerical calculations simpler. We introduce a method to in-corporate capture histories with uncertain stage and sex identifications, which allows theuse of capture history data with incomplete information. We introduce a simple functionthat allows multinomial transition probabilities to be written as functions of covariates (timeor environmental factors). Finally, we show how to convert transition probabilities estimatedby the MSMR method into a matrix population model. These methods are applied to dataon the North Atlantic right whale (Eubalaena glacialis).
capture–recapture studies; Eubalaena glacialis; multi-stage mark–recapture statistics; Markov chain; matrix population models; North Atlantic right whale; population projection matrix;survival probability; transition probability. developed to estimate probabilities of movementamong spatial locations (Arnason 1972, 1973, Brownie Mark–recapture estimates of survival probability et al. 1993, Lebreton 1995). For the MSMR method, have been applied to many animal populations (e.g., in addition to information on whether or not each in- Lebreton et al. 1992, Forsman et al. 1996, Weimer-skirch et al. 1997, Hastings and Testa 1998, Caswell dividual was captured, the capture history data must et al. 1999, Pease and Mattson 1999), and this method also include the stage of captured individuals at each has become an important tool in population manage- capture occasion. MSMR models account for inter- ment. Mark–recapture estimates are based on capture group heterogeneity in survival and capture probability histories of individually identified animals, which con- by grouping similar individuals into stages. The de- tain information on whether or not each individual was velopment from single-stage to multi-stage mark–re- captured at each sampling occasion. For example, cap- capture statistics parallels the development from un- ture history data may be obtained by annual observa- structured to structured population models. In fact, one tions of banded birds or photographically identified motivation for the statistical development was the need whales. When such data are available, mark–recapture to estimate parameters in stage-structured matrix pop- statistics are considered one of the best approaches for ulation models from mark–recapture data (Nichols et Modern demographic analysis goes beyond calcu- The analysis is based on maximization of a likeli- lating survival, by breaking the life cycle into stages hood function that depends on all of the possible se- (which may be based on age, size, developmental or quences of stage transitions compatible with an ob- behavioral states, physiological condition, spatial lo- served capture history. There can be very many of these cation, or any other property that divides individuals sequences, and one of the most complicated parts of into subgroups). The fate of individuals is described in the method of Nichols et al. (1992) is writing them all terms of transition probabilities among these stages, down with their associated probabilities. In this paper, and those transition probabilities form the basis for we describe the life cycle as a Markov chain, and take matrix population models (Caswell 2001). Nichols et advantage of this description to write the likelihood in al. (1992) introduced a method to estimate transition a simple matrix notation. A sketch of this method was probabilities among stages from mark–recapture data, given in Caswell (2001: Section Here we give which we call the multi-stage mark–recapture (MSMR) a complete presentation, and extend the method to in- method. This method extends the method originally corporate uncertainty in stage and sex identifications,which allows the use of capture histories containing Manuscript received 30 April 2001; revised 13 March 2002; incomplete information. We also introduce a simple accepted 5 April 2002; final version received 25 April 2002.
function that allows multinomial transition probabili- 1 Present address: Department of Ecology, Evolution and Marine Biology, University of California, Santa Barbara, Cal- ties to be written as a function of covariates (e.g., en- ifornia 93106 USA. E-mail: vironmental variables or time). Finally, we show how 1), immature individuals (stage 2), and mature indi-viduals (stage 3). In addition to these three stages, fe-males also have a stage for individuals nursing a calf(stage 4); we call the individuals in this stage‘‘mothers.’’ Stage 0 corresponds to death, and the prob-abilities ␾ associated with the arrows going to stage 0 are stage-specific mortality rates. As usual, ‘‘mor-tality’’ includes both death and permanent emigration.
The objective of the MSMR approach is to estimate the transition probabilities associated with each arrowand the capture probabilities of each stage. In the nextsection, we will show how to construct matrices con-taining the transition and capture probabilities, and toaccount for uncertainty in the assignment of individualsto stages. Then we will show how to calculate the like-lihood in terms of these matrices.
Transition and capture probability matrices The transition matrix is constructed by first putting the transition probability ␾ from (living) stage i to (living) stage j in the ( j, i) position. To this matrix isappended a row containing the probabilities of tran-sition from each stage to stage 0 (death) and a column A stage structure for (a) female and (b) male right containing the probabilities of transition from stage 0 whales. This structure is used as an example for the MSMR to each stage. Because we treat death as a stage, the result is the transition matrix of an absorbing Markovchain, with death as an absorbing state. The matrix is to convert the estimated transition probabilities into a column stochastic. The ability to treat transitions as a Markov chain is critical to our analysis. The transitionmatrix for females, corresponding to the stage structure The MSMR method involves three main steps: (1) constructing an appropriate stage structure; (2) ex-pressing the likelihood function in terms of parameters, based on available capture histories; and (3) finding the best parameter estimates using maximum likelihood theory. The parameters in the MSMR model are those that define the capture probabilities of each stage ateach sampling occasion and the transition probabilities where ␾ (t) is the probability of females making tran- among stages between consecutive sampling occasions.
sition from stage i to j between time t and t ϩ 1. The The method assumes that individuals in the same stage upper left block of the matrix describes the transition are identical and independent, but that individuals in among live stages; the lower left block of the matrix different stages may differ in their transition and cap- contains stage-specific probabilities of death. The 1 in ture probabilities. The MSMR method is very flexible the (5, 5) entry is the probability of dead individuals and can be applied to almost any stage structure. Con- remaining dead in the following year. The notation ␾ (t) structing a useful stage structure that is compatible with in this paper corresponds to ␾ in Nichols et al. (1992).
the life cycle of populations requires experience, in Similarly, the transition matrix for males, corre- addition to sufficient mathematical and biological sponding to the stage structure in Fig. 1, is knowledge, and different stage structures are exten-sively reviewed in Caswell (2001). In this paper, meth- ods for expressing the likelihood function and esti- mating parameters are described, assuming that an ap- propriate stage structure has been constructed.
To make our discussion more concrete, we will dem- onstrate the method using a stage structure (Fig. 1) developed to describe the life history of the North At- of parameters ␪. These parameters can be the ␾ them- lantic right whale (Eubalaena glacialis). This is a two- selves, or lower level parameters from which the ␾ji sex, multi-stage model that distinguishes calves (stage can be calculated. The objective is to estimate ␪.
Capture probability matrices P , defined for females
If the stage of the individual is known with certainty, and males separately, contain stage-specific capture its stage-assignment matrix contains a one in the cor- probabilities on the diagonal and zeros elsewhere: responding diagonal entry and zeros elsewhere. On theother hand, if the stage of an individual is completely unknown, the identity matrix can be used for specifies a uniform probability distribution over the possible stages. Alternatively, if an independent as- sessment of the probability is available, it can be en- tered into the matrix. For example, in an age-structured model of fish, the age of fish is sometimes determined from their length using age–length keys (e.g., Fournier and Archibald 1982, Deriso et al. 1985, Quinn and Deriso 1999). Such a key could provide the probability distribution of ages of the fish, which can be entered into the stage-assignment matrices, for an age-struc-tured model.
where p (t) is the probability of capturing individuals in stage j at time t. This notation corresponds to p in assume that dead individuals are never captured ( (f) The likelihood of the parameter vector ␪ contains contributions from the capture history of each individ- ϭ 0), but such captures could be included.
Transition and capture probability matrices can be ual. We denote by l (␪) the contribution to the likeli- defined separately for females and males when infor- hood from individual k; it is proportional to the prob- mation on sex identification is available, as in our ex- ability of the capture history. That probability is the ample, but it is not always possible or necessary to sum of the probabilities of all possible sequences of have a two-sex model. In such cases, only a single transitions that could have been taken by the individual transition and capture probability matrix is needed.
k. There may be many such possibilities. Their sum,however, can be calculated using the transition, capture probability, and stage-assignment matrices by the fol- A stage-assignment matrix is defined for each in- lowing algorithm. We assume that the individual is first dividual each time it is captured. The diagonal elements of the matrix are proportional to the certainty of stageidentification at time t (i.e., to the probability that the 1) Categorize individual k by its stage at its first individual is in a given stage when it is captured). This capture, taking uncertainty in stage assignment probability should be known prior to estimating tran- sition and capture probabilities. In our example, indi- vidual k is a female; its stage-assignment matrix is where e is a vector of ones. This product is a
vector whose entries are proportional to the prob- abilities of the initial stage of the individual at t.
2) Calculate the probability distribution of the stage at t by multiplying this vector by the transition ⌽ U e. (8)
u (t) is the probability that individual k at time t is in stage j ( j ϭ 1, 2, 3, 4). Similarly, if individual 3) Calculate the probabilities of observation out- k is male, its stage-assignment matrix is comes at t . If individual k was captured at t , multiply by the sighting matrix P :
Pt2⌽ U e. (9)
Ut ϭ 
If individual k was not captured at t , multiply by t2 ⌽ U
Because we assume that the capture probability of dead where I is the identity matrix.
into the likelihood calculations. Multiplication of 4) Account for stage identification at t by multiply- by a scalar has no effect on the maximum likelihood ing by the stage assignment matrix. If individual Some possible capture histories of North Atlantic ⌽ U e. (11)
right whales corresponding to the example stage structurein Fig. 1 and their likelihood.
If individual k was not captured at t , 5) Repeat steps 2–4 until the end of the capture his- tory for individual k. The result is a vector whose ⌽ U(1)P
ith entry is proportional to the probability of all eT(I Ϫ P )⌽ U (2)P U (2)P U(2)
the pathways by which individual k could have eTU (3)P ⌽ (I Ϫ P )⌽ U (3)P U (3)
eTU (4)P U (4)P ⌽ (I Ϫ P )⌽ U (4)
moved from its initial stage at t to stage i at t eTU (5)P U (5)P U (5)
and that are compatible with its capture history.
eT(I Ϫ P )⌽ (I Ϫ P )⌽ U (6)P U (6)
6) The final step is to sum the resulting vector of eTU (7)P U (7)
eT(I Ϫ P )⌽ U (8)P U (8)
eTU (9)P ⌽ (I Ϫ P )⌽ I Ϫ P )⌽ U (9)
Ϫ P )⌽ U(10)P ⌽ (I Ϫ P )⌽ U(10)
␪) ϭ e U · · ·
U e. (13)
eTU (11)P ⌽ (I Ϫ P )⌽ U (11)
eT(I Ϫ P )⌽ U (12)
In this algorithm, the probability distribution of the eT(I Ϫ P )⌽ (I Ϫ P )⌽ U (13)
eT(I Ϫ P )⌽ (I Ϫ P )⌽ (I Ϫ P )⌽ U (14)
individual’s stage is updated sequentially over time,taking into account the new data available at each time Notes: When the stage of the captured individual is i, U (k) is a matrix with 1 in the
ith row of the ith column and step and possible stage transitions determined by the 0 elsewhere. Terms are as follows: ⌽ , transition probability stage structure. Therefore, the right-hand side of Eq.
matrix at time t; U(k) , stage assignment matrix for individual
13 is the probability of the capture history for indi- k at time t; P , capture probability matrix at time t; I, identity
matrix; e, vector containing 1’s in its entries.
vidual k, taking into account all possible transition se- † X indicates that the individual was not captured; numbers quences compatible with that history.
indicate the stage of captured individuals.
The likelihood l (␪) is calculated using only female or male matrices if the sex of individual k is known.
If the sex of individual k is uncertain, algorithm (13) Here, we assume that individuals are captured and make stage transitions independently, but based on female- and male-specific matrices, respectively. Then, identical probability distributions (i.e., we assume that the number of outcomes falling into the possible cap- ture history sequences is multinomial).
l (␪) ϭ p l (␪) ϩ (1 Ϫ p )l (␪) Maximum likelihood estimates (␪ˆ) are found by where p is the probability that the individual k is fe- maximizing L(␪). The likelihood function can be max- male. The probability p is 1 or 0 when the sex of the imized numerically using software such as MATLAB individual is known to be female or male, respectively.
(1999). For example, the MATLAB routine ‘‘fminu()’’ If the sex of the individual is unknown, a probability can be used to find the maximum likelihood by mini- must be provided to calculate the likelihood.
Some examples of probabilities of the capture his- tories of individuals with four capture periods are shown in Table 1. Because our example contains mul-tiple stages, many possible capture histories exist, of Transition probabilities ␾ (t) may change over the which only a few are shown in Table 1. For simplicity, course of a study, and the changes may be correlated we assume that the sex of all individuals is known to with various factors. We would like to model the prob- abilities as functions of covariates measuring those fac- None of the likelihoods in Table 1 contains P , be-
tors. For example, population density and sampling ef- cause the probability of a capture history is always fort were used to model the survival and capture prob- conditional on the first capture; therefore, capture prob- abilities in studies of the roe deer (Capreolus capreo- ability at the first sampling time cannot be estimated.
lus) and the common lizard (Lacerta vivipara), For the same reason, the likelihoods of individuals 5, respectively (Lebreton et al. 1992), and time has been 7, 8, 11, 12, and 13 do not begin at time t ϭ 1, because used to model the survival probability of the Northern capture histories prior to the first capture of an indi- Spotted Owl (Strix occidentalis caurina; Forsman et vidual do not enter into probability calculations.
al. 1996) and the North Atlantic right whale (Caswell Given the likelihood functions l (␪) for all individ- uals, the likelihood associated with the data consisting Covariates are incorporated in the transition proba- of n capture histories is proportional to the product of bility using a link function. The link function must satisfy the constraint that each column of the transitionmatrix sums to 1, and each entry of the matrix must lie between 0 and 1. A flexible function that satisfies these properties is the polychotomous logistic function, which is derived by expressing the log of the odds ratio individual enters this stage, it gives birth; therefore, as a linear function of the covariates (Hosmer and Le- transition probabilities into the fertile stage are also probabilities of giving birth. If the number of female time t. The polychotomous logistic function is and male births at each reproductive event are b and b , respectively, the fertility terms in the projection matrix are given by the product of the number of off- F (t) ϭ b ␾ (t) F (t) ϭ b ␾ (t) where ␣ is an intercept parameter, and (d) parameter associated with the dth covariate. When all of the slope parameters are zero for all d, i, and j, the F (t) ϭ b ␾ (t).
transition matrix is constant over time. The simple lo- An important assumption in these fertility terms is that gistic function that is often used in mark–recapture mothers and their newborns have the same probability literatures (e.g., Burnham et al. 1987, Lebreton et al.
of being captured during sampling. To ensure that this 1992) is a special case of the polychotomous logistic condition is satisfied, when mothers are captured, their offspring should also be captured and entered into the database as new individuals. Similarly, when newborns Population projection matrices contain both transi- are captured, their mothers should be captured and tion probabilities and fertilities (see Caswell 2001). Be- identified as mothers. Later, we will show one example cause the transition probabilities are estimated by the of remedial methods when the equal ‘‘capturability’’ MSMR method, we can construct the projection matrix if we know the fertility terms. In this section, we show Model (17) is female-dominant; males do not affect an example of how those terms might be obtained, and population dynamics. This assumption is often legiti- how to compute confidence intervals for population mate when the population size of males is large enough growth rate calculated from the population matrix.
that searching for a partner does not limit reproductionby females. Thus, for calculation of population growth Conversion from a transition matrix to a population rate, the two-sex matrix may be reduced to the female The right whale example provides enough infor- mation to write a two-sex model. To do so, we renumber the male stages in Fig. 1 as 5, 6, 7. Letting ␾ (t) denote At ϭ 
the transition probability as before, the projection ma- ␾ (t) ␾ (t) ␾ (t) Confidence intervals for population growth rate The long-term population growth rate implied by a projection matrix A is given by the dominant eigen-
␾ (t) ␾ (t) ␾ (t Η) 0 value ␭ of A . A confidence interval for ␭ can be ap-
proximated from the MSMR statistics, using the ei-genvalue sensitivity formula and the covariance matrix The upper-left and lower-right blocks describe produc- tion of females by females and males by males, re- spectively. The entries in the lower-left block describeproduction of males by females.
where v and w are the left- and right-dominant eigen-
When constructing a population projection matrix, vectors of the population projection matrix (Caswell transition probability and fertility terms are often es- 1978, 2001). If the a are functions of some other pa- timated from two separate data sets (Caswell 2001), rameters ␪ , the sensitivity of ␭ to ␪ is: but the fertility terms can be estimated directly using the MSMR method if the stage structure includes moth- ers that give birth between two consecutive sampling periods (i.e., stage 4 in our example). Each time an Now let ␪ be a vector of parameters estimated by the MSMR method. An approximate 95% CI for ␭ is cal- When the stage of a captured individual is uncertain, the (2, 2) entry of the stage-assignment matrix is the probability that the individual is immature, given that the stage is uncertain. Similarly, the (3, 3) entry is the Ίq,r ΂ץ␪ ΃q΂ץ␪΃r probability that the individual is mature, given that the where is the (qr)th entry of the estimated covariance stage is uncertain. The other entries are all zero. To ˆ . The covariance matrix C can be estimat-
express these probabilities in mathematical form, let X ed by inverting the Hessian matrix (the information be a random variable giving the stage of an individual matrix; e.g., Burnham et al. 1987). This method of and let Y be a random variable taking the value 1 if constructing the confidence interval is an application the stage is known and 0 if the stage is uncertain. Then of the delta method (see Seber 1982: Chapter 1), taking advantage of the existence of the eigenvalue sensitivity u2 ϭ Pr(X ϭ 2 ͦ Y ϭ 0) u ϭ Pr(X ϭ 3 ͦ Y ϭ 0) ϭ 1 Ϫ u .
To calculate Pr(X ϭ 2 ͦ Y ϭ 0), we use Bayes’ Rule toderive We have applied the MSMR method to data on the North Atlantic right whale (Eubalaena glacialis). The northern right whale is considered one of the most en- Pr(X ϭ 2) Ϫ Pr(X ϭ 2 ͦ Y ϭ 1)Pr(Y ϭ 1) dangered large whale species in the world (Waring et al. 1999). The current population in the western NorthAtlantic contains fewer than 300 individuals. They mi- Here, Pr(Y ϭ 1) is the probability that the stage of an grate from the Bay of Fundy, which is a summer feed- immature or mature individual is known, and can be ing ground, to the coast of Florida, which is a winter estimated from the capture history data as calving ground. Caswell et al. (1999) showed that thecrude survival probability of individuals in this pop- ulation has been declining since 1980.
Data on the North Atlantic right whale have been where N , N , and N are numbers of captures of im- collected by the New England Aquarium and consist mature, mature, and uncertain stages, respectively. Pr(X of annual sighting histories of photographically iden- ϭ 2 ͦ Y ϭ 1) is the probability that the stage of an tified animals from 1980 to 1997 (Crone and Kraus immature or mature individual is immature, given that 1990). For the purpose of our analysis, we consider the stage is known. This probability can be calculated individuals to have been marked on the occasion of their first identification, and recaptured when they wereresighted during a subsequent year. Of the 372 indi- viduals used for the analysis, 141 are known to be females and 143 to be males. We assumed the remain-der to be either female or male with 50% probabilities.
Finally, Pr(X ϭ 2) is the probability that the stage is A few sightings of dead individuals exist, but are not immature, given that the stage is either immature or mature, regardless of whether the stage is known oruncertain. To estimate this probability, we estimated the parameters for a time-invariant projection matrixfrom the subset of the data containing only certain cap- We attempted to assign each individual at each cap- tures. From the stable stage distribution w (i.e., the
ture to one of the stages shown in Fig. 1. A whale wasconsidered mature if it was known to be Ն9 yr old or, right eigenvector associated with the dominant eigen- for females, if it had been observed with a calf. Stages value) of this matrix, we calculated the proportion of could be assigned with certainty in 78% of the captures.
individuals in stage 2 among stages 2 and 3 and used The remainder were known to be either immature or mature; for these captures, we must calculate the entries u , u , u , and u of the stage-assignment matrices (5) and (6). In the absence of information to the contrary, we assume that these probabilities are constant over For males, the same method was applied to the male time and across individuals, but differ between females stages (5 and 6). It should be noted that these calcu- and males. Because we use different criteria to assign lations work best when the capture probabilities of females and males to stages, we expect that the prob- stages 2 and 3 (5 and 6 for males) are similar. Other- ability distribution of stages among the unknown- wise, each count in (29) and (30) should be divided by staged captures would differ for females and males.
the corresponding capture probability (Nichols et al.
Dependence of the best capture model for the North Atlantic right whale on effort level and time.
This matrix is the same as (22), but with a particular set of assumptions defining the fertility terms. Consider F (t) in (22). When a female moves from stage 2 to † The sighting probability of calves cannot be estimated stage 4 (with probability ␾ ), she gives birth; the new- because the capture of a calf is always the first capture of born is female with probability 0.5. To appear as a calf in stage 1 at t ϩ 1, the newborn calf must survive long ‡ The Northern region includes Bay of Fundy, Brown’s Bank, Great South Channel, and Massachusetts Bay.
§ The Southern region includes the coast of Florida and 1994). The end result of these calculations is in u ϭ 0.87, u ϭ 0.13, u ϭ 0.30, and u ϭ 0.70.
Capture probabilities were modeled as binary logis- tic functions of estimated sampling effort levels in thenorthern and southern regions, which are major feedingand calving grounds, respectively. These effort levelswere approximated by the number of sampling datesper year in each region. We created models by includingall possible combinations of effort levels for all pos-sible combination of stages. This resulted in 1024 mod-els. The best capture model among the 1024 candidatemodels was selected using Akaike Information Criteria,AIC (Akaike (1973)). Because the sample size is large,we did not use the small-sample adjustment to AIC(i.e., AIC in Burnham and Anderson [1998]). The dif- ference in AIC between the best and the second-bestcapture models was about 2, indicating that the supportfor the best model relative to the second best model ishigh (Burnham and Anderson 1998). Furthermore, thefour best models differ only in how capture probabilityof mothers depends on effort; in all cases, the captureprobability was consistently close to 1 throughout thesampling period. Therefore, we used only the best mod-el shown in Table 2. The capture probabilities of im-mature males and females did not differ significantlyin the best model, based on a likelihood ratio test.
Therefore, we set these two capture probabilities equaland used the resulting capture probability model forfurther analysis.
Although we know that the vital rates have varied over time (Fujiwara and Caswell 2001), for this ex-ample we fit a model in which the transition probabil- Stage-specific capture probabilities, 1981–1996, ities are constant over time (i.e., no covariates). We for (a) immature male and female, (b) mature female, and (c) also assumed that the survival probabilities of female mature male right whales. Error bars indicate point-wise 95% and male calves are the same. This model gives a time- confidence intervals (CI) estimated from 1000 parametricbootstrap samples generated assuming a multivariate normal averaged picture of right whale demography. Estimated distribution of the logit of parameters. The covariance matrix capture and transition probabilities are shown in Fig.
of the distribution was estimated as the inverse of the Hessian matrix (see Burnham et al. 1987, Lebreton 1995). Mothers The population projection matrix for female right had a constant capture probability of 0.99 (95% CI ϭ [0.98, Estimated transition probabilities for the North the use of mathematical software packages such as MATLAB, so the transition and capture probability models need not be limited to those available in mark– recapture packages such as MARK (White and Burn- ham 1999), MSSURVIV (Hines 1994), or SURVIV Our method permits the use of capture histories with uncertain stage and sex. Individuals with such uncer- tainties tend to have lower survival rates than the rest of a population, because individuals that survive longer have more chances for accurate assessment of stage and sex identification. For example, right whales are Notes: The confidence intervals were estimated from 1000 considered mature at 9 yr of age. If animals that die parametric bootstrap samples generated assuming multivar- within nine years from their first capture are excluded iate normal distributions of parameters. The covariance ma- because their stage is uncertain, then we would over- trix of the distribution was estimated as the inverse of the estimate the survival probability. Observations with Hessian matrix (Burnham et al. 1987, Lebreton 1995).
uncertain stage or sex should never be discarded inparameter estimation. Our approach is one way to deal enough to be catalogued. Although newborn calves have distinct markings, they are harder to distinguish The stage structure that we used in this paper con- individually than other stages. Therefore, calf survival tains as a stage females that have given birth between is estimated from the time when the calf is seen suf- consecutive sampling periods. This stage makes the ficiently well to permit identification, which is not nec- conversion of the transition matrix into a population essarily on its first sighting. We assumed that calves, projection matrix relatively simple. Because the pur- on average, are identified midway through their first pose of the MSMR statistics often is to estimate a pop- year, and that the mother must survive this long (with ulation projection matrix, we recommend the use of ␾ ) in order for the calf to survive. F (t) this type of stage structure when possible.
The polychotomous logistic function is a flexible From this matrix, we estimated the long-term pop- way to allow transition probabilities to decrease or in- ulation growth rate and its confidence interval. They crease with a covariate while satisfying the requirement are ␭ ϭ 1.01 (95% CI ϭ [1.00, 1.02]). This result shows that each column of the transition matrix sum to one.
that the North Atlantic right whale population has been When time is used as a covariate, the polychotomous growing by 1% annually, on average, from 1980 to function allows inferences about temporal trends in 1997. (In fact, a time-varying model estimated by this stage-specific transition rates. This approach has been same procedure concludes that the growth rate has de- applied to the North Atlantic right whale data (Fujiwara clined from ␭ ϭ 1.03 to ␭ ϭ 0.98 over this time period (Fujiwara and Caswell 2001).) This matrix can now be Multi-stage mark–recapture data arise in many ap- analyzed to obtain the stable stage distribution, repro- plications. For example, Nichols and Kendall (1995) ductive value, damping ratio, sensitivity and elasticityof ␭, and other demographic statistics.
use them in population genetics context to test trade-offs between survival and reproduction. Hestbeck et al. (1991) use them to estimate spatial movement of The method presented here estimates a population individuals. We have applied them to deal with the projection matrix from mark–recapture data, which is problem of temporary emigration (Fujiwara Caswell one of the most commonly available data types for 2002). We hope that the extensions of the analytical animal populations. Once the population projection method presented here will make them even more use- matrix is estimated, it is subject to complete demo- graphic analysis; such analyses provide powerful tools Mark–recapture data are expensive to collect, and for conservation biology (e.g., Casewell 1989, 2001, they should be analyzed as completely as possible. If Tuljapurkar and Caswell 1997). They can be used to information on the stage of individuals (e.g., age, size, assess the causes of past population declines and to other developmental stages, or geographic locations) predict the effect of possible future management ac- is collected in addition to the basic mark–recapture tions. Because population projection matrices contain data, then MSMR statistics can be applied. The stage many parameters, it has been difficult to estimate them information need not be complete because our method accurately. This has been especially true for animals incorporates uncertainties in stage identifications. The that are not captured at every sampling period.
value of being able to use matrix population models The likelihood calculations here are simpler than for conservation makes it worthwhile to collect stage- those described in Nichols et al. (1992). This allows Fujiwara, M., and H. Caswell. 2002. A general approach to temporary emigration in mark–recapture analysis. Ecology We thank S. Brault, M. Hill, M. Neubert, J. Nichols, and 83:3266–3275.
the participants of the first Woods Hole Workshop on the Hastings, K. K., and J. W. Testa. 1998. Maternal and birth Demography of Marine Mammals for discussions and sug- colony effects on survival of Weddell seal offspring from gestions. We also thank J. Bence, E. Cooch, and an anony- McMurdo Sound, Antarctica. Journal of Animal Ecology mous reviewer for careful reviews and constructive com- 67:722–740.
ments. This project was funded by the David and Lucile Pack- Hestbeck, J. B., J. D. Nichols, and R. A. Malecki. 1991.
ard Foundation, the Rinehart Coastal Research Center, and Estimates of movement and site fidelity using mark–resight the Woods Hole Oceanographic Institution Sea Grant Program data of wintering Canada Geese. Ecology 72:523–533.
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