Journal of Applied Statistics, Vol. 29, N os. 1- 4, 2002, 589- 607
BR ETT K . SA N DERCOC K
1,2
& STEVEN R. B EISSIN G ER
2
,
1
C entre for
Applied Conser vation B iology, D epartment of Forest Sciences, University of B ritish
Columbia, Canada and
2
Department of Environmental Science, Policy &
M anagement, University of C alifornia at B erkeley, USA
Robust methods for estimating rates of population change ( ) are necessary for applied and theoretical goals in conser vation and evolutionar y biology. Tradition ally, has been calculated from either ratios of population counts (obser ved or obs
), or population models based on projection matrices (asymptotic or asy
,). N ew markrecapture methods permit calculation of from mark- resighting infor mation alone (realized or rea
), but empirical comparisons with other methods are rare. In this paper, rates of population change were calculated for a population of green-rum ped parrotlets ( Forpus passer inus ) that have been studied for more than a decade in central Venezuela. First, a ratio method based on counts of detected birds was used to calculate obs
. Next, a temporal symmetry method based on mark- recapture data (i.e. the -parameterization introdu ced by Pradel, 1996) was used to calculate rea
. Finally, a stage-str uctured matrix model based on state-speci® c estimates of fecundity, immigration, local sur vival, and transition rates was used to calculate asy
. A nalyses were conducted separately for females and males.
O verall values of
à from the three methods were consistent and all indicated that the ® nite rate of population change was not signi® cantly diþ erent from 1. A nnual values of
à from the three methods were also in general agreement for a majority of years. However, rea from the temporal symmetr y method had the g reatest precision, and apparently better accuracy than asy
. Unrealistic annual values of asy could have been due to poor estimates of the transitional probability of becoming a breeder ( c actual and the asymptotic stable stage distribution. In this study, the trade-oþ between biological realism and accuracy was better met by the temporal symmetr y than the matrix
Correspondence : B. K. Sandercock, Division of Biology, Kansas State University, Manhattan, Kansas
66506-49 01, USA. E-mail: bsanderc@ ksu.edu
ISSN 0266-476 3 print; 1360-053 2 online/02/010589-1 9
DOI: 10.1080 /02664760 12010881 8
© 200 2 Taylor & Francis Ltd

590 B . K. Sandercock & S. R. B eissinger method. O ur results suggest that the temporal symmetry models can be applied with con® dence to populations where less infor mation may be available.
1 Introduction
Population grow th is a biological process of great interest for population studies of plants and anim als. Robust estimates of the ® nite rate of population change ( ) are essential for both applied and theoretical goals in ecology. In conservation biology, reliable estim ates of are essential for identifying declining populations (
<
1), and for evaluating m anagem ent actions that could lead to stable or increasing population num bers ( 1; Beissinger & Westphal, 1998). In population biology, estim ates of rates of population change are central to understanding the factors that regulate population num bers, and for investigating the dynam ics of spatially structured populations, including m odels of source-sink interactions (Diþ endorfer,
1998; Virgl & M essier, 2000). In evolutionary ecology, has been used as an index of ® tness (Caswell, 2001). If variation in vital rates is due to genetic variance, then can be used to explore the adaptive function of life-histor y tactics, and the selective forces that shape them (McGraw & C aswell, 1996).
T he intrinsic rate of population change is given by: r ( B
+
I ) ( D
+
E ) where population gains are determ ined by the per capita birth ( B ) and im m igration rates ( I ), and population losses by death ( D ) and emigration rates ( E ). T he ® nite rate of population change, or the rate of change per individual per unit tim e ( ), is related to r by: e r
A num ber of m ethods can be used to estim ate for open populations (Nichols,
1992; Nichols et al ., 2000; Nichols & Hines, 2002). The ratio method m akes no attem pt to determ ine B , I , D or E . Instead, if a com plete census of a population is attainable, then the obser ved rate of population change ( obs
) is sim ply the ratio of the population size ( N ) at two consecutive time periods ( t ): t
N t
+
1
/ N t
M ore often, only surveys are feasible, yielding counts that are related to the true population size by an unknown probability of detection ( p )
<
1:
C t
N t p t
Use of a ratio of counts to estim ate population change between two occasions: obs
C t
+
1
/ C t
( N t
+
1 p t
+
1
) /( N t p t
) requires that probabilities of detection do not var y across tim e (i.e.
p t
+
1 is expected to be biased if this assum ption is violated (Nichols, 1992).
p t
), and
T he need to estim ate p was one im petus that led to the developm ent of early statistical models for analysis of m ark- recapture data, such as the Corm ack- Jolly-
Seber (CJS) m odel ( t
, p t
; Jolly, 1965; Seber, 1965). In the temporal symmetr y method , Pradel (1996) presented a reparam eterization of the CJS m odel that can be used to calculate direct estim ates of the realized rate of population change ( rea
).
In a conventional CJS m odel, capture-histories are analysed from the ® rst capture forward, yielding estim ates of local survival ( t
) between consecutive occasions.

M ethods for calculating 591
The com plement of local survival is a measure of population losses, and no attem pt is m ade to distinguish between D and E . Analysis of the sam e capture-histories from the last capture backwards yields a seniority probability ( t
), de® ned as the probability that the individual did not enter the population between t 1 and t .
Thus, 1 Ã is a measure of population gains, although no attempt is made to determ ine whether the gains are due to B , I or to increases in banding eþ ort or study area (Franklin, 2001). In the -param eterization of Pradel (1996), a ratio of these two probabilities yields a tim e-speci® c rate of population change: rea t
/ t
+
1
In the matrix method , stage- or age-speci® c estim ates of vital rates are synthesized into a projection m atrix and used to calculate an asymptotic rate of population change ( asy
, Caswell, 2001). T he m atrix method diþ ers from the ratio and tem poral sym m etry m ethods in that it requires detailed inform ation on various dem ographic rates. Projection m atrices are often developed for what are assumed to be closed populations, w here changes in population size are due to births ( B ) and deaths
( D ), the latter m easured as the complem ent of local survival ( S ): log e
( t
) B t
(1 S t
)
In investigations of m arked vertebrates on sm all study plots, however, it is diý cult to distinguish between m ortality and perm anent emigration due to dispersal
(Lam brechts et al ., 1999; K oenig et al ., 2000). Therefore, the com plem ent of local survival is not deaths (1 S
<
1, variation in either D t t
D t
), but deaths and emigration (1 S or E t t
D t
+
E t
). If may be responsible. To obtain reliable estim ates of population change for an open population at a local site, it is im portant to incorporate im m igration rates into the m atrix m odel, although this is rarely done: log e
( t
) ( B t
+
I t
) (1 S t
)
A potential drawback of the m atrix m ethod is that estim ates of certain vital rates m ay be diý cult to obtain or m ay require long-term study. Nonetheless, asy may be a better characterization of population dynam ics than obs or rea because asy incorporates a m ore detailed, m echanistic understanding of underlying biological processes.
T he m etrics of population change from the three methods are subtly diþ erent in their properties. First, transitions, whereas asy obs and rea are direct estim ates of population change in past predicts what the future rate of population change would be for an obser ved set of vital rates. T his distinction has been referred to as retrospective versus prospective analysis, respectively, although use of these term s is presently controversial (C aswell, 2000; W isdom et al ., 2000). Second, unlike and rea
, asy is asym ptotic in nature. For a given projection m atrix, asy obs is the predicted rate of population change if the population is at a stable stage or age distribution. Population projections based on determ inistic matrix m odels are som etim es criticized because such m odels do not incorporate the likely eþ ects of stochasticity and negative density-dependence, but the eþ ects of these processes m ay be of less concern if asy is used for short-term projections of a few years.
In this paper, we use three m ethods to calculate for a sm all neotropical bird, the green-rum ped parrotlet ( Forpus passerinus L., hereafter `parrotlet’ ). This species has been the focus of a long-term study since 1988 and has been used to investigate questions regarding the social behaviour (Beissinger et al ., 1998), hatching asynchrony (Stoleson & Beissinger, 1997, 1999) and dem ography of a tropical ver-

592 B . K. Sandercock & S. R. B eissinger tebrate (Beissinger & Waltm an, 1991; Sandercock et al ., 2000). Previously, we have shown that the adult population has a m ale-based sex ratio, and that the sexes have m arkedly diþ erent rates of survival, breeding propensity and natal dispersal
(Sandercock et al ., 2000). M ost demographic m odels of birds are female-based or com bine both sexes into one m atrix m odel (Caswell, 2001). To control for heterogeneity due to sex and to facilitate com parisons with past work, all analyses and m odelling were conducted separately for both sexes. T he objective of this paper was to use ® eld data collected over a decade of study to evaluate the relative perform ance of the ratio, tem poral sym m etry and m atrix m ethods for calculating .
2 M ethods
2.1 Study species
Green-rum ped parrotlets were studied at H ato M asaguaral, a cattle ranch 45 km south of C alabozo, in G uaÂrico, Venezuela (8ë 34 N, 67ë 35 W ). Field work was conducted over a 12-year period from 1988 to 1999. Analyses were restricted to data collected from 1990 to 1999 because sam ples of birds and nesting attem pts were som etim es sm all in the ® rst two years. Parrotlets are sm all parrots (25 - 34 g) that are m onom orphic in body size and readily sexed by their dichrom atic plum age.
They are seasonal breeders, and nesting occurs during the wet season from M ay to Novem ber. Pairs raise 1 - 2 or rarely 3 broods per year (Waltm an & Beissinger,
1992) and are unusual among tropical birds for laying a large clutch (mean 7 eggs, range 4 - 11, Beissinger & Waltm an, 1991). This species is both socially and genetically m onogam ous (extra-pair paternity
<
8%; M elland, 2000), and the number of young ¯ edged is a good indicator of reproductive success for both parents. Dispersal is high am ong juvenile fem ales, m oderate am ong juvenile m ales, and negligible am ong adults (Sandercock et al ., 2000; Beissinger, unpublished data). Parrotlets do not reproduce in their natal year, but both sexes breed as yearlings. T he study population has a m ale-biased sex ratio, and a high proportion of parrotlets are non-breeders, especially am ong m ales (
>
50% , Sandercock et al .,
2000). Non-breeding parrotlets are often obser ved actively searching for nest sites and m ating opportunities, som etim es harassing breeding pairs (Beissinger et al .,
1998). During the dry season, parrotlets roost and forage in large ¯ ocks (Waltm an
& Beissinger, 1992).
2.2 Field methods
Fecundity inform ation was derived from birds breeding in arti® cial nestboxes; the few nests that occurred in natural cavities were also monitored each year. T he number of nestboxes in the study area was 40 in 1988 and 100 to 106 during 1989 to 1999. Nestboxes were checked daily throughout the breeding season. To calculate the num ber of attem pts per pair, we included only nests where the identity of the fem ale was known. Each year we observed a few sm all clutches (1 - 4 eggs) that were never incubated, and failed for unknown reasons. A sm all number of fem ales also laid eggs in two nestboxes concurrently or consecutively, and then abandoned one or both attempts. We considered a nesting attem pt to be a full clutch that was laid in a single nestbox.
Nests were considered successful if they ¯ edged at least one chick. In several years, egg and chick num bers were experim entally m anipulated as part of separate

M ethods for calculating 593 investigations (Stoleson & Beissinger, 1997, 1999). Nest m anipulations did not lead to higher rates of depredation (Stoleson & Beissinger, 2001), so experim ental nests were included in analyses of nest success. We used unadjusted rates of nest success instead of M ay® eld estim ators because clutches were found at the onset of laying. M anipulations of oþ spring num ber had som e eþ ect on num ber of ¯ edglings produced, and analyses of num ber of ¯ edglings per successful nest were restricted to unm anipulated nests.
We attem pted to capture all nesting and prospecting parrotlets, and individual birds were banded with num bered m etal rings and unique com binations of coloured leg-bands. Adult parrotlets were captured in mistnets at roosting sites and in the vicinity of nestboxes, or were trapped in the nestbox while feeding young.
No parrotlet has been detected breeding during its natal year, and adults were assum ed to have sur vived at least one dry season. Nestlings were captured by hand in the nestbox, 3 to 5 days prior to ¯ edging. Parrotlets were resighted near nest boxes, at comm unal roosts and in feeding ¯ ocks. Band com binations and som etim es band num bers were read with binoculars (10 ) and spotting scopes
(40 - 60 ) at distances of 10 - 75 m. Field eþ ort consisted of daily sur veys over a 7-month period and was sim ilar in all years. From 1995 to 1999, however, we m ade a m ore concerted eþ ort to resight and identify non-breeding birds in the study population.
A parrotlet was considered to be a breeder if it was associated with at least one nesting attem pt during the study year, regardless of the fate of the nest. It is possible that a few breeders were overlooked each year. We think this is unlikely because nest boxes were monitored intensively and few nesting attem pts were m ade in natural nest cavities (Beissinger & Bucher, 1992; Stoleson & Beissinger, 1997).
Non-breeders were identi® ed by their social interactions, and were never associated with a nesting attempt. Non-breeding m ales som etim es form ed male- male pairs or remained in sm all ¯ ocks; non-breeders also included m ated pairs that did not nest (Beissinger et al ., 1998). Non-breeders were frequently yearlings but also included older birds that failed to obtain a m ate (Beissinger, unpublished data).
2.3 G eneral approaches to statistical analyses
All analyses of m ark- recapture data were im plem ented with Program M ARK
(version 1.9, W hite & Burnham , 1999) and m odel selection was based on the inform ation theoretic approach described by Burnham & Anderson (1998). As a
® rst step, we used a parametric bootstrap m ethod to test the goodness-of-® t (GO F) of the global m odel to the data. A distribution of expected deviances was generated under the assum ptions of no heterogeneity and full independence ( n 1000 simulations). A variance in¯ ation or overdispersion factor ( cà ) was calculated by dividing the observed deviance of the global m odel by the m ean expected deviance.
GO F tests for m ultistrata m odels are not yet available in Program M ARK, and were calculated with Program MSSURV IV instead (version 1; Brownie et al .,
1993).
Next, m odel structure was speci® ed using param eter index m atrices, and m aximum likelihood m odels were constructed using the sine-link function. M odel
® t was described by the deviance (Dev 5 2 2 ln ) and the number of param eters in the m odel ( K ). Relative m odel ® t was assessed with a m odi® ed version of
Akaike’s Information C riterion that included corrections for sm all sam ple size
(AIC c
), and in most analyses, overdispersion as well (quasi-AIC c or QAIC c
). M odels

594 B . K. Sandercock & S. R. B eissinger were considered equally parsim onious if the diþ erence in AIC c or Q AIC c from the best-® t model was 2.
Annual and overall estim ates of demographic rates were calculated with two procedures. First, annual m eans and variances of population change ( survival (
à ) and the transitional probability of becoming a breeder ( c rea
), local
à ) were derived from unconstrained global m odels, w hich included time-dependence. In analyses where cÃ
>
1, the variances of parameter estim ates were corrected by cà and were taken from the estim ate output of Program M ARK . Second, overall m eans and variances for our 10-year time series were calculated by applying the variance com ponents procedure of Program MARK to the annual param eters. Term inal and starting transitions that were non-identi® able were not included. T he variance com ponents procedure allowed us to rem ove the eþ ects of sampling variation, leaving the process variation due to biological processes (Gould & Nichols, 1998).
Statistical analyses of other vital rates were perform ed with SAS procedures
(version 8, SAS Institute Inc., Cary, NC 27513, U SA). The eþ ects of year on nesting success and the annual num ber of nesting attem pts per pair were analysed with contingency tests (Proc FR EQ ). Few pairs produced m ore than two clutches per year so we avoided sparse cells by com paring the frequencies of birds that produced one versus two or more clutches. The eþ ects of sex, social status and year on im m igration rates were analysed w ith logistic analysis (Proc C AT MO D).
The number of ¯ edglings per successful nest was norm ally distributed and year eþ ects were analysed with single classi® cation analysis of variance (Proc GLM ).
All tests were two-tailed and considered signi® cant at probability levels
<
0.05.
For graphical clarity, m eans of vital rates are presented 1 SE, and means of rates of population change are presented 95% CI.
2.4 Ratio method for estimating
Annual rates of obs were estim ated by using a ratio of the raw counts of m arked individuals that were detected on the study area each year. H ence, an annual rate was: t
( N non
+
N brd
) t
+
1
/( N non
+
N brd
) t where N non and given year ( t ).
N brd the num ber of non-breeders and breeders detected in a
2.5 Temporal symmetr y method for estimating
Annual rates of rea were estim ated using m ark- recapture data only. Capture histories were coded for adult parrotlets using the following criteria for each occasion: 1 banded, recaptured or resighted on the study area as either a breeder or a non-breeder, 0 not detected. H andling records of juvenile parrotlets were discarded. Rates of population change were estim ated using the -parameterization of Pradel’ s (1996) m odels. T he global model contained tim e-dependence in population change, local survival and resighting rates ( t
, t
, p t
). Goodness-of-® t
(GO F) tests are not yet available for this model. Instead, we calculated G OF to the CJS m odel ( t
, p t
) because the -param eterization m odel is derived from the
CJS m odel and is based on the sam e set of assum ptions (Pradel, 1996; Nichols &
Hines, 2002). O ur 10-year study period spanned nine transitions, and the global m odel yielded seven values of rea because the ® rst and last transitions were non-

M ethods for calculating 595 identi® able (Franklin, 2001). In reduced m odels, and p were treated as nuisance param eters and were not constrained.
T he rea
-values from tem poral sym m etry m odels are determ ined by changes in the num bers of newly encountered individuals, whether they are due to local recruits, individuals im m igrating onto to the study area or to changes in sam pling eþ ort or area. In this study, all newly captured birds were presum ably true im m igrants because the size of our study area and num ber of nestboxes m onitored rem ained constant during 1991 to 1998. M oreover, the ® rst annual value of rea was preceded by three years of banding eþ ort, and m ost breeder parrotlets are detected ( p
>
0.98) although non-breeders m ay be overlooked ( pà 0.31 to 0.44,
Sandercock et al ., 2000).
2.6 M atrix method for estimating
2.6.1 Estimation of vital rates.
Annual rates of asy were estim ated using a stagestructured matrix model that synthesized inform ation on births, imm igrants, and losses due to m ortality and perm anent em igration.
Births were calculated in two steps. First, fecundity ( F ) or the expected num ber of ¯ edglings per pair was calculated as the product of three com ponents of reproductive output:
F N A Succ FPN 0.5
where N A number of nesting attem pts per fem ale, Succ the percentage of nests
¯ edging at least one young, FPN the num ber of ¯ edglings per successful nest and 0.5 is the correction factor for sex ratio at ¯ edging (i.e. a 1:1 ratio, Beissinger, unpublished data). Secondly, the local survival of ¯ edglings in their ® rst year of life was estim ated from m ark- recapture data. Capture histories for nestling parrotlets that successfully ¯ edged were coded using the following criteria: 1 banded as nestling, or later recaptured or resighted on the study area as either a breeder or a non-breeder, 0 not detected. Juvenile survival rates were then estimated using recapture-only m odels. H ere, the global m odel was a two age-class m odel that separated local survival in the transition in the year after ® rst banding ( juv ) from local survival in all subsequent years ( adt ). It contained time-dependence in both local survival and resighting rates (
( t juv , t adt , p t
). Local survival rates of adults adt ) and resighting rates ( p ) were treated as nuisance param eters and were not constrained.
Per capita im m igration rates can be estim ated using the f -param eterization of the tem poral sym m etry models. Unfortunately, state inform ation cannot yet be included in these m odels, and our matrix m odel required separate estim ates of im m igration rates for non-breeders and breeders. Per capita im m igration rates were calculated for birds of each social state by dividing the number of newly m arked birds by the number of returning birds. Newly marked birds were presum ed to be imm igrants whereas returning birds included philopatric young and adults showing breeding site-® delity. T his approach relies on the untested assum ption that the detection rates of unbanded and m arked birds are equal. However, it is not aþ ected by known diþ erences in resighting rates between birds of diþ erent social status (Sandercock et al ., 2000).
Local survival and transition rates of adults were calculated from m ark- recapture data. T he com plem ent of local survival is com prised of losses due to m ortality and perm anent em igration, but no attem pt was m ade to distinguish these processes

596 B . K. Sandercock & S. R. B eissinger here. Capture-histories were coded for adult parrotlets using the following criteria:
B detected on the study area as a breeder, N detected on the study area as a non-breeder, or 0 not detected. Handling records of juvenile parrotlets were discarded. State-speci® c estim ates of local survival ( S non
, S brd
) and the probabilities of becom ing ( c non-brd
) or rem aining a breeder (1 c brd-non
), and resighting rates
( p non
, p brd
) were estim ated using dynam ic m ultistate m odels. Here, the global m odel included time-dependence within all param eters and strata ( S status* t
, c status* t
, p status* t
).
Previous analyses dem onstrated large diþ erences in S and c between non-breeders and breeders (Sandercock et al ., 2000). Thus, social status was retained in all m odels and no attem pt was m ade to test for interactions between strata. Resighting rates ( p ) were treated as nuisance param eters and were not constrained.
2.6.2 Parameterization of the stage-str uctured matrix model.
To synthesize the diþ erent com ponents of parrotlet demography, we developed a stage-structured population m odel based on prebreeding surveys. A life-cycle diagram was constructed (Fig. 1(a)), and the vital rates were cast into a 2 2 projection matrix
(Fig. 1(b)). This tim e-invariant m odel contained nodes for two stages: breeders and non-breeders. Age-structure was not included because diþ erences in social status have a greater eþ ect on parrotlet dem ography than absolute or relative age.
Density-dependence was not included because the relationship between bird numbers and vital rates was unknown.
T he two nodes of the life-cycle diagram were strongly connected by six arcs
(Fig. 1(a)). Arcs 1 and 2 describe the fecundity of breeders, weighted by the subsequent survival of the ¯ edglings produced and the probability that the juveniles become non-breeders or breeders by the next pre-breeding census. Arcs 3 and 4 parallel arcs 1 and 2, but are the transition rates of adult breeders, consisting of the survival rates of breeders and their subsequent social status. Finally, arcs 5 and
F ig . 1. The life-cycle diagram (a) and associated projection matrix (b) for a stage-structured population model of green-rumped parrotlets based on pre-breeding censuses. Notation includes: F fecundity measured as ¯ edglings per pair, I per capita immigration rate, S probability of local survival, c probability of becoming or remaining a breeder. Subscripts include: juv juvenile, non nonbreeder, brd breeder.
S juv is equivalent to juv of Table 3.

M ethods for calculating 597
6 are the life-history pathways of non-breeders, and are com posed of the survival of adult non-breeders and their breeding status in the follow ing year. T he nonbreeder node did not have additional arcs for alternative life-history tactics because genetic analyses have show n that intraspeci® c nest parasitism does not occur and extrapair fertilizations by non-breeders are rare (M elland, 2000).
2.6.3 C alculation of matrix properties.
M atrix properties were calculated using form ulae and the general m ethods described by Caswell (2001). T he stagestructured m odel was analysed as a tim e-invariant determ inistic m atrix: n ( t
+
1) An ( t ) where n ( t ) and n ( t
+
1) are vectors giving the abundance of each stage in the population at times t and t
+
1, and A is the projection m atrix (i.e. Fig. 1(b)).
Asym ptotic estim ates of the ® nite rate of population change ( or
1
, the dom inant eigenvalue) were derived with the power m ethod (Caswell, 2001), and validated with algorithms of Program M AT L AB (student version 11, M athworks, Natick,
M A 01760, USA). The sensitivity ( s i j
) of to instantaneous changes in a given m atrix elem ent ( a i j
) was calculated by: where w the stable stage distribution, and v the stage-speci® c reproductive value, and w , v the scalar product of these two vectors. T he sensitivity ( s x
) of to instantaneous changes in individual vital rates ( x ), or `lower-level’ sensitivities was calculated by: s x s i j
/ x
/ a ij ij v i w j
/ w , v
[( / a ij
) ( a ij
/ x )] where / a ij is the sensitivity of a m atrix elem ent and a i j
/ x is the partial derivative of a m atrix elem ent with respect to x (i.e. the product of vital rates other than x in term s containing x , Caswell, 2001). Finally, the variance of the rate of population change was calculated by: var( ) var( x ) ( / x )
2 where var( x ) is the variance of vital rate x and / x is the lower-level sensitivity of the vital rate x (Alvarez-Buylla & Slatkin, 1994). The 95% con® dence intervals of was then calculated by:
1.96
var( )
3 Results
3.1 Ratio method
Annual values of obs based on raw counts of individuals ranged from 0.757 to
1.124 in fem ales (in 1996 - 97 and 1995 - 96) and 0.707 to 1.360 in m ales (in 1996 -
97 and 1994 - 95, Fig. 2). T he geometric m ean of obs was 0.964 (0.833 - 1.095
95%C I) in females and 0.995 in males (0.809 - 1.181, Fig. 2).

598 B . K. Sandercock & S. R. B eissinger
F ig . 2. Mean estimates of population growth rates ( 95%CI) for female and male green-rumped parrotlets calculated with the ratio method (observed
à or r ea
à or o bs
), the temporal symmetry method (realized
) and the matrix method (asymptotic
à or asy
).
3.2 Temporal symmetr y method
A total of 1624 adult parrotlets (fem ales: n 602, m ales: n 1022) were marked during 1990 to 1998. T he param etric bootstrap GO F test indicated that a conventional C JS m odel ( fem ales ( P t
, p t
) was a relatively poor ® t to the capture histories of both
<
0.001) and m ales ( P
<
0.004). However, a poor ® t was expected because breeders and non-breeders were pooled, and social status is a known source of heterogeneity in parrotlet capture-histories (Sandercock et al ., 2000).
Nevertheless, we proceeded to use the -param eterization to estim ate rea because overdispersion factors were low (fem ales: cà 1.304, m ales: cà 1.310). T he cà -values were used to adjust the m odel selection with Q AIC global m odel t
, t
, p t indicated that rea c and the 95%C I of rea
. T he varied from 0.844 to 1.181 in females and
0.688 to 1.277 in m ales (Fig. 2). Subsequent m odelling revealed that the global m odel w ith annual variation in rea was the best ® t for m ales (Table 1). In fem ales, the best ® t m odel was one where -values did not vary annually but collapsed to a constant. O verall, rea calculated with the variance com ponents procedure indicated that the long-term rate of population change was close to one for both fem ales
( rea
0.978, 95% CI
<
0.001) and m ales ( rea
1.004, 0.849 - 1.159 95% CI,
Fig. 2).
3.3 M atrix method
Nesting pairs of green-rum ped parrotlets produced, on average, between 1 and 2 com plete clutches per year (Table 2). T he proportion of pairs producing more

M ethods for calculating
T able 1. Temporal symmetry mark- recapture models used to estimate population grow th rates of parrotlets
M odel structure a Model statistics b
Sex
Females
Males t c c t t t t t p t t t t
Dev
2875.9
2871.3
4935.4
4965.0
K
20
26
26
20
QAIC
0.0
9.1
0.0
10.3
c w i
0.989
0.011
0.995
0.005
a Population growth rates ( ) were estimated using the -parameterization model of Pradel (1996) . Local survival ( ) and resighting rates ( p ) were treated as nuisance parameters and were not constrained. M odel notation b includes: t time or annual variation, c constancy.
Model ® t is described by the deviance (D ev), number of parameters in the model ( K ), and Akaike’s Information Criterion corrected for both small sample sizes and overdispersion (QAIC c
). M odels are presented in order of
® t relative to the best ® t model ( QAIC c
0). Variance in¯ ation factors were low in females ( cà 1.304) and males ( cà 1.310).
599
T able 2. Annual variation in components of reproductive performance of breeding parrotlets
Year
Nesting attempts per pair n
Nest success a n
Fledglings per successful nest n b
1991
1992
1993
1994
1995
1996
1997
1.47
0.06
1.34
0.05
1.27
0.06
1.60
0.07
1.41
1.56
1.39
0.07
0.08
0.06
87
91
81
85
71
59
69
0.846
0.781
0.660
0.735
0.613
0.628
0.758
130
128
100
132
106
86
95
4.65
4.50
0.25
4.56
0.50
c
0.52
3.86
0.36
4.78
0.47
4.57
0.22
52
16
10
29
23
69 a
Nest success at least one chick was successfully ¯ edged from the nest. May® eld estimators were not used because all nests are found early in laying at 1 - 3 eggs.
b
Samples based on unmanipulated nests only.
c
No nests were left unmanipulated in 1993 . An estimate of ¯ edging success based on pooled years was used in the matrix model ( FPN 4.51
0.13).
than one nesting attem pt varied across years (contingency test, 2
6
19.7,
P 0.003). Nesting success for this cavity-nesting species was generally high
(Stoleson & Beissinger, 2001) and, in all years, m ore than 60% of nests produced at least one ¯ edgling. Rates of nest success varied signi® cantly among years
(contingency test,
2
6
25.1, P
<
0.001), ranging from 61.3% in 1995 to 84.6% in
1991. Annual variation in num ber of ¯ edglings produced per successful nest was not signi® cant (ANOVA, F
5,193
0.84, P 0.52), and usually averaged about 4.5
young.
During the period 1990 to 1998, a total of 3569 juvenile parrotlets (fem ales: n 1738, m ales: n 1831) were m arked as nestlings and subsequently ¯ edged.
GO F tests based on bootstrap sim ulations indicated that an age-structured global m odel ( t juv , t adt , p t
) provided a relatively poor ® t to the capture histories of juveniles
(fem ales: P 0.011, m ales: P
<
0.001). Nonetheless, we retained this starting m odel because the overdispersion factors were low in both sexes (fem ales: cà 1.344, m ales: cà 1.373). M ost of the overdispersion was likely due to heterogeneity in the param eters of older age-classes. Such departures were unim portant because

600 B . K. Sandercock & S. R. B eissinger
F ig . 3. Probability of local survival and the transitional probability of either becoming or remaining a breeder for green-rumped parrotlets of diþ erent ages and social status. Annual estimates were derived from the unconstrained global models presented in Tables 3 and 5. M eans are presented 1 SE.
our estim ates of adult sur vival were calculated with m ultistrata m odels and were based on a larger sample of parrotlets that included birds banded as adults.
T he global m odel t juv , t adt , p t indicated that local survival rates of juvenile parrotlets varied from 0.109 to 0.246 in fem ales and 0.386 to 0.547 in m ales (Fig.
3). Natal dispersal m ovem ents suggest that juvenile m ales had greater local sur vival than juvenile fem ales because males had stronger philopatry to their natal areas, but a possible sex diþ erence in true survival cannot be discounted (see Sandercock et al ., 2000). In both sexes, the best ® t m odel was one in which the local survival of juveniles ( juv
) was constant across and did not vary annually (Table 3). Overall, juv from the variance com ponents procedure was 0.141 in fem ales (SE and 0.465 in m ales (SE
<
0.001).
<
0.001)
T able 3. Age-structured mark- recapture models used to estimate local survival of juvenile parrotlets (females n 1738, males: n 1831)
M odel structure a
Model statistics b
Sex juv adt p Dev K QAIC c w i
Females
Males c t c t t t t t t t t t
175.0
165.2
437.0
431.6
18
25
18
25
0.0
7.0
0.0
10.3
0.971
0.030
0.994
0.006
a Local survival rate of juveniles ( juv
) was estimated using a two age-class model that separated local survival in the transition in the year after ® rst banding from all subsequent years ( a dt
). Local survival rates of adults ( adt
) and resighting rates ( p ) were treated as nuisance parameters and were not constrained. M odel notation includes: t time or annual variation, b c constancy.
See caption of Table 1 for de® nition of terms. Variance in¯ ation factors were low in females ( c 1.344) and males ( c 1.373).

M ethods for calculating 601
T able 4. Annual counts of birds and per capita immigration rates for adult parrotlets of diþ erent social status
Nonbreeders Breeders
Sex Year Outside a Local b
Immigration rate c Outside Local
Immigration rate
Females
Males
1992
1993
1994
1995
1996
1997
1991
1992
1993
1994
1995
1996
1997
1991
10
4
8
3
11
15
3
7
1
11
14
15
15
12
12
12
11
13
24
35
17
89
55
72
51
124
152
87
0.833
0.333
0.727
0.231
0.458
0.429
0.177
0.079
0.018
0.153
0.275
0.121
0.099
0.138
33
33
20
23
34
19
30
25
15
20
20
18
11
12
52
65
72
71
46
50
47
60
85
68
85
68
67
67 a
Outside birds that originated from outside the study area and were newly banded as immigrating adults.
b Local birds of local origin that included philopatric young that originated on the study area and returning adults that were banded in a previous year.
c Immigration rate outside/local.
0.635
0.508
0.278
0.324
0.739
0.380
0.638
0.417
0.176
0.294
0.235
0.265
0.164
0.179
As a consequence of sexual diþ erences in local survival of juveniles, per capita im m igration rates of adult parrotlets were generally higher in females (overall m eans, non-breeders: 0.455
0.093 SE, breeders: 0.500
0.067) than males (nonbreeders: 0.126
0.030, breeders: 0.247
0.034, Table 4). The proportion of birds that were im m igrants was signi® cantly higher am ong fem ales than m ales (maineþ ects logistic analysis, sex: non-breeders (social status: 2
1
2
1
58.0, P
13.9, P
<
0.001), and am ong breeders than
<
0.001), but annual variation was not signi® cant (year: 2
6
9.9, P 0.13).
Local sur vival rates of adults were estimated from the sam e capture-histories that were used to calculate rea in the tem poral sym metry analyses above. In these analyses, however, inform ation on social status of parrotlets was also included in the capture-histories. G O F tests calculated in Program M SSU RVIV indicated that the global m odel S
( P status* t
, c status* t
, p status* t was a good ® t to the data in both sexes
>
0.99). Annual estim ates of local survival and transition rates from the global m odel varied am ong years (Fig. 3). Con® dence intervals for fem ale param eter estim ates were generally wider than those of m ales (Fig. 3) because the m alebiased sex-ratio am ong adults resulted in a sm aller sample of females, am ong which non-breeders were less com m on (Table 4). M ark- recapture m odelling revealed that the best-® t m odel in fem ales was one where and c collapsed to constants, but a m odel that retained annual variation in the local survival of breeders was equally parsim onious (Table 5). In m ales, the best ® t m odel was one without annual variation in the probability of becom ing a breeder. T he ® t of this m odel was a twofold im provement over the global m odel where and c varied annually for both strata, but the global m odel was an equally parsim onious
® t to the data.

602 B . K. Sandercock & S. R. B eissinger
T able 5. M ultistate mark- recapture models used to estimate local survival and transition rates of adult parrotlets (females n 602, males: n 1022)
M odel structure a Model statistics b
Sex
Females
Males no n t t c c t t brd t t c t t t c n on-br d c brd-no n t c c c t c t c c c t t
P n on t t t t t t
P brd t t t t t t
Dev
307.7
291.9
262.6
681.1
650.8
669.4
K
22
30
52
37
52
45
AIC c
0.0
1.1
19.8
0.0
1.4
5.1
w i
0.63 0
0.36 7
0.00 0
0.61 4
0.30 3
0.04 9 a Local survival rates of non-breeders ( no n
) and breeders ( brd
) and the probability of changing social status ( c no n-br d
, c brd-no n
) were estimated using multistate models that included information on parrotlet social status. Resighting rates of non-breeders ( P no n
) and breeders ( P br d
) were treated as nuisance parameters and were not constrained. Model notation includes: t time or annual variation, c constancy.
b
See caption of Table 1 for de® nition of terms. Of the models tested, only the global model and models with moderate support ( AIC c
<
6) are presented.
O verall estim ates from the variance com ponents procedure indicated that local survival ( S ) of breeders was greater than that of non-breeders, and that males had higher local survival than fem ales (fem ale breeders: 0.594, SE non-breeders: 0.393, SE
<
0.001, fem ale
<
0.001, m ale breeders: 0.659
0.016 SE, m ale nonbreeders: 0.512
0.030). T he transitional probability of becoming or rem aining a breeder ( c ) was also higher am ong breeders but was higher in fem ales than in m ales (fem ale breeders: c 0.875
0.020
SE, fem ale non-breeders: c c
0.212
0.287
0.046, m ale breeders: c 0.725
0.040). In both sexes, resighting rates ( p
0.036, male non-breeders:
) were generally at or close to unity for breeders, but were m ore variable for non-breeders.
Annual estim ates of the m ean and variance of the three com ponents of reproductive output ( NA , Succ , FPN ), two im m igration rates ( I non
, I brd
), three survival rates
( S juv
, S non
, S brd
) and two transition rates ( c non
, c brd
) were then input into the m atrix m odel (Fig. 1(b)). We used annual estim ates to calculate variation in those vital rates was not signi® cant. Annual asy asy even if the annual
-values ranged from
0.669 to 1.266 in fem ales and 0.898 to 2.077 in m ales (Fig. 2). O verall, asy from m atrices based on m ean annual param eters indicated the long-term rate of population change should be stable in fem ales ( and increasing in m ales ( asy asy
1.000, 0.717 - 1.272 95% CI)
1.223, 0.8077 - 1.6381, Fig. 2).
3.4 Com parison of methods
The three analytical m ethods yielded overall values of that were not signi® cantly diþ erent. In both sexes, the 95% CI of obs
, rea and asy showed alm ost com plete overlap. Furthermore, the three estim ates of were close to, or not signi® cantly diþ erent from , a value of 1 in both sexes (Fig. 2). This result supported our qualitative im pression that population num bers of parrotlets were stable during the
10-year study period.
T he precision and accuracy of annual estim ates of were also fairly consistent.
The ratio m ethod perform ed quite well; obs fell within the 95%C I of rea and asy in 5 of 7 years in fem ales and 6 of 7 years in m ales (Fig. 2). The tem poral sym m etry

M ethods for calculating 603 m ethod yielded -values with the greatest precision. The 95% con® dence intervals of rea were narrower that the 95% CI of asy in at least 6 of 7 years in both sexes.
Extensive overlap of the 95% CI of rea and asy demonstrated that annual values of from the two m ethods were not statistically diþ erent in most years. However, annual values of asy rea in at least 5 of 7 years for both sexes (Fig. 2). O verall, the m ean degree of relative bias (i.e. (
+ in fem ales ( 0.5% ) but not m ales (
1 asy rea
) (( asy rea
) /2) ) was close to zero
+
16.8%). Years with a large discrepancy between asy and rea tended to be intervals where the transitional probabilities of becoming or rem aining a breeder were particularly low or high. Thus, in fem ales, the largest diþ erences between
1997 - 98 ( 35% , other years rea and asy were obser ved in 1994 - 95 ( 55%) and
<
28% , Fig. 2). These were intervals where the transitional probability of becom ing a breeder was apparently low for non-breeding fem ales (Fig. 3). In m ales, asy was
>
2 w hereas rea was
<
1 in 1991 - 92, a percentage diþ erence of
+
81% (Fig. 2). This was an interval where the transitional probabilities of becom ing or rem aining a breeder were particularly high for m ales
(Fig. 3). O ther years where diþ erences between
35%, 1996 - 97: rea and asy were large (1992 - 93:
+
41% , other years
<
21% , Fig. 2) also coincided w ith outlying values of c non-brd and c brd-brd
(Fig. 3).
4 D iscussion
In this study, three analytical methods were used to calculate estim ates of population change of green-rumped parrotlets from ® eld data collected over a decade. T he results were encouraging in that -values calculated w ith ratios of counts of population numbers, m ark- recapture m odels and projection m atrices were in general agreem ent. To our know ledge, this study is one of the ® rst attem pts to com pare sim ultaneously these three m ethods, and comparative data from other vertebrate taxa are not yet available. The tem poral sym m etry m ark- recapture m odels introduced by Pradel (1996) are relatively new and have not received w ide use. T his study is an early em pirical application of the -param eterization, although the -parameterization has been used to calculate recruitm ent rates for a few populations of birds (Pradel et al ., 1997; Schaub et al ., 1999; Oro & Pradel, 2000).
This study also joins the sm all num ber of m atrix m odels that have explicitly included population gains due to im m igration (see also Wootton & Bell, 1992).
M atrix models have been used to estim ate rates of population change for a wide range of vertebrate taxa (e.g. Cunnington & Brooks, 1996; Hiraldo et al ., 1996;
Crooks et al ., 1998; Woodworth, 1999), but populations are usually treated as closed and the problem of distinguishing m ortality and em igration is overlooked
(Haydon et al ., 1999). We ® rst com pare from the ratio, tem poral sym m etry and m atrix m ethods and then make recom m endations from our empirical results.
Evaluating the precision and accuracy of from the three diþ erent m ethods is diý cult without benchm ark values for com parison. A true can be set in a simulation study (e.g. Bart, 1995), but is usually im possible to m easure for a wild population of free-living vertebrates. In the rare cases where a complete population census is possible, then use of mark- recapture and m atrix m ethods becom es unnecessary because p 1 and obs true , respectively. T hus, a consideration for empirical com parison is that ® eld projects will only oþ er alternative estim ates of .
Despite the potential pitfalls of using the ratio m ethod, obs performed quite well and fell within the 95%C I of rea and asy in a m ajority of years. This m ay have been the case because resighting rates were close to unity for breeders and did not

604 B . K. Sandercock & S. R. B eissinger vary among years for non-breeders (this study, Sandercock et al ., 2000). Nonetheless, our analyses illustrate the inherent problem s of using uncorrected counts to estimate because the lowest and the highest values of increased eþ ort to resight non-breeders in 1995 and 1996.
obs coincided with an
In our initial comparisons of from the three m ethods, we expected that asy would be best-suited as a benchm ark for com parisons because the projection m atrix incorporated all of our current understanding of the social system and demography of green-rumped parrotlets. However, the overall and annual estim ates of rea from the tem poral sym m etry m ethod were m ore precise than m atrix m ethod. The 95% CI of asy m ay have been wider than rea asy from the because the m atrix m odel required estimates of 10 vital rates (or 13 including p ) whereas the tem poral symm etry m odels required estim ates of only one param eter (or three including and p ). The 95%C I of asy m ethods for calculating the variance of could also have been in¯ uenced by our asy
. We used analytical approach to calculate var( asy
), but the variance could also have been estim ated by bootstrapping annual estim ates of the vital rates (Ebert, 1999). However, con® dence intervals from analytical and bootstrapping approaches are generally in agreem ent if the coeý cients of variation for vital rates are relatively small (Alvarez-Buylla & Slatkin,
1994), which was the case here.
In certain years, asy appeared to be an unrealistic prediction of population change (e.g.
asy
2.08 in 1991 for m ales), and was m arkedly diþ erent from
(percentage diþ erence
>
30% ). W hile asy rea was based on a ® ner understanding of parrotlet demography, asy also appeared to be sensitive to poor param eter estim ates for a handful of vital rates, particularly the transitional probabilities of becom ing a breeder. T hus, the trade-oþ between biological realism and accuracy in was better m et by the tem poral sym m etry than the m atrix m ethod.
Two caveats should be considered in evaluating our results. First, we applied the three m ethods to ® eld data on fem ales and m ales from the sam e study population.
Som e congruence in annual estim ates of population change m ight have been expected because all analyses were based on the sam e dataset and because the dynam ics of the two sexes could not be considered independent. Second, population num bers of parrotlets were apparently stable for the duration of our study. In conser vation problem s, however, it is in scenarios w here populations are declining that these m ethods could have their greatest utility. W hile from the tem poral sym m etry and matrix m ethods were congruent for a stable population, this m ight not be the case if rates of population change were m arkedly diþ erent from 1.
W ith these cautions, som e recom m endations can be m ade regarding the relative m erits of the ratio, tem poral sym m etry and m atrix m ethods for estim ating rates of population change. T he ratio m ethod required the least am ount of inform ation and was simple to calculate. Our observation that obs perform ed well should not be taken as an endorsem ent of this approach. Nevertheless, this m ethod m ay be useful for certain conservation applications w here lim ited resources preclude m ore rigorous analyses. If so, obs should only be presented if accom panied by appropriate caveats.
T he m atrix m ethod required the m ost information but also allowed ¯ exibility in specifying life-stages and their rates. An advantage of m atrix m ethods is that they yield fam iliar m etrics such as the stable stage distribution ( w ), the stage-speci® c reproductive value ( v ), and elasticities ( e ), which can be used to assess population status, and to predict the likely eþ ects of changes in vital rates (Doak & M orris,
1999; de Kroon et al ., 2000). The poor perform ance of asy in a subset of years is

M ethods for calculating 605 not an indication that m atrix m ethods are faulty, rather that our particular model of parrotlet demography m ay have been overparam eterized. Better success m ight have been obtained if the matrix m odel had been param eterized diþ erently: by pooling birds of diþ erent social status, or by using an age- instead of stagestructured m odel. Another consideration in using matrix m ethods is the asym ptotic nature of the m etrics derived from projection m atrices. Diþ erences between rea and asy are to be expected if there is a m ism atch between the actual frequency of non-breeders and breeders in a given year and the asym ptotic stable stage distribution. However, this bias is likely to be sm all for parrotlets because the dam ping ratio is high for this population m odel (Sandercock & Beissinger, unpublished m anuscript).
T he tem poral sym m etry m ethod was easier to compute than the m atrix method because only detections were used for direct estim ation of . Like the CJS m odel from which it is derived, parameter estimates from the tem poral sym m etry m odel m ay be biased if or p contain heterogeneity due to age-structure or trap response
(Nichols & Hines, 2002). Our results suggest that if the assum ptions of the tem poral sym m etry m odel are m et, rea m ay be as good or superior to asy
. A current advantage of m atrix m ethods, however, is that they allow for a mechanistic analysis of the processes responsible for population change (Caswell, 2001).
Em pirical exam ples are scarce because tem poral sym m etry methods are relatively new, but m ark- recapture m ethods can also be used to evaluate dem ographic contributions to . For exam ple, the tem poral sym m etry m odels can be used to m odel rea as a function of environmental covariates. Nichols et al . (2000) and
Nichols & Hines (2002) have also shown that seniority probabilities ( ) are analogous to the elasticities derived from m atrix methods. Unlike m atrix m ethods, direct m easures of and à from mark- recapture data are not based on asymptotic assum ptions, Thus, rea m ay be m ore useful for characterizing populations over short periods of tim e or in variable environm ents (Nichols et al ., 2000). In conclusion, the greater biological realism oþ ered by m atrix m ethods has trade-oþ s, and temporal sym m etry m odels are a useful shortcut for obtaining robust estim ates of rates of population change.
Acknowledgem ents
We dedicate this paper with gratitude to Evan G. Cooch for his encouragem ent and m entorship, and for introducing B. K. Sandercock to m ark- recapture m ethods in a graduate seminar in 1994. We thank Sr Tom aÂs Blohm for encouraging us to study parrotlets on his private ranch in Venezuela, and the m any ® eld assistants and graduate students who worked hard to help us to collect the demographic inform ation that is sum marized here (listed in Sandercock et al ., 2000). A. B.
Franklin and R. G reen kindly invited us to participate in the session of direct estim ation of at the EU R IN G 2000 meeting. J. D. Nichols and J. E. Hines shared an unpublished m anuscript, H. Caswell and L. S. M ills critiqued our m atrix m odel, and E. G. Cooch, K. M artin and two anonym ous reviewers m ade constructive com m ents on previous versions of this manuscript. B. K . Sandercock was supported by a Killam Postdoctoral Fellowship from the University of British
Colum bia, and ® eld research was supported by grants to S. R. Beissinger from the
National G eographic Society and the National Science Foundation (DEB-
9503194, IBN-9407349).

606 B . K. Sandercock & S. R. B eissinger
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