Content uploaded by Jean-Dominique Lebreton
Author content
All content in this area was uploaded by Jean-Dominique Lebreton on Oct 30, 2017
Content may be subject to copyright.
The Canadian Journal of Statistics 85
Vol. 36, No. 1, 2008, Pages 85–98
La revue canadienne de statistique
The potential of integrated modelling in
conservation biology: a case study of the
black-footed albatross (Phoebastria nigripes)
Sophie V´
ERAN and Jean-Dominique LEBRETON
Key words and phrases: Albatross; conservation biology; Kalman filter; state-space models.
MSC 2000: Primary 62P10; secondary 92D40.
Abstract: Conservation biology aims at assessing the status of a population, based on information which
is often incomplete. Integrated population modelling based on state-space models appears to be a power-
ful and relevant way of combining into a single likelihood several types of information such as capture-
recapture data and population surveys. In this paper, the authors describe the principles of integrated popu-
lation modelling and they evaluate its performance for conservation biology based on a case study, that of
the black-footed albatross, a northern Pacific albatross species suspected to be impacted by longline fishing.
Le potentiel de la mod´
elisation int´
egr´
ee en biologie de la conservation :
une ´
etude de cas de l’albatros `
a pieds noirs (Phoebastria nigripes)
R´
esum´
e : La biologie de la conservation vise `
a´
evaluer l’´
etat d’une population `
a l’aide d’informations
souvent incompl`
etes. La mod´
elisation int´
egr´
ee des populations `
a l’aide de mod`
eles `
a espaces d’´
etats est
une m´
ethode puissante qui permet de combiner de fac¸on ad´
equate en une seule vraisemblance plusieurs
types d’informations telles que des donn´
ees de capture-recapture et des inventaires de population. Dans
cet article, les auteurs d´
ecrivent les principes de la mod´
elisation int´
egr´
ee des populations et ils en ´
evaluent
la performance pour la biologie de pr´
eservation `
a l’aide d’une ´
etude de cas portant sur l’albatros `
a pieds
noirs, une esp`
ece d’albatros vivant dans l’Oc´
ean Pacifique Nord vraisemblablement affect´
ee par la pˆ
eche
palangri`
ere commerciale.
1. INTRODUCTION
Conservation biology often aims at assessing the population status and proposing management
actions for endangered or harvested species (Soul´
e 1987; Gauthier & Lebreton 2004). To this
purpose, two main approaches have been traditionally used: analyses of successive surveys of
population size and structure on the one hand, and population dynamics modelling on the other.
These two approaches are based on quite different sources of information.
In the first approach, the pattern of population change is investigated through more or less
sophisticated time series methods to determine the rate of change and variability of population
size (Miller & Botkin 1974; Clark 2003). This approach requires reliable repeated surveys,
which are often unavailable. Furthermore, observation errors are difficult to incorporate in the
classical time series models and are unfortunately often ignored, with, for instance, disastrous
consequences on the detection of density-dependence in the rate of population change (Dennis
et al. 2006).
Demographic models provide a more process-oriented insight into population dynamics
(Caswell & John 1992; Beissinger & Westphal 1998). Such models require estimates of de-
mographic parameters, such as survival probabilities and fecundity (Caswell 2001). Estimates
of annual survival probabilities in natural animal populations are usually obtained from capture-
mark–recapture sampling, in which animals receive unique individual marks. Subsequent ob-
servations of these marked animals, either alive or dead, result in data which are analysed using
models incorporating annual probabilities of survival (Lebreton, Burnham, Clobert & Ander-
son 1992) together with probabilities of detection. These models are the counterpart when there
86 V´
ERAN & LEBRETON Vol. 36, No. 1
is incomplete detection of individuals of the discrete time survival models used in human health
studies (Lebreton, Pradel & Clobert 1993). The quality and relevance of demographic model re-
sults can be highly affected by uncertainties in vital rates estimations and by problems of model
validation and structure (Beissinger & Westphal 1998).
Hence, each of these two approaches has severe shortcomings when used by itself. Moreover,
so-called “integrated population monitoring” often provides both survey information and demo-
graphic data. In this case the two types of information have been until recently always combined
in an ad hoc fashion, by visually matching model predictions to population size estimates (Nel,
Taylor, Ryan & Cooper 2003; Arnold, Brault & Croxall 2006) without resorting to any analytical
tool (Besbeas, Freeman & Morgan 2005). This situation is particularly annoying in conservation
biology, in which data are often sparse or fragmentary, needs are urgent, and an optimal use of
all existing information is thus badly needed.
Recently, integrated population modelling has appeared as a powerful and relevant way of
blending several types of information, by offering the possibility of combining into a single like-
lihood the information brought by surveys and individual demographic data. Integrated popula-
tion modelling is based on state-space models (Harvey 1989, pp. 10–11), with various methods of
treatment and estimation (Besbeas, Freeman, Morgan & Catchpole 2002; Besbeas, Lebreton &
Morgan 2003; Buckland, Newman, Thomas & Koesters 2004; Thomas, Buckland, Newman &
Harwood 2005). As it is a recent and still developing methodology, Integrated population mod-
elling has been little used apart illustrative examples (see, however, Gauthier, Besbeas, Lebre-
ton & Morgan 2007). The potential of integrated population modelling for conservation biology
is strong, as diagnosis and management actions often have to be taken with the information at
hand, stakeholders having limited time and budgets. Integrated population modelling is espe-
cially relevant because, in this context, the data are generally incomplete: Integrated population
modelling can thus be the glue to assemble in a coherent fashion various pieces of information.
The purpose of this paper is to review the basic ideas of integrated population modelling, to
discuss its efficiency, to recommend some specific tools and to propose perspectives for improv-
ing its expected impact on conservation biology. As an example a case study we use through-
out is that of the impact of incidental by-catch by long-line fisheries of black-footed albatross
(Phoebastria nigripes), a north Pacific ocean albatross (Cousins & Cooper 2000; Lewison &
Crowder 2003). In terms of information available, this example is typical of cases studies in con-
servation and management of vertebrate populations: while a capture-recapture study provided
evidence for a relationship between adult survival and fishing activity (V´
eran et al. 2007), there
is no straightforward estimate available for survival in the first part of life, as is often the case
for vertebrates (Clobert & Lebreton 1991). In such a situation can population surveys provide
enough information for estimating such a parameter? This question is particularly critical for a
long-lived species such as albatross, in which the population growth regime is not very sensi-
tive to immature survival. Then, do population surveys bring further information and do they
reinforce the conclusions drawn from the capture-recapture analysis of adult survival?
We present first the albatross case study and the available data (Section 2), then state-space
models (Section 3), their statistical treatment by Kalman filtering (Section 4), and their applica-
tion to the albatross case study (Section 5). The discussion (Section 6) covers perspectives for
integrated population modelling in conservation biology.
2. THE BLACK-FOOTED ALBATROSS CASE STUDY
Industrial long-line fishing has been suspected since the 1990s to impact upon black-footed
albatross populations by inducing a biologically significant amount of mortality (Cousins &
Cooper 2000), as proven for albatross species in the southern Hemisphere (Weimerskirch, Broth-
ers & Jouventin 1997). However, in particular because of the wide range of the species at sea and
the diversity of fleets concerned, no precise estimate of by-catch mortality has been available to
ascertain this suspicion (Cousins & Cooper 2000; Lewison & Crowder 2003) although educated
2008 INTEGRATED MODELLING 87
guesses of the number caught seem to point to deleterious effects on the population (Niel & Le-
breton 2005). As other albatross species, the black-footed albatross is a long-lived species with
delayed maturity, first breeding taking place at age 5. But for the breeding season, the albatross
spend most of their life at sea. Breeding pairs are faithful for life, and forming a new pair af-
ter death of the partner may require some time, a biological feature with key consequences on
population dynamics.
The only two sources of information are (a) partial surveys of the number of breeders,
(b) capture-recapture data.
(a) The surveys by the US Fish and Wildlife Service concerned breeding numbers in the three
main colonies of the Hawaiian Archipelago: French Frigate Shoals, Midway and Laysan
Island. They account for about 75% of the world population of black-Footed albatross
(Cousins & Cooper 2000). Unavoidably, the investigation is restricted to this segment
of the population. The extrapolation to the entire world population implicitly assumes
that the portion of the population on the sampled islands did not vary over time. A key
feature is that only breeders are surveyed, a large number of non breeders being at sea
and not amenable to any kind of count. The survey data provide some evidence of a
decline. However, the relationship with total population size depends in a complex fashion
of the population structure, and, in the absence of an estimation of survey uncertainty, the
significance of the decline cannot be assessed in a straightforward fashion.
(b) In total, 13854 black-footed albatross chicks have been ringed (with a metal ring from the
United States Fish and Wildlife Service) since 1980 on Tern Island (23◦45′N, 166◦15′W),
in the north western Hawaiian Islands. Regular recaptures of breeding birds started
in 1992. The resulting data set made available to us consisted of 2046 capture histories
of known age breeding birds over T= 12 years (1992–2003), covering thus 11 yearly
intervals for survival. These data were analysed by V´
eran et al. (2007), who related the an-
nual adult survival probability to covariates characterising fishing effort in the Pacific using
a linear regression built as a constraint into the capture-recapture model (Clobert & Lebre-
ton 1985; Lebreton, Burnham, Clobert & Anderson 1992). In relation with the sparseness
of the data, the low number of years of study (11 years), and the large number of largely
collinear candidate covariates, strict rules for protecting the quality of regression were
applied, by reducing 8 candidate covariates to 3 uncorrelated ones using principal com-
ponents analysis, and by using a Bonferroni correction to test for the effect of these 3
resulting covariates. The annual adult survival probability was significantly linked to the
second principal component, which, among the 8 original variables, was most strongly cor-
related with the tonnage of swordfish over the north Pacific. Hence, the results indicated,
as expected, a decrease in survival with increasing fishing effort. However, the estimated
survival for a fishing effort equal to 0 (corresponding to a negative value of the second
principal component) was larger than the baseline survival one could expect for such a
species (∼0.95) and even larger than 1, suggesting a nonlinear relationship for low levels
of fishing effort, discussed by V´
eran et al. (2007).
Again, this case study is thus quite typical of conservation biology, in that both survey and
demographic information are sparse and lead only to uncertain answers. A state space model
will serve as a link between the two pieces of the puzzle.
3. STATE-SPACE MODELS
A state-space model is made of a state and of an observation equation. The state equation de-
scribes the state of the system, here a population vector, over a discrete time scale, while the
observation equation relates this state to the measurements of the system, here the partial survey.
State-space models are generally used in engineering to estimate the state of the system based
88 V´
ERAN & LEBRETON Vol. 36, No. 1
on incomplete observations (through the observation equation) and assuming a perfect knowl-
edge of the behaviour of the system, i.e., of the parameters of the state equation (Harvey 1989,
pp. 10–11).
We present here and comment on the specific state equation and observation equation for the
black-footed albatross study case.
3.1. State equation.
The state equation relates the population vector of the number of females at time t,Nt, to that
at time t+ 1,Nt+1. The expected change over one time step E (Nt+1 |Nt)is given, as in other
state space models for animal populations (Gauthier et al. 2007), by a stage-structured matrix
model (Caswell 2001, ch. 2, ch. 4), based on the life cycle of the black-footed albatross (see
Table 1). The stages are mutually exclusive. They consist of 8 age classes and 2 adult states,
breeder (B), and nonbreeders (NB), respectively. The parameters are presented in Table 2; the
transitions between states is illustrated in Figure 1.
TABLE 1: The matrix Mused in the state-space model for the black-footed albatross Phoebastria
nigripes. The expected vector E (Nt+1 |Nt)of population size at time t+ 1 is obtained as M×Nt. The
parameters are described in the text and in Table 2.
Class 1 2 3 4 5 6 7 8 NB B
10 0 0 0 0 0 0 0 0 0.5×s1×f
2simm 0 0 0 0 0 0 0 0 0
30simm 0 0 0 0 0 0 0 0
40 0 simm 0 0 0 0 0 0 0
50 0 0 simm 0 0 0 0 0 0
60 0 0 0 sa(1 −r5)0 0 0 0 0
70 0 0 0 0 sa(1 −r)0 0 0 0
80 0 0 0 0 0 sa(1 −r)0 0 0
NB 0 0 0 0 0 0 0 0 sa(1 −b)sa(1 −sa)(1 −a)
B0 0 0 0 sa×r5sa×r sa×r sasa×b sa(1 −sa)a+s2
a
Breeders produce young females that enter the population at the next time step in age class 1:
The resulting net fecundity 0.5×s1×f, expressed in females of age 1 produced per female,
is the product of the fecundity f(in young fledged per female) by the proportion of female at
birth, 0.5, and the survival probability over the 6 months from fledging, i.e., the time they leave
the nest, to age 1, s1. Survival is assumed to be constant from fledging, at 6 months of age,
until age 5. We denote the corresponding annual survival probability as simm. One thus has
s1=s0.5
imm since s1covers 6 months. The overall survival from fledging to age 5, sI, is thus
arbitrarily represented as s4.5
imm. The assumption of constancy of survival between fledging and
age 5 inherent in this decomposition has no effect on the growth rate and adult structure of the
model, entirely determined by s1. Breeders and nonbreeders over age 5 (i.e., in stages 5, 6,
7, 8, B, NB) are assumed to have the same adult survival probability sa. Individuals “recruit”
by progressively moving to state B, between age 5, the age at first breeding at the population
level, and age 8, with respective rates r5(at age 5), r(at age 6 and 7), and 1 at age 8, implying
full recruitment. Each of the corresponding transition probabilities is obtained as recruitment
probability ×survival probability (Table 1). Once they become breeders, individuals breed every
year if both members of the couple survive, or if they loose their partner but repair immediately
sa(1 −sa)a+s2
a. The widow individuals unable to repair in the same year move during the next
year to the nonbreeder state sa(1 −sa)(1 −a). Nonbreeding individuals breed again when they
repair sa×b; otherwise they remain in the nonbreeding state sa(1−b). The relationship taken into
2008 INTEGRATED MODELLING 89
account in the transitions just described between the death of the partner and future reproduction
is a specific feature of this model (V´
eran in preparation). It accounts for a peculiar albatross
trait of specific interest in the context of the impact of longline by-catch, as the demographic
cost of widowing, in terms of time to form a pair and reproduce again, will tend to increase the
demographic impact of by-catch beyond the direct effect on survival. When using the model in
practice, annual survival probabilities, and in particular, the adult one sawill vary over time, for
example, in relation to fishing effort. The resulting matrix model is thus E (Nt+1 |Nt) = MtNt
where Mtis given in Table 1.
FIGURE 1: Life cycle of the black-footed albatross (P. nigripes). From fledging to age 5, the annual
survival probability is assumed to be simm. From 5 years old onwards, the annual survival probability is
assumed to be the adult one sa. A proportion r5of 5 year old individuals breeds for the first time. This
proportion is rfor 6 and 7 year old individuals, and 1 for 8 year old individuals. Once they become
breeders, if individuals loose their partner, they form a new pair immediately with probability a. The
widows unable to repair the same year become nonbreeders the following year. Nonbreeding individuals
have a probability bto pair again and hence to breed again; otherwise they remain in the nonbreeding state.
0.5×ffemale chick per female (i.e., pair) fledge at the age of 6 months. They are assumed to have a
probability s1=s0.5
imm to survive over the 6 months that follow, i.e., until one year old.
In the state equation, a random term εtaccounts for demographic stochasticity, leading to
Nt+1 =
N1
N2
N3
N4
N5
N6
N7
N8
NB
B
t+1
=Mt
N1
N2
N3
N4
N5
N6
N7
N8
NB
B
t
+εt.(1)
90 V´
ERAN & LEBRETON Vol. 36, No. 1
The demographic stochasticity for survival is binomial. An albatross laying only 1 egg, i.e.,
f < 1, that for the fecundity component too. Even with small populations made up of a few hun-
dred individuals, a normal approximation is quite acceptable. Hence the components of εtare
assumed to be independent and normally distributed with zero means and appropriate binomial
variances (see also Besbeas, Freeman, Morgan & Catchpole 2002). For instance the variance for
the first component, corresponding to the net fecundity, was Btp(1 −p)with p= 0.5×f×s1.
A key feature is that the state equation is linear. Nonlinearity would correspond biologically
to density-dependence; Positive density-dependence, corresponding to a deterioration of demo-
graphic performances with increasing population size, will in general not be a priority to consider
in conservation biology, dealing with small, often decreasing, largely environment driven popula-
tions. Inverse density-dependence at low population size, known as the Allee effect (Courchamp,
Clutton-Brock & Grenfell 1999) is on the contrary quite topical in conservation biology for very
small populations, although not of concern in our case. It could for instance result from difficul-
ties in finding a partner, as a result of randomly unbalanced sex-ratio. It may require nonlinear
state equations and then preclude the formal use of Kalman filtering (see below). An alternative
is to think then of population analysis in terms of pseudo-extinction, considering the population
viability is definitely impaired below some threshold, corresponding for instance to a deterio-
ration of demographic performance through the Allee effect. Linear state equations seem thus
to have a wide applicability in Conservation Biology. For vertebrate population conservation or
management concerns, matrix models are commonly built from the knowledge of the life cycle
(Caswell 2001, p. 56 ff.), and thus it will be relatively easy to build a linear state equation.
TABLE 2: The demographic parameters of the state space model for the black-footed albatross
Phoebastria nigripes (V´
eran & Lebreton, in preparation).
Parameter Notation Estimate Origin
Fecundity f0.67 US Fish and Wildlife Service
(E. Flint, unpublished data)
Annual Immature survival simm 0.771 Based on cohort analysis
then estimated (V´
eran & Lebreton, in prep.)
in this paper
Adult survival saEstimated in
this paper
Probability of remaining a0.34 Based on data of time of repairing
breeder after loss of (Cousins & Cooper 2000;
partner V´
eran & Lebreton, in prep.)
Probability of breeding b0.62 Based on data of time of repairing
for nonbreeder birds (Cousins & Cooper 2000;
V´
eran & Lebreton, in prep.)
Recruitment rate for r50.28 Based on cohort analysis
5 years old birds (V´
eran & Lebreton, in prep.)
Recruitment rate for r0.61 Based on cohort analysis
birds >5years (V´
eran & Lebreton, in prep.)
2008 INTEGRATED MODELLING 91
3.2. Observation equation.
The observation equation links the vector of population size, Nt, with counts of breeding pairs yt,
corresponding to the 10th component of Nt, accounting for survey uncertainty through a random
term ηt.
Hence the observation equation is:
yt=ANt+ηt(2)
with A= (0,0,0,0,0,0,0,0,0,1).
The random component on population size ηtis assumed to follow a Normal distribution,
with mean 0 and variance σ2=cB2
t. This amounts to assuming a constant coefficient of vari-
ation √cfor the survey, quite a logical assumption in practice and not a very constraining one
given the relatively narrow variation in the number of surveyed breeders over the study years.
Moreover, these components are assumed to be independent over time. Gauthier, Besbeas, Le-
breton & Morgan (2007) show that replacing the state vector component by the observation, i.e.,
approximating the variance as cy2
tgives quite satisfying results, although it is formally incor-
rect. At this stage, the only quantities observed are the surveyed numbers of breeders over time
YT= (y1,...,yT), traditionally called the “observations.”
4. KALMAN FILTERING AND OVERALL LIKELIHOOD
4.1. Kalman filter and survey likelihood.
In the simplest case of a linear state-space model, the Kalman filter is a method for recursively
forecasting the values of the state vector given the past observations. It is interesting to note that
the first mention of a possible application of the Kalman filter in population biology appears
in Brillinger (1981), a paper published in The Canadian Journal of Statistics. The Kalman
filter can be used to build a likelihood of the survey data LK(Y, θ), in order to estimate the
parameters θof the matrix model in (1) and the variance parameter in (2). Doing so, one attempts
to investigate process (the demographic flows) based on patterns (the surveys) and it is clear that
many components of θwill not be separately identifiable. This is why this likelihood will be
combined later with other pieces of information on the parameters.
All variables in (1) and (2) being normally distributed, the state vector at time tis also nor-
mally distributed, conditional on past observations y1,...,yt−1. In order to fully determine these
conditional distributions, it is thus sufficient to obtain their first two moments, which are given
directly by the usual form of the Kalman filter as recurrence equations (Harvey 1989, p. 106),
and to provide a density function g(N0)for the distribution of the initial population vector N0.
With YT= (y1, . . . , yT), the probability density of YTcan be expressed as
f(YT) = T
Y
t=2
f(Yt|Yt−1, N0, θ)f(Y1|N0, θ)g(N0).
To initialize the Kalman filter, we ran the time-dependent matrix model over the entire study pe-
riod, taking the final age-stage distribution and the observed initial number of breeders to produce
the expectation of N0, following Gauthier, Besbeas, Lebreton & Morgan (2007). The covariance
matrix was obtained by assuming a coefficient of variation equal to 10%, independently on all
components. A multivariate normal density with this expectation and covariance matrix was used
for the probability density of the initial state vector, g(N0).
A derived technique, Kalman smoothing (Harvey 1989) provides estimates of the state
vector, and in turn, the surveyed population size based on all past and future observation
YT= (y1,...,yT). The latter estimates appear as more relevant to an analysis such as ours
than the similar one derived from Kalman filtering which only takes into account the information
of past surveys.
92 V´
ERAN & LEBRETON Vol. 36, No. 1
4.2. Overall likelihood.
The purpose of the integrated population modeling is to estimate parameters by combining dif-
ferent sources of information. Based on the independence of the capture-recapture and the survey
data, an overall likelihood can be obtained here as the product of a likelihood for the capture-
recapture data and that for the survey data (Besbeas, Freeman, Morgan & Catchpole 2002).
As often, the capture-recapture models and the resulting likelihood considered in this study
were unavoidably complex, mostly because of a strong capture heterogeneity that had to be
accounted for in the model structure (V´
eran et al. 2007). The data did not fit a simple Cormack–
Jolly–Seber model, and showed strong signs of heterogeneity of capture. V´
eran et al. (2007)
had to use a two-state capture-recapture model which accounted for temporary emigration, with
age-dependence in the emigration rate. Survival was considered equal for the two states. This
model approximately fitted the data, and was further constrained. As a starting point we use a
model with time-dependent annual adult survival probabilities.
To combine the capture-recapture likelihood with the likelihood for the survey data, follow-
ing (Besbeas, Morgan & Lebreton 2003), we used the approximation of the capture-recapture
likelihood based on the asymptotic normal distribution of the estimates. Namely, we used the
estimates of the parameters ˆ
θof the final capture-recapture model and their estimated covariance
matrices b
Σ(V´
eran et al. 2007) to approximate the likelihood of the capture-recapture data by
Lc(X, θ)≈φ(θ;ˆ
θ, b
Σ), where φ(X;µ, Σ) is the probability density of a vector Xdistributed as
N(µ, Σ) (Besbeas, Morgan & Lebreton 2003).
Among the parameters of the state equation, i.e., of the matrix population model, estimates
of the parameters a,b,f, and recruitment rates riand r5were available without information on
their uncertainty (V´
eran & Lebreton, in prep.). Given the high generation time of albatross, these
parameters assumed to be known without uncertainty were indeed low sensitivity parameters in
the matrix model at the core of the state equation (see Gaillard et al. 2005). As a consequence,
assuming a zero variance for these parameters had negligible consequences on the likelihood, as
checked by Gauthier, Besbeas, Lebreton & Morgan (2007).
Finally, immature survival, for which no estimate was available, was looked at in two differ-
ent fashions:
•First, a guess estimate was used, and supposed to be known without uncertainty, to check
the behaviour of the overall likelihood by comparison with the capture-recapture analysis,
in particular to determine what degree of improvement could be obtained in the mod-
els relating adult survival to fishing effort. The value used, sI= 0.310, was deduced
from the only available estimate of annual immature survival simm = 0.771 (Cousins &
Cooper 2000) using sI=s4.5
imm which also induced s1=s0.5
imm = 0.878.
•Second, in a more tentative fashion, immature survival was considered as part of the pa-
rameters to estimate. In this case, the information on immature survival could only come
from the survey data.
The overall log-likelihood was obtained as
log Lc(X, θ) + log Lk(Y, θ )≈log Lc(X, θ) + log φ(θ;ˆ
θ, b
Σ).
Parameter estimates ˆ
θand their variance-covariance were obtained by maximizing the overall
likelihood. We used the MATLAB code kindly provided by P. Besbeas for the Kalman filter
calculations. A quasi-Newton method (function fminunc in MATLAB) was used to minimize
with respect to θthe approximated relative deviance −2{log Lc(X, θ) + log φ(θ;ˆ
θ, b
Σ)}.
4.3. Model selection.
In a first step, we estimated the annual adult survival probability saand the squared coefficient
of variation of the observed number of breeders c, by minimizing the approximated relative
2008 INTEGRATED MODELLING 93
deviance of the integrated model above under different constraints; all other parameters were
assumed to be known with uncertainty, in particular the guess estimate of immature survival. We
considered 3 different models:
•In the first model, denoted sa(i), the annual adult survival probability was considered as
constant over time.
•In the second model, denoted sa(t), it varied over time.
•In the third model, denoted sa(E), it was linked to fishing effort Eas: sa=β0−
β1E. The measure of fishing effort was, as explained in the Introduction, the 2nd principal
component of 8 fishing effort covariates, selected as the best predictor of adult survival in
capture-recapture models by V´
eran et al. (2007).
In the latter model the parameters of the regression equation, β0and β1, replaced the time-
dependent annual survival probabilities as parameters in the deviance submitted to minimiza-
tion. The linear equation above can be derived as an excellent approximation for high baseline
survival and relatively low rates of by-catch from standard theory on the dynamics of exploited
populations (see, e.g., Lebreton 2005), assuming the number of individuals caught by long lines
is proportional to fishing effort.
Model selection was based on the Akaike information criterion (AIC; Lebreton, Burnham,
Clobert & Anderson 1992).
The effect of longline fishing was also assessed using a formal test of the null hypothesis H0:
β1= 0 versus the alternative hypothesis H1:β16= 0, by analysis of deviance (Skalski 1996).
This statistic is
F=Dev(sa(i)) −Dev(sa(E))
Dev(sa(E)) −Dev(sa(t)) ×n,
in which nis the difference in the number of identifiable parameters between models sd(t)
and sa(E). It approximately follows under H0:β1= 0 a Fisher–Snedecor distribution with 1
and ndegrees of freedom even in presence of unexplained environmental variation (Lebreton &
Gimenez, in prep.).
In a second step, we also estimated immature survival, and obtained a profile likelihood
95% confidence interval. As for adult survival, we considered three different models, under the
preferred model for adult survival:
•The first one with constant immature survival, denoted as model simm(i).
•The second with time varying immature survival, denoted as model simm(t).
•The third one with immature survival function of fishing effort, denoted as model
simm(E). For the sake of parsimony, when used under sa(E), we used the same slope
for young and adults.
5. RESULTS
Among the three model structures for adult survival, model sa(E)with survival related to
fishing effort had the lowest AIC (Table 3). The analysis of deviance was highly significant
(F1,9= 7.03; p= 0.0013). The estimations of the number of breeders based on the Kalman
smoother fit the observations quite satisfactorily (see Figure 2): the smoothed breeding pop-
ulation size was almost always within the 95% confidence interval of the observed breeding
population size, except for year 2000 where the smoothed breeding population size was esti-
mated to 46822 individuals for an upper limit of the confidence interval equal to 46109. The
estimated coefficient of variation of the surveyed number of breeders was √ˆc= 0.073. The
closeness between the smoothed and the surveyed numbers of breeders (Figure 2) confirmed that
94 V´
ERAN & LEBRETON Vol. 36, No. 1
the approximation σ2=cy2
tinstead of σ2=cB2
twas quite reasonable in (2). Among the three
models for immature survival, the lowest AIC model was that with constant immature survival
(Table 4), although the model with immature survival function of fishing effort (sa(E)simm(E))
and a same slope for adults and immatures came very close (AIC=202.48 vs. 201.36). Although
the sign of the slope was as expected negative, the evidence for an effect of on immature survival
was not significant (analysis of deviance, as a one-tailed student’s test, t9= 1.33; p= 0.1081).
The estimate of the overall immature survival probability sIwas 0.230 (with a profile likeli-
hood 95% confidence interval equal to [0.155; 0.305], leading for the annual immature survival
probability simm to an estimate equal to 0.721, with a profile likelihood 95% confidence interval
equal to [0.661,0.768].
TABLE 3: Model selection for adult survival probabilities based on the combined likelihood. np is the
total number of identifiable parameters, including the coefficient of variation of the census. The preferred
model (lowest AIC) is shown in bold.
Model Deviance np QAIC
sa(t)186.60 12 210.60
sa(i)202.24 2 206.24
sa(E)195.38 3 201.38
0
10000
20000
30000
40000
50000
60000
70000
1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
FIGURE 2: Surveyed (∗) and smoothed (plain line) with 95% confidence interval (dotted line) number of
breeding pairs for the surveyed part of the Hawaii population size of the black-footed albatross. The
smoothed values are obtained by the Kalman smoother.
TABLE 4: Model selection for immature survival probability under a model for adult survival assuming a
linear relationship with fishing effort. np is the total number of identifiable parameters, including the
coefficient of variation of the survey. The preferred model (lowest AIC) is shown in bold. In model
sa(E)simm(E)(last line), the slope of survival as a function of fishing effort is assumed to be the same
for young and adults: this model thus has the same number of parameters as model sa(E)simm (i)(line
above).
Model Deviance np QAIC
sa(E)simm(t)188.80 14 236.80
sa(E)simm(i)193.36 4 201.36
sa(E)simm(E)194.48 4 202.48
2008 INTEGRATED MODELLING 95
6. DISCUSSION
Our first conclusion is that, in the black-footed albatross example, the integrated population
modelling (IPM) was quite efficient in providing an estimate of immature survival, a missing
link in the comprehensive view of population dynamics provided by the matrix population model
at the core of the state equation. This conclusion is extremely useful and relatively unexpected:
the elasticity (Caswell 2001), or relative sensitivity, of the asymptotic growth rate one would
deduce from a constant parameter matrix model to changes in immature survival is indeed the
inverse of generation time (Lebreton & Clobert 1991; Gaillard et al. 2005). For the black-footed
Albatross, with a generation time close to 25 years (Niel & Lebreton 2005), this sensitivity is
around 0.04, i.e., a 25% change in immature survival is needed to generate a 1% change in the
population growth rate. Obviously this context results in a fairly wide confidence interval for
immature survival, carefully estimated here by profile likelihood. However, even when direct
capture-recapture approaches such as those based on dead recoveries can be used to estimate
immature survival, the precision will often be low. Hence, it is clear that in many cases, IPM will
be able to help in estimating parameters difficult to estimate by regular demographic approaches.
TABLE 5: Comparison of parameter estimates and their precision: capture-recapture analysis only versus
integrated population modelling (IPM) without estimation of immature survival, and the integrated
population modelling with estimation of immature survival. In all cases, adult survival is linearly linked to
fishing effort E, as sa(i) = β0+β1E(i)for interval i, a model denoted sa(E). Since fishing effort is
expressed as a standardized covariate, the intercept corresponds to the average fishing effort.
Parameter estimate
Model Intercept β0(s.e.) Slope β1(s.e.)
Capture-recapture 0.930 (0.0043) −0.036 (0.0047)
IPM, sa(E), without 0.926 (0.0029) −0.034 (0.0044)
estimation of immature survival
IPM, sa(E)simm(i), with 0.930 (0.0028) −0.033 (0.0044)
estimation of immature survival
Our second conclusion is that IPM confirmed the capture-recapture based evidence for a
relationship between adult survival and fishing effort. The survival estimates were comparable
to those obtained by capture-recapture and their precision was slightly increased. One could
have expected that the estimation of immature survival would have used all the information
available in the population survey. In fact, immature survival was largely determined by the
overall population growth rate. In that sense IPM formalizes the ad hoc practice of model tuning,
according to which a parameter can be estimated by matching the growth rate of a matrix model
to the growth rate estimated from the surveys. However, the surveys have ups and downs which
provide information distinct from that in the growth rate and which can translate in the estimates
in two different fashions:
1. If these ups and downs do no match changes in adult survival, the most sensitive pa-
rameters, they will contribute to increase the estimate of the variance parameter of the
observation equation. This will also be the case if the variation in survival estimates is
spurious.
2. If they do match changes in survival, they will increase theprecision of survival estimates,
and lead to a low estimate of the variance parameter in the state equation.
We are clearly in the second case here, as confirmed by the good match between the observed
survey of the number of breeders and the smoothed estimates, and the low estimated coefficient of
96 V´
ERAN & LEBRETON Vol. 36, No. 1
variation of the survey. This good match also dismisses another possible mechanism, a variation
in the number of breeding adults being only the result of a variation in the number of individuals
skipping reproduction.
IPM appears thus as efficient and promising for consolidating relationships expressing a vari-
ation in demographic parameters explained by environmental covariates. A key perspective for
integrated modelling concerns variation over time in parameters unexplained by environmental
covariates. When such a time variation will be confirmed by IPM, i.e., when concomitant ev-
idence for such a variation comes from the survey, a capture-recapture model with a random
effect will be relevant. Based on the alternative described above, IPM would then appear as nat-
ural way to separate a process variance (e.g., in survival) from a sampling variance both in the
capture-recapture sampling and in the surveys.
More generally, in conservation biology as well as in other population dynamics studies, IPM
will be useful every time there are problems with parameter identifiability. In particular multi-
event models (Pradel 2005) which generalise multistate capture-recapture models to uncertain
state attribution, appear to be very promising by widely broadening the type of biological infor-
mation that can be recovered from individual capture-recapture data. The price to pay in terms
of parameters identifiability could be to a great extent alleviated by combining these models with
survey information by IPM.
Finally, this study confirms that integrated modelling is a very promising method in demogra-
phy and particularly in conservation biology, a field characterized by often sparse and incomplete
data. We expect improved population estimates resulting from IPM to help make population vi-
ability analysis, in particular extinction risk analyses, more precise and robust.
ACKNOWLEDGEMENTS
This project was funded by Cooperative Agreement NA17RJ1230 between the Joint Institute for Marine
and Atmospheric Research (JIMAR) and the National Oceanic and Atmospheric Administration (NOAA).
The views expressed herein are those of the authors and do not necessarily reflect the views of NOAA of
any of its subdivisions. The U.S. National Marine Fisheries Service and the U.S. Fish and Wildlife Service
provided data. The field research was done on a National Wildlife Refuge by refuge staff and volunteers.
We would like to thank the field workers for their hard work banding and reading bands. We thank also
P. Besbeas for making available the MATLAB code for the Kalman filter. We also thank an anonymous
referee for helpful comments that greatly helped to improve this paper.
REFERENCES
J. M. Arnold, S. Brault & J. P. Croxall (2006). Albatross populations in peril: a population trajectory for
Black-Browed Albatrosses at South Georgia. Ecological Applications, 16, 419–432.
S. R. Beissinger & M. I. Westphal (1998). On the use of demographic models of population viability in
endangered species management. Journal of Wildlife Management, 62, 821–841.
P. Besbeas, S. A. Freeman & B. J. T. Morgan (2005). The potential of integrated population modelling.
Australian and New Zealand Journal of Statistics, 47, 35–38.
P. Besbeas, S. A. Freeman, B. J. T. Morgan & E. A. Catchpole (2002). Integrating ark-recapture-recovery
and census data to estimate animal abundance and demographic parameters. Biometrics, 58, 540–547.
P. Besbeas, J.-D. Lebreton & B. J. T. Morgan (2003). The efficient integration of abundance and demo-
graphic data. Journal of the Royal Statistical Society Series C: Applied Statistics, 52, 95–102.
D. R. Brillinger (1981). Some aspects of modern population mathematics. The Canadian Journal of Statis-
tics, 9, 173–194.
S. T. Buckland, K. B. Newman, L. Thomas & N. B. Koesters (2004). State-space models for the dynamics
of wild animal populations. Ecological Modelling, 171, 157–175.
2008 INTEGRATED MODELLING 97
H. Caswell (2001). Matrix Population Models: Construction, Analysis, and Interpretation, Second Edition,
Sinauer, Sunderland, MA.
H. Caswell & A. M. John (1992). From individual to the population in demographic models. Individual-
Based Models and Approaches in Ecology (D. L. DeAngelis, ed.), London, Chapman & Hall, pp.
38–51.
J. S. Clark (2003). Uncertainty and variability in demography and population growth: A hierarchical ap-
proach. Ecology, 84, 1370–1381.
J. Clobert & J.-D. Lebreton (1985). D´
ependance de facteurs de milieu dans les estimations de taux de survie
par capture-recapture. Biometrics, 41, 1031–1037.
J. Clobert & J.-D. Lebreton (1991). Estimation of demographic parameters in bird populations. Bird Popu-
lation Studies: Relevance to Conservation and Management (C. M. Perrins, J.-D. Lebreton & G. J. M.
Hirons, eds.), Oxford University Press, Oxford, pp. 75–104.
F. Courchamp, T. Clutton-Brock & B. Grenfell (1999). Inverse density dependence and the Allee effect.
Trends in Ecology and Evolution, 14, 405–410.
K. L. Cousins & J. Cooper (2000). The population biology of the black-footed albatross in relation to
mortality caused by fishing. Technical report for the Western Pacific Regional Fishery Management
Council, Honolulu, Hawaii.
B. Dennis, J. M. Ponciano, S. R. Lele, M. L. Taper & D. F. Staples (2006). Estimating density dependence,
process noise, and observation error. Ecological Monographs, 76, 323–341.
J.-M. Gaillard, N. G. Yoccoz, J.-D. Lebreton, C. Bonenfant, S. Devillard, A. Loison, D. Pontier & D.
Allaine (2005). Generation time: A reliable metric to measure life-history variation among mammalian
populations. American Naturalist, 166, 119–123.
G. Gauthier, P. Besbeas, J.-D. Lebreton & B. J. T. Morgan (2007). Population growth in snow geese: A
modeling approach integrating demographic and survey information. Ecology, 88, 1420–1429.
G. Gauthier & J.-D. Lebreton (2004). Population models for Greater Snow Geese: a comparison of different
approaches to assess potential impacts of harvest. Animal Biodiversity and Conservation, 27, 503–514.
A. C. Harvey (1989). Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge Uni-
versity Press.
J.-D. Lebreton (2005). Dynamical and statistical models for exploited populations. Australian and New
Zealand Journal of Statistics, 47, 49–63.
J.-D. Lebreton, K. P. Burnham, J. Clobert & D. R. Anderson (1992). Modeling survival and testing biolog-
ical hypotheses using marked animals: a unified approach with case studies. Ecological Monographs,
62, 67–118.
J.-D. Lebreton & J. Clobert (1991). Bird population dynamics, management, and conservation: the role of
mathematical modelling. Bird Population Studies: Relevance to Conservation and Management (C. M.
Perrins, J.-D. Lebreton & G. J. M. Hirons, eds.), Oxford University Press, pp. 105–125.
J.-D. Lebreton, R. Pradel & J. Clobert (1993). The statistical analysis of survival in animal populations.
Trends of Research in Ecology and Evolution, 8, 91–95.
R. L. Lewison & L. B. Crowder (2003). Estimating fishery bycatch and effects on a vulnerable seabird
population. Ecological Applications, 13, 743–753.
R. S. Miller & D. B. Botkin (1974). Endangered Species - models and predictions. American Scientist, 62,
172–181.
D. C. Nel, F. Taylor, P. Ryan & J. Cooper (2003). Population dynamics of the wandering albatross Diomedea
exulans at Marion Island: longline fishing and environmental influences. African Journal of Marine
Science, 25, 503–517.
C. Niel & J.-D. Lebreton (2005). Using demographic invariant to detect overharvested bird populations
from inomplete data. Conservation Biology, 19, 826–835.
98 V´
ERAN & LEBRETON Vol. 36, No. 1
R. Pradel (2005). Multievent: an extension of multistate capture-recapture models to uncertain states.
Biometrics, 61, 442–447.
J. R. Skalski (1996). Regression of abundance estimates from mark-recapture surveys against environmental
covariates. Canadian Journal of Aquatic Sciences, 53, 196–204.
M. E. Soul´
e (1987). Viable Population for Conservation. Cambridge University Press.
L. Thomas, S. T. Buckland, K. B. Newman & J. Harwood (2005). A unified framework for modelling
wildlife population dynamics. Australian and New Zealand Journal of Statistics, 47, 19–34.
S. V´
eran, O. Gimenez, E. Flint, W. Kendall, P. Doherty & J.-D. Lebreton (2007). Quantifying the impact
of longline fisheries on adult survival in the black-footed albatross. Journal of Applied Ecology, 44,
942–952.
H. Weimerskirch, N. Brothers & P. Jouventin (1997). Population dynamics of wandering albatross
Diomedea exulans and amsterdam albatross D. amsterdamensis in the indian ocean and their relation-
ship with long-line fisheries: conservation implications. Biological Conservation, 79, 257–270.
Received 28 January 2007 Sophie V´
ERAN: sveran@nature.berkeley.edu
Accepted 24 October 2007 Department of Environmental Science, Policy and Management
137 Mulford Hall #3114, University of California
Berkeley, CA 94720-3114, USA
Jean-Dominique LEBRETON: jean-dominique.lebreton@cefe.cnrs.fr
Centre d’´
ecologie fonctionnelle et ´
evolutive, UMR 5175
Centre National de la Recherche Scientifique
1919, route de Mende, FR-34293 Montpellier cedex 5, France
