Using Demographic Invariants to Detect
Overharvested Bird Populations from
Incomplete Data
COLIN NIEL∗ † AND JEAN-DOMINIQUE LEBRETON∗ ‡
∗
Centre d’Ecologie Fonctionnelle et Evolutive (CEFE), UMR 5175 Centre National de la Recherche Scientifique (CNRS),
1919 Route de Mende, 34 293 Montpellier Cedex 5, France
Abstract: Conservation biology must be able to provide guidelines even when available data are incomplete,
because data on rare and endangered species are usually limited. For instance, data on the effect of additional—
human-induced—sources of mortality on vertebrate populations, such as bycatch of seabirds by longline
fisheries, are typically incomplete. The importance of an additional source of mortality can be evaluated
by comparing it with the maximum annual growth rate of the species of concern, and various authors have
attempted to determine the maximum growth rate from incomplete data. We developed a procedure we call the
“demographic invariant method” (DIM). First we determined that the maximum growth rate per generation
does not vary, by recalling that it is a dimensionless number primarily independent of body weight and also by
empirically establishing its near constancy over a restricted set of bird species for which reliable demographic
information was available. This first step provided an implicit function linking generation time and maximum
annual growth rate, from which we obtained the maximum annual growth rate as a simple function of
generation time. From several different ways of obtaining estimates of generation time, we derived in turn
several ways to estimate the maximum annual growth rate of a bird species from incomplete demographic
data set. We applied our approach to the Black-footed Albatross (Phoebastria nigripes) and determined from
incomplete data that longline fishery bycatch has a biologically significant impact on the growth potential of
Black-footed Albatross populations. Our method can be applied broadly to the conservation and management
of harvested bird populations.
Key Words: annual growth rate, Black-footed Albatross, generation time, seabird bycatch
Uso de Invariantes Demográficas para la Detección de la Sobreexplotación de Poblaciones de Aves Utilizando Datos
Incompletos
Resumen: La biologı́a de la conservación debe ser capaz de proporcionar directrices aun cuando los datos
están incompletos porque usualmente hay pocos datos de especies raras y en peligro. Por ejemplo, los datos sobre
el efecto de fuentes – inducidas por humanos – adicionales de mortalidad sobre poblaciones de vertebrados,
como la captura incidental de aves marinas por pesquerı́as, están incompletos tı́picamente. La importancia
de una fuente adicional de mortalidad puede ser evaluada mediante la comparación de la fuente con la tasa
máxima de crecimiento anual de la especie en cuestión. Varios autores han tratado de determinar la tasa de
crecimiento máximo a partir de datos incompletos. Desarrollamos el método demográfico invariante (MDI).
Primero determinamos que la tasa de crecimiento máximo por generación no varı́a, demostrando que es un
número adimensional primariamente independiente del peso corporal y estableciendo su cuasi constancia en
un conjunto restringido de especies de aves de las que se disponı́a de información demográfica confiable. Este
†Current address: Direction de l’Agriculture et de la Forêt de Guyane, Parc Rebard, BP 5002, 97 302 Cayenne Cedex, French Guiana.
‡Address correspondence to J.-D. Lebreton, email lebreton@cefe.cnrs-mop.fr
Paper received July 8, 2003; revised manuscript accepted August 13, 2004.
826
Conservation Biology, Pages 826–835
Volume 19, No. 3, June 2005
Niel & Lebreton
Demographic Invariants and Overharvest
827
primer paso proporcionó una función implı́cita que eslabonó el tiempo generacional y la tasa máxima de
crecimiento anual, de la que obtuvimos la tasa máxima de crecimiento anual como una función simple del
tiempo generacional. A partir de varias formas diferentes para obtener estimaciones del tiempo generacional,
derivamos varias formas para estimar la tasa máxima de crecimiento anual de una especie de ave a partir
de un conjunto incompleto de datos demográficos. Aplicamos nuestro método al albatros Phoebastria nigripes
y, a partir de datos incompletos, determinamos que la captura incidental por pesquerı́as tiene un impacto
biológico significativo sobre el potencial de crecimiento de las poblaciones de P. nigripes. Nuestro método puede
ser aplicado ampliamente en la conservación y manejo de poblaciones de aves.
Palabras Clave: captura incidental de aves marinas, tasa de crecimiento anual, tiempo generacional, Phoebastria
nigripes
Introduction
Conservation biology must be able to provide guidelines
for managing ecological systems even in the absence of
complete data. Incomplete data on rare and endangered
species are common, often because these species have
a limited range and are difficult to study (examples in
Collar & Andrew 1988). “Even among the named species
only a minute fraction, less than 1%, have been studied
beyond the essentials of habitat preference and diagnostic
anatomy” (Wilson 2000).
The deleterious effect of additional—human-induced
—sources of mortality on vertebrate populations is a
typical example of such conservation issues, where demographic information is needed but is rarely available
in enough detail and precision. For instance, various albatross (Diomedeidae) populations have been suffering
over the last 20 years from incidental bycatch by longline
fisheries (Croxall 1998). For some of these species, such
as the Black-footed Albatross (Phoebastria nigripes), neither a series of censuses nor monitoring of marked individuals are available to determine with enough certainty
the impact of bycatch on the population dynamics of the
species (Cousins & Cooper 1999).
The central question in all such cases is to decide
whether or not the rate of additional mortality is sustainable (i.e., whether or not the population is being overharvested). A key quantity in this context is the maximum
annual growth rate of the species in question (λ max , Odum
1971; Caughley 1977), which is the annual growth rate of
a population of this species without limiting factors and at
low population density. It can be measured, for example,
when a population is colonizing a new environment. Even
in optimal demographic conditions, a population cannot
increase by a proportion higher than λ max −1 (i.e., the
sustainable incidental mortality cannot exceed λ max −1).
To assess overharvest from incomplete data, Robinson
and Redford (1991) and Wade (1998) propose (1) estimating λ max in the absence of complete life-table data, based
on a simplified version of the Euler-Lotka equation (e.g.,
Caswell 2000), which links demographic parameters and
the growth rate of the population and (2) comparing the
actual harvest to Nβ(λ max −1), where N is an estimate of
population size and β is a number smaller than 1 that
takes into account the effect of density on the growth
rate (Wade 1998).
The simplified version of the Euler-Lotka equation used
by Robinson and Redford (1991) was first proposed by
Cole (1954). It considers only the age at first reproduction, the adult fecundity, and the age of last reproduction,
and assumes that all survival probabilities are equal to 1
prior to the age of last reproduction (i.e., no mortality).
Because of this unrealistic assumption, the resulting estimates of growth rate (and in turn of sustainable humaninduced mortality) are unreliable (Hayssen 1984). They
tend to be biased by excess, a troublesome characteristic for conservation purposes. Slade et al. (1998) propose
another version of the equation, which incorporates survival rates and brings some improvements to the Robinson and Redford (1991) approach.
The comparative approach (e.g., Harvey & Purvis
1991), however, shows that many life-history traits covary in a predictable way among wide sets of species. In
particular, the allometric variation of many traits related
to the production and use of energy with body size is
well known (e.g., Mc Mahon 1973). Similar relationships
exist for demographic traits such as adult life expectancy,
survival, and fecundity (Gaillard et al. 1989). Although
these relationships have received considerable attention
from evolutionary biologists (e.g., Pianka 1970; Western
& Semakula 1982; Charnov 1993), their potential for conservation biology has been neglected.
To estimate the maximum annual growth rate of species
from incomplete data, we used a life-history invariant
(Charnov 1993), the maximum growth rate per generation, (λ max )T , where T is the species generation time.
First, we tested the invariance of (λ max )T in birds for a
restricted set of well-known species for which accurate
estimates of maximum annual growth rate and generation
time were available. We show how to estimate the maximum annual growth rate of any bird species when only
estimates of the age at first reproduction and adult annual
survival probability are available. Next, we applied these
approaches to the conservation of Black-footed Albatross.
Conservation Biology
Volume 19, No. 3, June 2005
828
Demographic Invariants and Overharvest
Niel & Lebreton
Because of the central role of demographic invariance
in our approach, we call it the “demographic invariant
method” (DIM).
Methods
Maximum annual growth rate (Blueweiss et al. 1978;
Western 1979; Charnov 1993) and generation time are
linked to body mass M (Bonner 1965; Millar & Zammuto
1983; Gaillard et al. 1989; Douzery et al. 1995) by allometric relationships sharing a common exponent in absolute
value, close to 0.25:
rmax ≈ ar M −0.25
(1)
T ≈ aT M 0.25 ,
(2)
and
where r max is the intrinsic rate of natural increase
(r max = ln[λ max ]; Odum 1971).
Several authors deduced from these equations that the
maximum growth rate, expressed with generation as time
unit, is approximately independent of body mass (Lebreton 1981; Fowler 1988; Charnov 1993):
T ln λmax
= rmax T ≈ ar aT M ◦ = ar aT .
(3)
This maximum growth rate per generation is a dimensionless number because T is measured in years and r max
in years−1 . Dimensionless numbers are commonly used
in physics to study properties of systems independently
of their dimensions (Langhaar 1951). In biology Stahl
(1962) emphasizes the usefulness of dimensionless numbers, which he calls “similarity criterions.” Dimensionless life-history characteristics are used to discuss ecophysiological characteristics of species (Calder 1983) and
to characterize effective population size (Waite & Parker
1996). The residual allometric exponent of ln(λ max T ) is
close to 0 and λ max T is a dimensionless number, suggesting that λ max T can be a demographic invariant remaining
stable within homogeneous taxonomic groups and independent of variations in body mass, generation time, and
maximum annual growth rate among the species of these
groups. Indeed, Heron (1972) provides results at the scale
of all living organisms corresponding to a near constancy
of λ max T .
To further test for this invariance and to determine the
value of λ max T in bird species, we collected from the literature complete demographic parameters for 13 bird
species over a large range of body weights (from 19 to
7400 g) and from 10 families. These data were selected
to correspond to optimal demographic conditions. Some
were obtained from populations growing in presumably
optimal conditions: the Griffon Vulture after reintroduction under complete protection (Ferrière et al. 1996); the
Herring Gull (Chabrzyk & Coulson 1976), the Great Cormorant (Frederiksen et al. 2001), and the Northern Ful-
Conservation Biology
Volume 19, No. 3, June 2005
mar (Dunnet & Ollason 1978; Ollason & Dunnet 1988)
during invasion processes; the Snow Goose during the
most favorable years of a long-term study (Gauthier &
Brault 1998); and other growing populations (Barnacle
Goose, Larsson et al. 1988; Atlantic Puffin, Harris & Wanless 1991). Scientific names are given in Table 1. In these
cases, λ max was effectively realized in the population.
We then collected maximum values of survival and
fecundity from the literature on other well-studied bird
species ( White Stork, Lebreton 1978, Barbraud et al.
1999; Black-headed Gull, Lebreton & Landry 1979, Lebreton 1981, Clobert et al. 1987, Lebreton et al. 1990,
Prévot-Julliard et al. 1998; Great Tit, McCleery & Perrins
1989; Rock Sparrow, Tavecchia 2000; Black-legged Kittiwake, Thomas & Coulson 1988; Caspian Tern, Gill &
Mewaldt 1983) and considered these parameters as potential maxima. In these cases, λ max is then a theoretical,
unobserved value. We neglected the differences between
these theoretical values and “real” maximum growth rates
of the species. Actual maximum rates can be slightly
higher if the demographic parameters considered were
not estimated in optimal conditions. The rates can be
lower if some demographic parameters had been measured in several distinct populations and thus were not
subject to potential trade-offs between survival and fecundity. In the absence of precise information, we assumed
that there was no senescence (i.e., fecundity and survival
remain stable from some age onward; Lebreton & Clobert
1991).
We used the ULM program (Legendre & Clobert 1995)
to obtain from each data set (Table 1) the maximum annual growth rate (λ max ) as the dominant eigenvalue of a
Leslie matrix model, accounting for age dependence in
demographic parameters (Caswell 2000). The ULM program also gave two values of generation time, the cohort
generation time (Tc) and the mean generation length (T ,
Leslie 1966) according to the following formulas:
∞
fi li λ−i
max = 1,
(4)
∞
if i li
Tc = i=1
,
∞
i=1 fi l i
(5)
i=1
and
T =
∞
ili fi λ−i
max ,
(6)
i=1
where f i is the annual fecundity at age i and l i is the probability of surviving from birth to age i. Because generation
time varies with demographic conditions, it is important
to note that generation times were obtained here under
optimal conditions. Consequently, we call them “optimal
generation times” and denote them as Tc op and T op . The
preferable measure of generation time in our context is
Niel & Lebreton
Demographic Invariants and Overharvest
829
Table 1. Demographic data used in matrix models to obtain values of annual maximum growth rate and generation time.a
Species
Northern Fulmar
(Fulmarus glacialis)
Atlantic Puffin
(Fratercula artica)
Griffon Vulture
(Gyps fulvus)
Black-legged Kittiwake
(Rissa tridactyla)
Herring Gull
(Larus argentatus)
Snow Goose
(Anser caerulescens)
Barnacle Goose
(Branta leucopsis)
Great Cormorant
(Phalocracorax carbo)
Black-headed Gull
(Larus ridibundus)
Age-specific survival
probabilities
s 1 = 0.950
s i = 0.972 for i ≥ 2
s 1 = 0.638
s i = 0.950 for i ≥ 2
s 1 = 0.858
s 2 = 0.858
s 3 = 0.987
s i = 0.975 for i ≥ 4
s 1 = 0.79
s i = 0.90 for i ≥ 2
s 1 = 0.83
s i = 0.935 for i ≥ 2
s 1 = 0.5
s i = 0.83 for i ≥ 2
s 1 = 0.83
s i = 0.95 for i ≥ 2
s 1 = 0.601
s 2 = 0.873
s 3 = 0.882
s 4 = 0.889
s 5 = 0.893
s 6 = 0.896
s 7 = 0.897
s i = 0.896 for i ≥ 8
s 1 = 0.5
s 2 = 0.8
s i = 0.9 for i ≥ 3
White Stork
(Ciconia ciconia)
s 1 = 0.74
s i = 0.78 for i ≥ 2
Caspian Tern
(Sterna caspia)
s1
s2
s3
si
s1
si
s1
si
Great Tit
(Parus major)
Rock Sparrow
(Petronia petronia)
= 0.82
= 0.79
= 0.87
= 0.89 for i ≥ 4
= 0.365c
= 0.73 for i ≥ 2
= 0.46
= 0.77 for i ≥ 2
Age-specific breeding
successb
Age-specific
fecundities
f i = 0.491 for i ≥ 12
References
a i = 0.9 for i ≥ 5
f i = 0.84 for i ≥ 5
Dunnet & Ollason 1978
Ollason & Dunnet 1988
Harris & Wanless 1991
a i = 0.430 for i ≥ 4
f i = 1 for i ≥ 4
Ferrière et al. 1996
a 5 = 0.9
a i = 1 for i ≥ 6
f i = 1.36 for i ≥ 5
Thomas & Coulson 1988
f i = 1 for i ≥ 5
Chabrzyk & Coulson 1976
f i = 2.75 for i ≥ 2
Gauthier & Brault 1998
f i = 1.425 for i ≥ 3
Larsson et al. 1988
a 2 = 0.35
a 3 = 0.77
a i = 0.85 for i ≥ 4
a 3 = 0.336
a i = 0.672 for i ≥ 4
a 2 = 0.403
a 3 = 0.620
a 4 = 0.797
a 5 = 0.905
a 6 = 0.958
a 7 = 0.983
a i = 0.992 for i ≥ 8
f2
f3
f4
fi
= 1.3
= 1.5
= 1.8
= 2 for i ≥ 5
Frederiksen et al. 2001
a 2 = 0.3
a i = 0.7 for i ≥3
f i = 2.2 for i ≥ 2
a 2 = 0.2
a 3 = 0.4
a i = 1 for i ≥ 4
f 2 = 2.5
f i = 3.2 for i ≥ 3
Lebreton & Landry 1979
Lebreton 1981
Clobert et al. 1987
Lebreton et al. 1990
Prévot-Julliard et al. 1998
Lebreton 1978
Barbraud et al. 1999
f i = 1.61 for i ≥ 2
Gill & Mewaldt 1983
f i = 5 for i ≥ 1
McCleery & Perrins 1989
f 1 = 6 for i ≥ 1
Tavecchia 2000
a Variables:
s, survival; i, age; a, breeding success; f, fecundity.
took a i = 1 in the absence of detailed data, assuming that this parameter is included in the estimation of fecundity.
c Assuming s = s /2.
1
2
b We
T op because T is generally less affected than Tc by considering senescence patterns in Leslie matrix models for
growing populations (λ > 1; C.N., unpublished results).
Mathematically, this is probably because T measures the
increase in the current average generation number per
year (Leslie 1966)—in a growing population, the new
generations produce more individuals than old ones and
accelerate the change in the average generation number.
As a result, T is not very sensitive to changes in survival
in old age classes and as a consequence to senescence.
We first tested the invariance of the maximum growth
rate per generation based on a linear regression model
between ln(r max ) and the logarithm of generation time
(model 1) with the GLIM program (Crawley 1993):
ln(rmax ) = aT ln(T op ) + bT + ξT
(7)
ln(rmax ) = aTc ln(Tcop ) + bTc + ξTc ,
(8)
and
where aT and a Tc are the slopes, bT and b Tc the intercepts, and ξT and ξ Tc the usual random terms of the linear regression model applied to T op and Tc op . Because
we neglected senescence, we expected that the regression would be more accurate if we used T op .The slope a
Conservation Biology
Volume 19, No. 3, June 2005
Demographic Invariants and Overharvest
830
Niel & Lebreton
Table 2. Maximum annual growth rate (λ max ) and optimal generation times (Tc op and T op ) obtained from matrix models (MM) (λ max [MM],
T op [MM], and Tc op [MM]) and estimated from the demographic invariant method (DIM) (λ max [DIM] and T op [DIM]) for 13 bird species.∗
Species
Northern Fulmar
Atlantic Puffin
Griffon Vulture
Black-legged Kittiwake
Herring Gull
Snow Goose
Barnacle Goose
Great Cormorant
Black-headed Gull
White Stork
Caspian Tern
Great Tit
Rock Sparrow
∗ See
λ max (MM)
Top (MM)
Tc op (MM)
λ max (DIM)
Top (DIM)
1.06
1.09
1.09
1.12
1.13
1.17
1.18
1.19
1.19
1.21
1.29
1.64
2.15
22.92
11.81
12.33
9.18
9.80
5.01
7.55
6.28
5.97
4.98
4.22
1.80
1.56
46.97
24
43
14.09
19.39
7.49
22.49
12.08
11.94
6.84
10.07
3.70
4.35
1.04
1.08
1.08
1.11
1.09
1.21
1.15
1.21
1.20
1.21
1.21
1.52
1.48
26.99
12.20
14.24
9.35
11.00
4.66
6.84
4.79
5.00
4.83
4.79
1.92
2.09
Methods for a complete explanation of variables.
T
op
under the hypothesis of stability of λmax
is −1. Thus, we
compared the estimate of the slope with this predicted
value based on a z test before fitting the alternative linear
model (model 2, T op being Tc op or T op ):
ln(rmax ) = − ln(Top ) + b + ξ,
(9)
which is equivalent to
rmax Top = eb+ξ ≈ c
(10)
Top
i.e., λmax
= ermax Top ≈ ec .
(11)
From the 95% symmetric confidence interval on the estimate of b, we then deduced an asymmetric confidence
Top
interval for r max T op and for λmax
by back transforming the
bounds.
To test the robustness of the result on the invariance of
r max T op , we modified demographic parameters in each
matrix model by systemically reducing fecundities in all
age classes (50% reduction) or adult survival probabilities
(10% reduction), therefore simulating “nonoptimal” matrix models. Using the ULM program with these data, we
obtained rates of increase by generation values (rT) that
could arise in hypothetically nonoptimally growing populations. We then compared these rT values to maximum
rates r max T op . For the sake of clarity we denoted simple
linear correlation coefficients in our statistical analyses as
R rather than r.
Results
Invariance of Maximum Growth Rate per Generation
The range of maximum annual growth rate λ max obtained
from matrix models was large (from 1.06 to 2.15; Table 2).
Conservation Biology
Volume 19, No. 3, June 2005
If smaller birds have generally higher growth rates, body
mass cannot explain all the differences between species.
For example, the lowest growth rate was for the Northern
Fulmar, a species that is smaller than the Griffon Vulture,
Herring Gull, Snow Goose, Barnacle Goose, and Great
Cormorant. The Northern Fulmar, however, belongs to
Procellariiforms, which are known for their low reproductive output (Lebreton et al. 1987).
The two linear regressions between ln(r max ) and the
logarithm of generation time (Tc op or T op ) were highly
significant. They remained significant even if, to account
for a potential phylogenetic inertia, a low number of degrees of freedom was used (e.g., n = 6: z = 10.20 and z =
3.64 lead to p = 0.0000 and 0.0002 for the two measures
of generation time, respectively).
As expected, the two measures of generation time gave
different results: the mean generation length (T op ) explained a larger part of the variation in r max among species
(R2 = 0.963) than cohort generation time (Tc op : R2 =
0.778). We then continued our analysis with T op . Because
they are obtained from the same demographic parameter estimates, λ max and T op are interrelated by a negative
sampling correlation (Eq. 6). Based on this formula, any
increase in λ max as a consequence of sampling variability
results in a decrease in T op . But the sampling variation
on, for example, T op in the Northern Fulmar was narrow
(T op being at any rate in the order of 20 years), and could
not explain the overall negative correlation over a wide
range of T op among bird species.
Because the slope a did not differ significantly from −1
(â = −0.9209, SE = 0.1070, z = 0.7393, p = 0.4598),
the original regression model did not differ significantly
from model 2 (Eq. 9), which still explained a large part of
the between-species variation in r max (Fig. 1; R2 = 0.956).
Based on the confidence interval on the estimate of b, we
deduced the following 95% confidence intervals: rmax T op
T
op
= (0.9783, 1.1483) and λmax
= (2.67, 3.153).
Niel & Lebreton
Demographic Invariants and Overharvest
831
Table 3. Comparison between maximum growth rate per generation
T
op
(λmax
) of 13 bird species and growth rates per generation (λT ) that
could arise in hypothetical “nonoptimally’’ growing populations,
obtained by reducing fecundity values (50 % reduction: λTf ) and
annual survival probability (10 % reduction: λsT ) in matrix models.
Figure 1. Relationship between the intrinsic rate of
natural increase (r max ) and the optimal generation
time ( Top ) of 13 bird species.
These results confirm the invariance of the maximum
growth rate per generation among bird species, with a
value close to 3. For the sake of simplicity, we assumed
T
op
that rmax T op ≈ 1 and λmax
≈ 3 in what follows.
Hence, although the range of maximum annual growth
rate λ max was very large among bird species, the maximum growth rate per generation was almost constant in
this taxonomic group. As predicted by dimensional analysis, this dimensionless number characterizes bird population dynamics independently of longevity and body mass
differences between species and independently of phylogenetic position. Biologically, this result means that in
optimal demographic conditions, no bird population can
multiply its size by more than three in one generation.
The test of robustness (Table 3) showed that our result
was not a mathematical consequence of the definition
of T op because growth rates per generation that could
arise in nonoptimal demographic situations were all much
smaller than 3.
T
Species
op
λmax
λTf
λTs
Northern Fulmar
Atlantic Puffin
Griffon Vulture
Black-legged Kittiwake
Herring Gull
Snow Goose
Barnacle Goose
Great Cormorant
Black-headed Gull
White Stork
Caspian Tern
Great Tit
Rock Sparrow
3.90
2.75
2.96
2.82
3.31
2.17
3.52
2.90
2.79
2.58
2.91
2.44
3.30
2.46
1.75
2.15
1.65
2.08
1.25
2.43
1.83
1.77
1.46
1.93
1.56
2.23
0.40
0.89
0.90
1.19
1.30
1.41
1.73
1.64
1.62
1.68
2.02
2.17
2.99
λ max can be deduced from an estimate of T op , which still
requires a detailed knowledge of demographic parameters.
A more straightforward approach is possible when
even less information is available. Assuming constant
adult survival probability s and fecundity after the age
at first reproduction α, the generation time reduces to
T op = α +
s
λmax − s
(14)
(see Appendix).
This formula and the previous relation (Eq. 13) provide
a set of two equations with two unknowns. Knowing α
Explicit Links between Longevity and Growth Potential
The invariance of maximum growth rate per generation
links maximum annual growth rate to generation time, as
ln(λmax )T op ≈ 1,
(12)
which, for high values of T op , reduces to
λmax ≈ 1 +
1
T op
.
(13)
This result (Fig. 2) gives a simple and explicit form to
the often implicit or qualitative statement that long-lived
species have a lower growth potential than short-lived
species (e.g., Stearns 1992). At this stage, an estimate of
Figure 2. Maximum annual growth rate (λ max ) of
bird species as a function of optimal generation time
( Top ) according to the formula λmax = 1 + 1/Top .
Conservation Biology
Volume 19, No. 3, June 2005
832
Demographic Invariants and Overharvest
and s, we deduce that
λmax ≈
(sα − s + α + 1) +
Niel & Lebreton
(s − sα − α − 1)2 − 4sα 2
2α
(15)
and
T op ≈
1
.
λmax − 1
(16)
It is then possible to estimate both λ max and T op from
estimates of the age at first reproduction (α) and adult
survival probability (s) only, provided these estimates are
valid under optimal growth conditions.
To test the quality of the estimation, we applied the
method (Eqs. 15 & 16) to the 13 bird species studied and
compared the results with those obtained from the matrix model (MM) with complete demographic data (Table 2). The strongest differences between λ max (MM) obtained from the matrix model and the DIM estimate rate
λ max (DIM) concerned the two Passerine species (Great
Tit and Rock Sparrow) and can be explained by the
approximation of Eq. 13, which is valid only for high
generation-time values. The quality of the prediction of
λ max (MM) by λ max (DIM) was good for the remaining 11
species (R = 0.884). For these 11 species (the first 11
species in Table 2), the regression slopes of λ max (MM)
with respect to λ max (DIM) and of λ max (DIM) with respect to λ max (MM) were similar (0.8820, SE = 0.1524, and
0.8740, SE = 0.1454, respectively) and did not differ significantly from 1. As a consequence, the slope of the major
axis (1.0265) was close to 1, indicating a relationship of
the form λ max (MM) = λ max (DIM) + a. The mean difference λ max (MM)–λ max (DIM) (0.0118, SE = 0.0319) did not
differ significantly from 0 (z = 0.3699, p = 0.355), indicating that the relationship λ max (MM) = λ max (DIM) was
acceptable. Quite similar results and conclusions were
reached when we used logarithm of growth rates. Altogether for these 11 species, λ max (DIM) gave a sound prediction of the maximum annual growth rate, with a slight
nonsignificant underestimation that fitted well with the
precautionary principle.
For short-lived species such as the Rock Sparrow and
Great Tit, the method was less accurate, and λ max was
better estimated by solving the equation:
−1 s
λmax = exp α +
.
(17)
λmax − s
Application to Conservation Biology
The Demographic Invariant Method
The previous relations, which link λ max to T op , lead to
the simple DIM being used to estimate the key parameter
Conservation Biology
Volume 19, No. 3, June 2005
λ max and then to detect overharvested bird populations.
When estimates of generation time T op are available, a
quick prediction of λ max , as 1 + T1 becomes possible.
op
As we showed, however, this formula requires that T op
be measured under optimal conditions, and such data are
generally lacking.
To solve this problem, a first option is to use λmax =
1 + 1/T with any available estimate of generation time
T . But estimates of T under less than optimal conditions
will generally be larger than T op , although this will not
be the case if survival parameters are reduced far below
their optimal values. This approach tends to give a lower
bound to the maximum annual growth rate, a result that
adheres to the precautionary principle.
A second option when data are lacking is to deduce
λ max from estimates of the age at first reproduction (α)
and adult survival probability (s), using Eqs. 15 or 17 for
short-lived species. Here too, if these estimates do not
correspond to optimal conditions, the estimate of λ max –1
will generally, but not always, be conservative, as in the
previous option.
After having estimated λ max , Wade (1998) proposes to
compare the estimated number of individuals killed by the
additional source of mortality with the potential excess
growth P:
P = Nβ(λmax − 1),
(18)
where N is the estimated population size and β is a parameter introduced to account for the effect of density on demographic performance (β = 0.5; Wade 1998). In applications to conservation, Wade’s idea is valid for large populations submitted to large-scale additional mortality, (e.g.,
the Wandering Albatross [Diomedea exulans]; Weimerskirch et al. 1997). In small populations, even if negative density dependence is generally negligible, it seems
safe to recommend using such a parameter for several
other reasons: (1) other sources of additional mortality
may be present although they cannot be quantified (e.g.,
ingestion of plastic debris by the Black-footed Albatross;
Cousins & Cooper 1999); (2) the impact of an additional
source of mortality may be increased by an imbalance in
favor of age classes with the highest reproductive values
(reproductive adults); (3) a small population can be more
sensitive to catastrophes if it is restricted to a few sites
(e.g., the Short-tailed Albatross [Phoebastria albatrus];
Hasegawa 1984); (4) positive density-dependent reproductive success can occur because of reduced mating efficiency (Allee effect; Legendre 1999); and (5) the price of
a wrong decision in a small population of a slow-growing
species is high. Considering these four points, the value
of the parameter β in small populations should be discussed for each case on the basis of expert knowledge of
the population of concern, with 0.5 being a strict maximal default value.
Niel & Lebreton
Demographic Invariants and Overharvest
Case Study: Impact of Longline Fishery Bycatch on the
Black-Footed Albatross
The Black-footed Albatross is a long-lived seabird. In the
past, its breeding range was from Japan to the Marshall
Islands, but because of harvesting of eggs and adults
this range has progressively shrunk to only the Hawaiian archipelago (Cousins & Cooper 1999). The remaining population has been submitted recently to incidental mortality resulting from longline fishing by Pacific
fleets. The number of birds killed each year in the 1990s
by the Hawaiian fishing fleet was estimated to be 2000
(Cousins & Cooper 1999), and the estimation did not
include killings by Japanese and Taiwanese fishers. Attempts to produce more exhaustive estimates indicate
that more than 12,000 individuals could have been killed
annually (E. Melvin, personal communication).
Because little is known about the population dynamics
of the Black-footed Albatross (Cousins & Cooper 1999),
it is difficult to evaluate the effect of this human-induced
mortality on the species. This is a good example to which
the DIM can be applied to contribute to debate on the
conservation of the species.
The Black-footed Albatross starts breeding at an average age close to 8.6 years (α = 8.6), and adults survive
annually with an average probability of 0.947 (s = 0.947;
Cousins & Cooper 1999). Equations 15 and 16 lead to
T op = 17.09 and λmax = 1.059.
Thus, even under ideal demographic conditions, the population cannot grow at a rate higher than 5.9% per year.
The estimated number of breeding pairs is 60,000. Based
on a Leslie matrix model, the population size N, which includes nonbreeding individuals, is then close to 300,000
birds (Cousins & Cooper 1999).
Even with the maximum value of 0.5 for the safety
parameter β, any additional source of mortality that approaches the potential excess growth of 8,850 individuals
(Eq. 18) will be a source of serious concern. Therefore,
the strict minimum of 2,000 individuals caught annually
by the Hawaiian fishing fleet is already close to a quarter of this maximum additional mortality. Knowing that
albatrosses move to fish all over the Pacific (Cousins &
Cooper 1999), the impact of other fishing fleets must be
in the same range, as is apparent in the overall estimate
of 12,000 individuals killed annually. Clearly, this impact
cannot be considered negligible and current fishing techniques can thus be considered a potential threat for the
species.
Discussion
The case of the Black-footed Albatross shows that the
DIM makes it possible to readily assess the effect of an
additional source of mortality on a bird population when
only limited data are available. Because DIM considers
833
maximum rates, its use must be limited to the detection
of overharvested populations (Slade et al. 1998). It could
be applied to predict whether an additional source of
mortality is unsustainable, but it cannot be used the other
way around (i.e., to predict that it is sustainable).
The application of the method to a small population
requires a discussion of the value of the safety parameter
β, considering all the factors that can affect population
growth besides human-induced additional mortality. If 0.5
is the maximum default value, further investigations could
lead to a more systematic method to determine this safety
factor. From the perspective of adaptive resource management (Lancia et al. 1996), such a method should include a
reevaluation of the parameter each year as knowledge of
the population increases and the uncertainty of the effect
of each factor decreases.
Compared with other methods, DIM can be applied
with fewer data than Slade’s method (Slade et al. 1998).
DIM does not require the estimation of prereproductive survival, which is notoriously difficult to estimate
in the field (Lebreton 2001), or of fecundity because it
takes these factors implicitly into account with the demographic invariant. Compared with Cole’s (1954) formula,
which assumes no mortality in the population before last
reproduction, DIM provides more realistic and more conservative growth rates because ignoring mortality leads
to overestimating growth rates.
Even in the absence of direct adult survival estimates,
a frequent case with threatened species, DIM can still be
useful. First, one can obtain an estimate by comparing
the species in question (i.e., resort to a comparative approach). For long-lived species, one could also estimate
optimal generation time and maximum annual growth
rate by using only age at first reproduction (α) and age at
last reproduction (ω), assuming no mortality in the population (s = 1). In general, this leads to an overestimation of
generation time and an underestimation of maximum annual growth rate, according to Eq. 13. These estimations
would be more conservative than Cole’s (1954) because
in our case, assuming no mortality leads to an underestimation of growth potential. The demographic invariant
method could thus help the maximum annual growth rate
of several tropical bird species, currently based on Cole’s
formula (e.g., O’Brien & Kinnaird 2000) and their conservation status.
The evolutionary reasons for the invariance of the maximum growth rate per generation, in particular in relation
to resource allocation (Calow 1983), deserve further research. To be more widely applicable in conservation biology, the invariance of the maximum growth rate per
generation also needs to be investigated for more bird
species and in other taxonomic groups.
Taking senescence patterns into account in the matrix
models is another way to test whether T op remains a better predictor than Tc op and to determine whether the estimates of λ max are strongly affected by this phenomenon.
Conservation Biology
Volume 19, No. 3, June 2005
834
Demographic Invariants and Overharvest
The central role of generation time in elasticity analysis
(Lebreton & Clobert 1991) and here in the prediction of
maximum annual growth rate make it a key demographic
statistic (see also Gaillard et al., in press). The ability to
estimate it, including ways to estimate adult survival in
conservation research programs, is thus a priority.
Acknowledgments
We thank J.-M. Gaillard for helpful comments on the
manuscript.
Literature Cited
Barbraud, C., J.-C. Barbraud, and M. Barbraud. 1999. Population dynamics of the White Stork Ciconia ciconia in Western France. Ibis
141:469–479.
Blueweiss, L., H. Fox, V. Kudzma, D. Nakashima, R. Peters, and S. Sams.
1978. Relationships between body size and some life history parameters. Oecologia (Berlin) 37:257–272.
Bonner, J. T. 1965. Size and cycle: an essay on the structure of biology.
Princeton University Press, Princeton, New Jersey.
Calder, W. A. 1983. Ecological scaling: mammals and birds. Annual Review of Ecology and Systematic 14:213–230.
Calow, P. 1983. Energetics of reproduction and its evolutionary implications. Biological Journal of the Linnean Society 20:153–165.
Caswell, H. 2000. Matrix population models. 2nd edition. Sinauer, Sunderland, Massachusetts.
Caughley, G. 1977. Analysis of vertebrate populations. John Wiley &
Sons, New York.
Chabrzyk, G., and J. C. Coulson. 1976. Survival and recruitment in the
Herring Gull Larus argentatus. Journal of Animal Ecology 45:187–
203.
Charnov, E. L. 1993. Life history invariants. Some explanations of symmetry in evolutionary ecology. Oxford University Press, Oxford,
United Kingdom.
Clobert, J., J. D. Lebreton, and D. Allaine. 1987. A general approach to
survival rate estimation by recaptures or resightings of marked birds.
Ardea 75:133–142.
Cole, L. C. 1954. The populational consequences of life history phenomena. Quarterly Review of Biology 29:103–137.
Collar, N. J., and P. Andrew. 1988. Birds to watch: the ICPB world checklist of threatened birds. International Council for Bird Preservation
(ICBP) technical publication 8. Smithsonian Institution Press, Washington, D.C.
Cousins, K., and J. Cooper. 1999. The population biology of the Blackfooted Albatross in relation to mortality caused by longline fishing.
Western Pacific Regional Fishery Management Council, Honolulu.
Crawley, M. J. 1993. GLIM for ecologists. Blackwell Science, Oxford,
United Kingdom.
Croxall, J. P. 1998. Research and conservation: a future for albatrosses?
Pages 269–290 in G. Robertson and R. Gales, editors, Albatross biology and conservation. Surrey Beatty & Sons, Chipping Norton, New
South Wales.
Douzery, E., J. D. Lebreton, and F. M. Catzeflis. 1995. Testing the generation time hypothesis using DNA/DNA hybridization between Artiodactyls. Journal of Evolutionary Biology 8:511–529.
Dunnet, G. M., and J. C. Ollason. 1978. The estimation of survival rate in
the fulmar, Fulmarus glacialis. Journal of Animal Ecology 47:507–
520.
Ferrière, R., F. Sarrazin, S. Legendre, and J.-P. Baron. 1996. Matrix population models applied to viability analysis and conservation: theory
and practice using the ULM software. Acta Oecologica 17:629–656.
Conservation Biology
Volume 19, No. 3, June 2005
Niel & Lebreton
Fowler, C. W. 1988. Population dynamics as related to rate of increase
per generation. Evolutionary Ecology 2:197–204.
Frederiksen, M., J. D. Lebreton, and T. Bregnballe. 2001. The interplay
between culling and density-dependence in the great cormorant: a
modeling approach. Journal of Applied Ecology 38:617–627.
Gaillard, J.-M., D. Pontier, D. Allainé, J.-D. Lebreton, J. Trouvilliez, and
J. Clobert. 1989. An analysis of demographic tactics in birds and
mammals. Oikos 56:59–76.
Gaillard, J.-M., N. G. Yoccoz, J. D. Lebreton, C. Bonenfant, S. Devillard,
A. Loison, D. Pontier, and D. Allainé. 2005. Generation time: a reliable metric to measure life history variation among mammalian
populations. American Naturalist in press.
Gauthier, G., and S. Brault. 1998. Population model of the greater snow
goose: protected impacts of reduction in survival on population
growth rate. Pages 65–80 in B. D. J. Batt, editor. The greater snow
goose: report of the Arctic Goose Habitat Working Group. Joint venture special publication. U.S. Fish and Wildlife Service, Washington,
D.C., and Canadian Wildlife Service, Ottawa, Ontario.
Gill, R. E. J., and R. L. Mewaldt. 1983. Pacific coast Caspian terns: dynamics of an expanding population. The Auk 100:369–381.
Harris, P. M., and S. Wanless. 1991. Population studies and conservation of Puffins, Fratercula artica. Pages 230–248 in C. M. Perrins,
J. D. Lebreton, and G. J. M. Hirons, editors. Bird population studies:
their relevance to conservation and management. Oxford University
Press, Oxford.
Harvey, P. A., and A. Purvis. 1991. Comparative methods for explaining
adaptations. Nature 351:619–624.
Hasegawa, H. 1984. Status and conservation of seabirds in Japan, with
special attention to the short-tailed Albatross. Pages 481–500 in J. P.
Croxall, G. H. Evans, and R. W. Schreiber, editors. Status and conservation of the world’s seabirds. Technical publication 2. International
Council for Bird Preservation, Cambridge, United Kingdom.
Hayssen, V. 1984. Basal metabolic rate and the intrinsic rate of increase:
an empirical and theoretical reexamination. Oecologia (Berlin)
64:419–424.
Heron, A. C. 1972. Population ecology of a colonizing species: the
pelagic tunicate Thalia democratica I. and II. Oecologia (Berlin)
10:269–293 and 294–312.
Lancia, R. A., et al. 1996. ARM! For the future: adaptive resource management in the wildlife profession. Wildlife Society Bulletin 24:
436–443.
Langhaar, H. L. 1951. Dimensional analysis and the theory of models.
John Wiley & Sons, New York.
Larsson, K., P. Forslund, L. Gustafsson, and B. S. Ebbinge. 1988. From
the high Arctic to the Baltic: the successful establishment of a Barnacle Goose Branta leucopsis population on Gotland, Sweden. Ornis
Scandinavia 19:182–189.
Lebreton, J.-D. 1978. Un modèle probabiliste de la dynamique des populations de cigogne blanche (Ciconia ciconia L.) en Europe occidentale. Pages 277–343 in J. M. Legay and R. Tomassone, editors. Biométrie et écologie. Société Française de Biométrie, Paris
(in French).
Lebreton, J. D. 1981. Contribution à la dynamique des populations
d’oiseaux. Modèles mathématiques en temps discret. Université
Lyon I, Villeurbanne (in French).
Lebreton, J. D. 2001. The use of bird rings in the study of survival. Ardea
89:85–100.
Lebreton, J.-D., and J. Clobert. 1991. Bird population dynamics, management, and conservation: the role of mathematical modeling. Pages
105–125 in C. M. Perrins, J. D. Lebreton, and G. J. M. Hirons, editors. Bird population studies: their relevance to conservation and
management. Oxford University Press, Oxford, United Kingdom.
Lebreton, J.-D., and P. Landry. 1979. Fécondité de la mouette rieuse,
Larus ridibundus, dans une colonie importante de la plaine du
Forez (Loire, France). Le Gerfaut 69:159–194 (in French).
Lebreton, J.-D., G. Hemery, J. Clobert, and H. Coquillart. 1990. The estimation of age-specific breeding probabilities from recaptures or
Niel & Lebreton
resightings in vertebrate populations. I. Transversal models. Biometrics 46:609–622.
Lebreton, J. D., D. Allainé, J. Clobert, J. M. Gaillard, D. Pontier, and J.
Trouvilliez. 1987. Comparative bird demography: methods, data and
preliminary results (extended abstract). Acta Ornithologica 23:77–
79.
Legendre, S. 1999. Demographic stochasticity: a case study using ULM
software. Bird Study 46:140–147.
Legendre, S., and J. Clobert. 1995. ULM, a software for conservation and
evolutionary biologists. Journal of Applied Statistics 22:817–834.
Leslie, P. H. 1966. The intrinsic rate of increase and the overlap of successive generations in a population of guillemots (Uria aalge pont.).
Journal of Animal Ecology 35:291–301.
McCleery, R. H., and C. M. Perrins. 1989. Great Tit. Pages 35–55 in I.
Newton, editor. Lifetime reproduction in birds. Academic Press, San
Diego.
Mc Mahon, T. 1973. Size and shape in biology. Science 179:1201–1204.
Millar, J. S., and R. M. Zammuto. 1983. Life histories of mammals: an
analysis of life tables. Ecology 64:631–635.
O’Brien, T. O., and M. F. Kinnaird. 2000. Differential vulnerability of
large birds and mammals to hunting in North Sulawesi, Indonesia,
and the outlook for the future. Pages 199–213 in J. G. Robinson and
E. L. Bennett, editors. Hunting for sustainability in tropical forests.
Columbia University Press, New York.
Odum, E. P. 1971. Fundamentals of ecology. Saunders, Philadelphia.
Ollason, J. C., and G. M. Dunnet. 1988. Variation in breeding success
in Fulmars. Pages 263–278 in T. H. Clutton-Brock, editor. Reproductive success: studies of individual variations in contrasting breeding
systems. University of Chicago Press, Chicago.
Pianka, E. R. 1970. On r and K selection. The American Naturalist
104:592–597.
Prévot-Julliard, A.-C., J.-D. Lebreton, and R. Pradel. 1998. Re-evaluation of
adult survival of Black-headed Gulls (Larus ridibundus) in presence
of recapture heterogeneity. The Auk 115:85–95.
Robinson, J. G., and K. H. Redford. 1991. Sustainable harvest of neotropical forest mammals. Pages 415–429 in J. G. Robinson and K. H. Redford, editors. Neotropical wildlife use and conservation. University
of Chicago Press, Chicago.
Slade, N. A., R. Gomulkiewicz, and H. M. Alexander. 1998. Alternatives
to Robinson and Redford’s method of assessing overharvest from
incomplete demographic data. Conservation Biology 12:148–155.
Stahl, W. R. 1962. Similarity and dimensional methods in biology. Science 137:205–212.
Stearns, S. C. 1992. The evolution of life histories. Oxford University
Press, New York.
Tavecchia, G. 2000. Potentialités et limites des analyses de recapture et
reprise en biologie des populations: une approche empirique. Ph.D.
thesis. Université Montpellier II, Montpellier, France (in French).
Thomas, C. S., and J. C. Coulson. 1988. Reproductive success of Kittiwake Gulls, Rissa tridactyla. Pages 251–262 in T. H. Clutton-Brock,
editor. Reproductive success: studies of individual variations in contrasting breeding systems. University of Chicago Press, Chicago.
Demographic Invariants and Overharvest
835
Wade, P. R. 1998. Calculating limits to the allowable human-caused mortality of cetaceans and pinnipeds. Marine Mammals Science 14:1–
37.
Waite, T. A., and P. G. Parker. 1996. Dimensionless life history and effective population size. Conservation Biology 10:1456–1462.
Weimerskirch, H., N. Brothers, and P. Jouventin. 1997. Population dynamics of Wandering Albatross Diomedea exulans and Amsterdam
Albatross Diomedea amsterdamensis in the Indian Ocean and their
relationship with long-line fisheries: conservation implications. Biological Conservation 79:257–270.
Western, D. 1979. Size, life history and ecology in mammals. African
Journal of Ecology 17:185–204.
Western, D., and J. Semakula. 1982. Life history patterns in birds and
mammals and their evolutionary interpretation. Oecologia (Berlin)
54:281–290.
Wilson, E. O. 2000. On the future of conservation biology. Conservation
Biology 14:1–3.
Appendix 1. Calculating generation time assuming
constant adult survival probability, S, and fecundity after the
age at first reproduction α.
∞
Generation time is defined as T = i=1
ili fi λ−i . In this formula and
those that follow, i and k are indices for age, si is the survival probability
from age i-1 to i, li = ik=1 sk is the survival probability from birth to
age i, λ is the population growth rate, and fi is the fecundity (in female
per female) for an animal of age i.
Basic assumptions: (1) constant adult survival s after age of first reproduction α and (2) constant adult fecundity f after age α.
T = fl α
∞
isi−α λ−i
i=1
∞ i−α
fl s
= αα
i
λ i=1 λ
=C
∞ i−α
s
i
,
λ
i=1
with C =
fl α
,
λα
which reduces to
s −1
s
s
T =C 1−
α+
with < 1.
λ
λ−s
λ
Then from the Euler-Lotka equation
∞ i−α
s
s −1
fl α = 1 or C 1 −
= 1,
−α
λ i=α λ
λ
which leads to
T =α+
s
.
λ−s
Conservation Biology
Volume 19, No. 3, June 2005