Estimating the Ability of Birds to Sustain Additional Human-Caused Mortalities Using a Simple Decision Rule and Allometric Relationships Peter W. Dillingham*, David Fletcher

B I O L O G I C A L C O N S E RVAT I O N

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available at www.sciencedirect.com

journal homepage: www.elsevier.com/locate/biocon

Estimating the ability of birds to sustain additional human-caused mortalities using a simple decision rule and allometric relationships
Peter W. Dillingham*, David Fletcher
Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, New Zealand

ARTICLE INFO

ABSTRACT

Article history:

Many bird species are subject to human-caused mortality, either through direct harvest

Received 7 January 2008

(e.g. game birds) or through incidental mortalities (e.g. fisheries-related bycatch of seabirds,

Received in revised form

impact with vehicles, wind turbines, or power lines). In order to assess the impact of addi-

20 April 2008

tional mortalities on birds, both the number of birds killed and their ability to sustain those

Accepted 21 April 2008

deaths must be estimated. Niel and Lebreton [Niel, C., Lebreton, J.-D., 2005. Using demo-

Available online 11 June 2008

graphic invariants to detect overharvested bird populations from incomplete data. Conservation Biology 19, 826–835] applied a simple decision rule [Wade, P.R., 1998. Calculating

Keywords:

limits to the allowable human-caused mortality of cetaceans and pinnipeds. Marine Mam-

Bycatch

mal Science 14, 1–37] to estimate the level of additional human-caused mortality or poten-

Population growth

tial biological removal (PBR) that can be sustained for bird species given only (1) estimates

Magpie goose

of the population size, adult survival, and age at first breeding, and (2) the current popula-

Snow goose

tion status and management goals. We provide guidelines for appropriate use of the

White-chinned petrel

method and case studies comparing results from this method to other approaches. Partic-

Procellariiformes

ular focus is placed on applying the method to Procellariiformes.
PBR limits may then be set without a population model and when monitoring levels are minimal, and in a computationally straightforward manner. While this approach has many advantages, there are limitations. The PBR rule was initially developed for cetaceans and pinnipeds and there have been no adaptations for the unique biology of birds which may need further consideration. Additionally, because this is a simplifying method that ignores differences in life stages, it may not be appropriate for very small populations or for those listed as ‘critically endangered’, and further work is needed for situations where mortalities have large gender or age bias.
Ó 2008 Elsevier Ltd. All rights reserved.

1.

Introduction

The annual growth rate, current status, and management objectives for a species determine the level to which it can sustain additional mortalities. Among bird species, harvests or incidental mortalities come from a variety of sources, such as indigenous harvest, recreational hunting, collision with

man-made objects (vehicles, wind turbines, power lines), and bycatch in fisheries. This work was motivated by our work on fisheries bycatch of seabird species (primarily Procellariiformes such as albatrosses, shearwaters, and petrels) in
New Zealand where limited demographic information is available. Typically, demographic information is limited to rough estimates of the population size, adult survival, and

* Corresponding author: Tel.: [phone].
E-mail address: [e-mail] (P.W. Dillingham).
0006-3207/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.biocon.2008.04.022 1784

B I O L O G I C A L C O N S E RVAT I O N

age at first breeding (Brooke, 2004). This constraint meant that a method for estimating potential biological removal from minimal information was needed.
In the marine mammal setting, Wade (1998) developed a simple rule for estimating allowable bycatch of pinnipeds and cetaceans. Potential biological removal (PBR), or the number of additional mortalities than can be sustained each year by a population, may be calculated given an estimate of the population size, the maximum annual recruitment rate
(Rmax), and a management objective. While the rule is simple, it allows for density-dependence, stochasticity, and the potential for bias in its estimate of PBR. Among similar rules, the Wade rule performed best over a range of conditions (Milner-Gulland and Akcakaya, 2001), including different growth–
¸
fecundity combinations, suggesting that it can be used for a variety of species. However, while methods to estimate population sizes are available for a wide variety of species, Rmax is only observable under optimal conditions. In order to use the
Wade rule for species under non-optimal conditions, Rmax must be estimated.
For bird species, Niel and Lebreton (2005) provide an estimate for Rmax given only adult survival (s) and age at first reproduction (a) under optimal conditions. In addition to being mathematically simple, this approach is relevant to the available data: while detailed knowledge of the biology of many species is limited, reasonable estimates for s and a are often either directly available or may be inferred from similar species. Niel and Lebreton (2005) apply the decision rule of Wade (1998) using point estimates for population size, s, and a, but do not explore the impact on the PBR of uncertainty in these estimates. The impact of uncertainty is important to consider, as s and a are typically estimated under nonoptimal conditions and population estimates for seabirds are imprecise. Combining the work of Wade (1998) and Niel and Lebreton
(2005) allows estimation of allowable harvest levels for birds given only management objectives, an estimated population size, adult survival (s) and age at first reproduction (a). Although the PBR approach tends to be conservative (Hall and Donovan,
2001), it has the benefit of allowing decision making with minimal information, providing a quick and simple method for calculating an estimate of PBR that can be compared to an estimate of human-caused mortalities. If the estimate of such mortalities is substantially greater than the PBR, the species might be over-exploited, suggesting a need for increased monitoring, additional analysis, and possible management intervention. Conversely, if the mortality estimate is substantially below the PBR estimate, it is likely that such mortality is not a substantial force on population trends. Note that this requires all sources of human-caused mortality to be considered: in practice, some or all sources of mortality may be difficult or impossible to assess, particularly for wide-ranging species such as seabirds. In the case of a declining population, knowing that a particular source of mortality is well below the PBR estimate, could lead to focussing on alternative sources of mortality, such as ecosystem change, pollution, or disease, more quickly than might otherwise be achieved. If the PBR estimate is close to the estimate of human-caused mortality, it suggests that further information is required. Overall, use of this approach means that minimal effort can provide valuable infor-

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mation for a large number of species, allowing research effort and resources to be focussed on those populations for which there is a clear need.
The primary objectives of this paper are to describe the
PBR method in detail and to explore the impact of uncertainty in the estimates used to calculate the PBR. We provide three example of its use. The first two involve the greater snow goose (Anser caerulescens atlanticus) and the magpie goose
(Anseranas semipalmata), and allow us to compare the results with those from a detailed population model. The third example involves white-chinned petrel (Procellaria aequinoctialis), where relatively little is known about the population, and for which there appears to be high level of human-caused mortality, from bycatch in longline fisheries.

2.

Methods

Potential biological removal (PBR), or the number of additional mortalities than can be sustained each year by a population, is estimated by
1
PBR ¼ Rmax Nmin f
2

ð1Þ

where Rmax is the maximum annual recruitment rate, Nmin is a conservative estimate of population size (Wade (1998) recommended the 20th percentile) and f is a recovery factor between 0.1 and 1 (Wade, 1998; Taylor et al., 2000; Hunter and
Caswell, 2005; Niel and Lebreton, 2005). This simple formula can be applied to a variety of management objectives, such as maintaining a species at or above the maximum net productivity level (MNPL; the population size at which the annual increase in population size is maximised) or minimizing time to recovery by setting different levels of f (Wade, 1998). This method provides a conservative estimate of PBR for the MNPL objective, assuming a convex (h > 1) or logistic (h = 1) densitydependent growth curve, given by
"
 h #
Nt
Ntþ1 ¼ Nt þ Rmax 1 À
K
where N is abundance, t is time, and K is the carrying capacity
(Wade, 1998). While the rule is simple, it allows for densitydependence, stochasticity, and the potential for bias in its estimate of PBR, and was developed through extensive simulation. While methods to estimate population sizes are available for a variety of species, and the recovery factor is based on a management decision, Rmax is only observable in optimal conditions. In order to use Eq. (1) for species in non-optimal conditions, Rmax must be estimated by other means. Niel and Lebreton (2005) use allometric relationships (Blueweiss
´
et al., 1978; Allaine et al., 1987; Gaillard et al., 1989) to estimate
Rmax in terms of adult survival (s) and age at first reproduction
(a) for a variety of bird species. This key result allows a mathematically simple approach to estimating the PBR for bird species. 2.1.

Estimating Rmax

The maximum recruitment rate (Rmax) and maximum annual population growth rate (kmax) are related by the equation

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Rmax = kmax À 1. With appropriate demographic information, matrix population models can be constructed to estimate kmax (Caswell, 2001). However, for many species too little data is available to construct such matrices.
The methods of Niel and Lebreton (2005) allow estimates of a theoretical maximum annual growth rate (kmax) and the mean optimal generation length (T op ) knowing only age at first reproduction (a) and adult survival (s) for bird species.
This approach assumes constant fecundity and constant adult survival after age of first reproduction. Two key relationships are used in the calculation by Niel and Lebreton (2005): lnðkmax ÞTop % 1 and Top ¼ a þ

s kmax À s

Combining these yields
"
À1 # s kmax ¼ exp a þ kmax À s

ð2Þ

which can be solved using numerical methods. Alternatively,
Niel and Lebreton (2005) provide a quadratic solution based on a first-order Taylor series approximation. Niel and Lebreton (2005) show for 11 bird species undergoing growth that the estimates from Eq. (2) are similar to estimates achieved from matrix approaches (R2 = 0.884). These species have a variety of life history traits, including early (a = 1) to late
(a = 12) reproduction, and low (s = 0.73) to high (s = 0.987) survival (Niel and Lebreton, 2005). Maximum growth rates for a variety of survival/age at first reproduction combinations are shown in Fig. 1.
In practice, population parameters (a, s) may not be available for all species of interest and would rarely be available for optimal conditions. If life history data for a similar species are available, it may be reasonable to use estimates from that species. Otherwise, plausible values for a and s may be reasonably estimated from data at hand, providing a plausible range for kmax.

2.2.

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lower bound of a 60% confidence interval. That is, the PBR decision rule incorporates both the population estimate and an estimate of the uncertainty surrounding it. Depending on the species, population estimates may be characterized in b several ways, such as an estimate ( N ) and standard error
(rb ), an estimate and a coefficient of variation
^
N
^b
(CV b ¼ rb = N ), or as an upper (NU) and lower limit (NL) for a
N
N
(1 À a)% confidence interval. In some cases these may be estimated directly by the researcher, or may be available in the literature. b
Wade (1998) assumed that the population estimate ( N ) followed a log-normal distribution with known coefficient of variation (CV N ¼ rN =N ), where the pth percentile estimate is given by
 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ð3Þ
Np ¼ N exp Zp lnð1 þ CV2 Þ
N
where Zp is the pth standard normal variate. For N0.2, the lower bound of a 60% confidence interval, p = 0.2, and Zp % À0.84. b In practice, percentile estimates ð N p Þ are based on an estimated coefficient of variation ðCV b Þ rather than a known
N
one. The ratio of the two percentile estimates is
 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b Np lnð1 þ CV2 Þ À lnð1 þ CV2 Þ
¼ exp Zp
N
b
N
Np
In practice, this difference may be small if CV b is reasonably
N
close to CVN, but can be substantial otherwise. For example, if b CVN = 0.5 and CV b ¼ 0:4, there is an 8% bias in N 0:2 , which inN creases to 21% for CV b ¼ 0:25.
N
In some cases, such as when decision makers must use estimates available in the literature, the only population estimates available may be upper and lower bounds of a (1 À a) confidence interval. In these cases, and assuming that the confidence interval is again based on a log-normal distribution pffiffiffiffiffiffiffiffiffiffiffiffi b ð4Þ
N ¼ NL NU and vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 
2 ! u lnðNU =NL Þ
À1
CV b ¼ texp
N
2Z1Àa=2

ð5Þ

Estimating Nmin

1.20

α =3 α =4 α =7 α = 12

1.00

λ max

Eqs. (4) and (5) may then be used in Eq. (3) to estimate Np.
A Taylor series approximation, lnð1 þ CV 2 Þ % CV 2 , may be
N
N b used in Eq. (3) to estimate N min ¼ N 0:2 as

α =2

1.10

1.30

To calculate the PBR requires a conservative estimate of the population size (Nmin), suggested by Wade (1998) to be the

0.80

0.85

0.90

0.95

1.00

Adult survival
Fig. 1 – Maximum annual growth rate (kmax) as a function of age at first breeding (a) and adult survival estimated by the methods of Niel and Lebreton (2005).

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b
Nmin ¼ N expðZ0:2 CV b Þ
N

ð6Þ

This approximation is valid for CVN < 0.6, resulting in a 0 to À4% bias in N0.2, and the approximation is reasonable up to CVN = 1
(À13% bias in N0.2). Combining Eq. (6) with Eq. (1) yields
1
b
PBR ¼ Rmax f N expðZ0:2 CV b Þ
N
2

2.3.

ð7Þ

Selecting f

The value selected for f can be used to implement alternative management strategies. For example, a value of 0.1 can be used to provide a minimal increase in recovery time for a depleted population, to maintain a population close to its carrying capacity, or to minimize the extinction risk for a population with limited range, while a value of 1 could be used to maintain a healthy, growing population at or above its maximum net productivity level (Wade, 1998; Taylor et al., 2000). Wade (1998) suggests a value of 0.5 for most healthy populations, as this provides protection against bias in population estimates, maximum growth rates, and mortality estimates. While this approach was designed to maintain a population at or above MNPL, a value of 1 < f < 2 could be used to control a population at a lower level, while b f > 2N min = N would be expected to reduce the population size no matter where it was in relation to carrying capacity.
The recovery factor f is selected based on a species’ population status, with a value of 0.1 suggested for threatened or endangered species (Wade, 1998; Taylor et al., 2000; Niel and
Lebreton, 2005). BirdLife International maintains the International Union for the Conservation of Nature and Natural Resources (IUCN) population status for birds. Birds are classified according to IUCN criteria (IUCN, 2001) as ‘least concern’, ‘near threatened’, or ‘threatened’. ‘Threatened’ species are further classified as ‘vulnerable’, ‘endangered’, or ‘critically endangered’. Without further information, it may be reasonable to set f = 0.5 for ‘least concern’ species, f = 0.3 for
‘near threatened’, and f = 0.1 for all threatened species. A value of f = 1.0 may be appropriate for ‘least concern’ species known to be increasing or stable.
Further, the value of f could be an important part of an adaptive management system (Williams et al., 2002). With ongoing monitoring, the value could be updated to reflect increasing knowledge of the system, with initial values set based on a variety of considerations but allowed to increase or decrease if warranted. For example, it may be reasonable to set f = 0.3 for a vulnerable species that had a large population and breeding range (i.e. some level of additional decline would not jeopardize the viability of the species), monitor the population, and determine if f needed to be lowered or could eventually be raised. Alternatively, it may be preferable to start with a conservative value (f = 0.1, say) and increase it after the species’ status improved.

2.4.

Harvest rates

In some cases it may be preferred to calculate an allowable harvest rate (ha), rather than the PBR. The relationship between the allowable harvest rate and PBR is

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ha ¼

PBR b N

or, substituting into Eq. (1)
1
Nmin ha ¼ Rmax f b 2
N
Using the estimate of Nmin from Eq. (6), this is re-written as
1
ha ¼ Rmax f expðZ0:2 CV b Þ
N
2

ð8Þ

The maximum harvest rate (hmax) using the PBR approach, applied to a non-threatened species with an increasing population trend and with a perfect census is
1
hmax ¼ Rmax
2

ð9Þ

Thus, combining Eqs. (8) and (9), the ratio between the allowable harvest rate and maximum harvest rate is ha =hmax ¼ f expðZ0:2 CV b Þ
N

That is, the allowable harvest rate is decreased from the maximum harvest rate by only two sources: the recovery factor and the variation in the population estimate.

2.5.

Sensitivity of the PBR estimate

The PBR estimate depends on several factors, and changing any of them changes the estimate. In some cases, this change may be straightforward, while it is more complex in others.
The impact of changes in different factors on the PBR may be examined through sensitivity or elasticity analyses (Caswell, 2001), or less formally, as done here. For example, increasing the recovery factor from f = 0.1 to f = 0.5 clearly results in a five-fold increase in the PBR, while an decrease in the CV from 50% to 49% would increase the PBR by approximately 0.4% (see Eq. (7)). Both of these results are intuitive, and are straightforward computationally. The relationships between the PBR and s and a are computationally more challenging and less intuitive.
For example, the Chatham albatross (Thallasarche eremita) is a critically endangered species which has an estimated survival of s = 0.87 (Robertson et al., 2003). However, other albatrosses typically exhibit survival rates of 0.95 or more
(Brooke, 2004). It is important to understand the impact on the PBR of using s = 0.87 versus using a survival estimate from other species in the genus; in general, the desire should be to use a survival rate that reflects survival during optimal conditions. Similarly, it is not always clear which value to use for a.
While the Niel and Lebreton model assumes constant fecundity from the age of first reproduction, this may not always be the case. For example, great cormorants (Phalocracorax carbo) begin breeding at a = 2 but fecundity and age-specific breeding success increase until age 5 (Frederiksen et al., 2001; Niel and
Lebreton, 2005). It is not immediately clear which value of a is appropriate to use in Eq. (2). For example, suppose s = 0.89. If a = 2 then kmax = 1.25, whereas if a = 3 then kmax = 1.18. The
PBR based on these estimates differs by over 30%. In practice, using a mid-point value for a produces similar results to modifying the population model to allow for increasing fecundity over time, low estimates of a may be reasonable if age at first breeding is density-dependent (i.e. fecundity is reduced in

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early age classes) and conditions are non-optimal, and high estimates of a produce the most conservative PBR.
In a Leslie matrix approach, which many researchers are familiar with, all else being the same, higher survival would lead to greater annual growth. However, in the Niel and Lebreton method, survival is implicitly tied to fecundity and generation length. That is, birds with the highest survival rates
(e.g. albatrosses) also have the lowest fecundities. The inverse relationship between fecundity and survival means that higher survival estimates are associated with lower annual growth
(Fig. 1), a counterintuitive relationship for those used to matrix models. Further, kmax quickly decreases as s approaches
1, suggesting that the validity of the Niel and Lebreton method may be questionable for very high survival estimates.
From a management perspective, this means that an underestimate of survival results in an overestimate of PBR.
Survival is typically estimated using capture–recapture methods which naturally incorporate emigration but not immigration (Nichols and Hines, 2002; Peery et al., 2006). This means that survival estimates tend to be negatively rather than positively biased. Further, most survival estimates are derived in non-optimal conditions. Consequently, if survival estimates are derived in non-optimal conditions or estimates have not been adjusted for possible emigration from the study area, conservative (i.e. high) survival estimates should be used to avoid over-estimation of kmax and PBR.
Consistent with expectations from a Leslie matrix approach, low values of a are associated with low annual growth. In general, the estimated value of a can be important for birds with early fecundity, especially if survival is also low
(Fig. 1). For birds with delayed fecundity and high survival, such as seabirds, changes in a lead to only small changes in kmax, and mid-point values for a are usually appropriate, while high values lead to conservative estimates of kmax and
PBR.

2.6.

Case studies

Three species were chosen to illustrate use of the PBR approach described in this paper. Greater snow geese are a well-studied population where kmax may be estimated using census data, matrix model approaches, or the Niel and Lebreton method. Magpie geese are a common waterfowl in tropical northern Australia with a history of indigenous and recreational harvest. Sustainable harvest rates were estimated in the late 1980s using census data (Bayliss, 1989) and more recently using a population model (Brook and
Whitehead, 2005b). Finally, like many Procellariiformes, the white-chinned petrel is a species for which there is limited demographic information, and which has suffered high mortality rates in fisheries; it therefore provides an application of the approach when alternative management methods are not available. 2.7.

Greater snow geese

Greater snow geese are an abundant and widespread North
American goose whose population was reduced to less than
10,000 birds in the early 1900s (Menu et al., 2002). Protection measures allowed the population to rebound to 100,000 birds

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1787

by 1970, and nearly 750,000 birds by 1998 (Menu et al., 2002).
After adjusting for known harvest rates, they are a good example of a species that is growing at near optimal rates.
There is now concern about overabundance, and research is focussed on finding a minimal harvest rate (hc) that would control the population (Gauthier and Brault, 1998; Menu et al.,
2002; Gauthier and Lebreton, 2004). They are well enough studied to allow population models (Gauthier and Brault,
1998; Gauthier et al., 2001; Gauthier and Lebreton, 2004) to be constructed. This, combined with accurate census data, makes this species a good one for which to compare different approaches to estimating harvest rates necessary to control the population. Available data includes census data and resulting growth rates, and harvest and/or survival rates for the periods 1970–1974, 1975–1983, and 1984–1998, corresponding to high growth/low harvest, low growth/high harvest, and moderate growth/moderate harvest periods, respectively.
Niel and Lebreton (2005) compared a matrix model estimate of kmax % 1.167 from Gauthier and Brault (1998), with a point estimate of kmax % 1.21 (Eq. (2)), using s = 0.83 and a = 3. This survival rate includes hunting mortality; both approaches may be improved by estimating survival in the absence of hunting (natural survival, s0). For the period 1990–
1998, Gauthier et al. (2001) estimated this as ^0 ¼ 0:91 coms pared to an estimate from Gauthier and Brault (1998) of
^0 ¼ 0:88. The higher estimate (all else the same) increases s the matrix model estimate to kmax % 1.257, and decreases the estimate to kmax % 1.164 (Eq. (2)); both approaches are sensitive to the estimate of s but in opposite directions. A minimum harvest rate of hc = 10.5% for adults was estimated to result in a stable or declining population (Gauthier and Lebreton, 2004).
Growth rates from census data and harvest rate estimates
(Menu et al., 2002) from the three periods were used to provide a census-based estimate of kmax. Since most birds that were harvested would have otherwise survived and had offspring at normal rates (i.e. assuming additive mortality, an assumption supported by Gauthier et al., 2001), the observed growth rate is approximately k % kmax ð1 À hÞ, or kmax % k=ð1 À hÞ. Hence, potential growth rates for each time period (kmax,p) were estimated as the ratio of the observed growth (kp) and the complement of the harvest rates. Adult harvest rates for each period (hp) were used, as the population is most sensitive to these, and juvenile survival rates were low (although it is important to note that young geese are disproportionately harvested; Menu et al., 2002; Gauthier and
Lebreton, 2004). Hunter and Caswell (2005) provide some discussion on this topic, although for purposes of illustration differential harvest rates between age classes are ignored. For each time period, the period-specific maximum growth rate was then estimated as
^max;p ¼ k ^p k ^
1 À hp

Assuming independence between kp and hp, and using the delta method (Rice, 1995)

^ rkmax;p vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!2
u u ^p k tr2 þ r2
^kp ^hp
%
^
^
1 À hp
1 À hp
1

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In order to weight the growth rate in each year equally, kmax was estimated as
^max ¼ k 3
X

k wp ^max;p

^k and r2max ¼

p¼1

3
X

w2 r2max;p p ^k

p¼1

where wp ¼ #fyears in periodg=#fall yearsg. A plausible
^
range was considered to be ^max Æ 2rkmax ; the use of the term k confidence interval is avoided due to uncertainty in the shapes of the distributions involved and to the simplifications and approximations used. From this, Eq. (9) was used to estimate hmax as a proxy for hc. However, the PBR method is designed to keep a population at or above MNPL; if the desired population level is below this value then the required harvest rate may be greater than hmax (e.g. f > 1). An empirical estimate is also available, noting that k % kmax ð1 À hÞ )
^c ¼ 1 À 1=^max by setting k = 1 with approximate standard k h error rhc % rkmax =^2 .
^
^ kmax In addition to the growth and harvest rates used above,
Menu et al. (2002) provided independent estimates of adult survival for the periods 1970–1974 and 1984–1998. This allows the uncertainty in adult survival and age at first breeding, and an adjustment for harvest rates, to be incorporated into the estimate of kmax from Eq. (2), generating a plausible range.
Once again assuming that most harvested birds would have otherwise survived, potential or natural survival for each period (^0;p ) is approximated as the ratio of adult survival for the s period (^p ) and the complement of the adult harvest, or s ^0;p ¼ s ^p s ^
1 À hp

From the delta method vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u ^p s 1 u2 tr þ r2
^kp ^hp rs0;p %
^
^
^
1 À hp
1 À hp
Natural mortality was then estimated in an analogous manner to kmax, with weights based on the period length. Finally, while some birds begin breeding by age two, breeding propensity increased from 0.35 at age 2 to 0.77 and 0.85 at ages 3 and
4, respectively (Gauthier and Brault, 1998). Values from a = 2 to a = 4 are reasonable for this species, with the best estimate being a = 3. Survival and breeding estimates were used in
Eqs. (2) and (8) to provide plausible ranges for kmax and hmax, where plausible ranges were the middle 95% of 10,000 samples, each generated by sampling from a 2 f2; 3; 4g with probs ^s ability pa = {0.25, 0.50, 0.25} and s0 $ N ð^0 ; r^0 Þ. Finally, estimates of kmax and the harvest rate necessary to control the population from the three approaches were compared.

2.8.

Magpie geese

Magpie geese are a common waterfowl in tropical northern
Australia, currently harvested at a rate of up to 18% of the population, and it is unclear if this rate is sustainable (Brook and Whitehead, 2005a). They number approximately 3.5 million (Brook and Whitehead, 2005a), with a reasonable lower bound of 2 million (Bayliss and Yeomans, 1990; Brook and
Whitehead, 2005b). Between 130,000 and 360,000 birds are harvested annually (Brook and Whitehead, 2005a). The primary source of harvest is indigenous (100,000–290,000 annu-

1 4 1 ( 2 0 0 8 ) 1 7 8 3 –1 7 9 2

ally), with another 30,000–70,000 per year harvested recreationally (Brook and Whitehead, 2005a). Thus, current harvest rates are likely between 4% and 10%, but may be as high as 18%.
Bayliss (1989) and Brook and Whitehead (2005b) estimated the maximum annual rate of population growth in order to estimate allowable annual harvest rates. The maximum annual rate of population growth was estimated by Bayliss
(1989) using aerial survey counts at kmax = 2.18 (i.e. a potential
118% annual population growth at low densities), with an allowable harvest rate of approximately 30% per annum.
The annual harvest appears to be well below the sustainable harvest rate calculated by Bayliss (1989).
Brook and Whitehead (2005b) used matrix methods to estimate kmax = 1.18, with an extreme upper bound of kmax = 1.65 and a more realistic upper bound of kmax = 1.32. For these methods, reasonable assumptions lead to an annual sustainable harvest rate of 5–14% with a best estimate of 8.5% (Brook and Whitehead, 2005b), far lower than the 30% estimated by
Bayliss (1989). Thus, their method suggests that current harvest rates may be too high.
Using the PBR approach, we only need estimates of age at first breeding, adult survival, conservation status, and population size to estimate kmax, ha, and the PBR. Consistent with the values used by Brook and Whitehead (2005b), a was set between 2 and 3 years, with an assumed minimum adult survival rate of 0.85, a best estimate of 0.93, and an assumed maximum of 0.95. The best population estimate of 3.5 million, with a lower bound of 2 million, was used to estimate Nmin
(Eqs. (4)–(6)). As the current population trend is unknown, a value of f = 0.5 is suggested by Wade (1998). In practice, the choice of f should incorporate impacts on stakeholders, such as impacts from lower harvest limits, along with conservation goals, and could also be influenced by the level of monitoring.
While less information is available for magpie geese than for greater snow geese, this still allows plausible ranges for kmax, ha, hmax, and PBR to be calculated.

2.9.

White-chinned petrel

Seabirds, such as the white-chinned petrel, have low fecundity and delayed maturity, making them vulnerable to any reduction in adult survival. Fisheries bycatch (and other related mortalities) are currently estimated to kill hundreds of thousands of seabirds each year (Baker et al., 2007), and there is general concern about the impact of bycatch and other threats (see Baker et al., 2002 for a good review of threats to Procellariiformes).
White-chinned petrels appear to be undergoing a population decline (Berrow et al., 2000), are commonly caught in longline fisheries, and the species is listed as vulnerable (BirdLife International, 2007). During 1997 and 1998, between 80,000 and
200,000 seabirds were killed in the unregulated Patagonian toothfish fishery, of which approximately 60% were whitechinned petrels (SC-CALMR, 1998). They are the most commonly caught seabird species in the Southern Ocean (Weimerskirch et al., 1999). Further, bycatch in the Patagonian toothfish fishery was heavily male-skewed (>80%), with nearly all birds killed in adult plumage (Ryan and Boix-Hinzen, 1999; Nel et al., 2002). They may also be affected by future changes in marine habitat in the southern oceans (Croxall, 2004).

B I O L O G I C A L C O N S E RVAT I O N

There are approximately 2.5 million breeding pairs of white-chinned petrels worldwide, and perhaps 7 million birds total (Brooke, 2004). These estimates are imprecise, so CV b ¼ 0:5 was assumed. Birds begin breeding around
N
a = 6.5 (Schreiber and Burger, 2001). One published survival estimate (s = 0.79; Schreiber and Burger, 2001) is clearly wrong: it apparently comes from the misapplication of s ¼ 0:79 þ 0:019 ln x where x is weight in grams (Croxall and Gaston, 1988). Based on a weight of 1350 g (Brooke,
2004), correct application of this equation suggests ^ ¼ 0:93 s although the predictive power of the equation was only
R2 = 0.27 (Croxall and Gaston, 1988). For the white-chinned petrel’s sister species, the spectacled petrel (Procellaria conspicillata), Ryan et al. (2006) suggested that plausible levels of adult survival range from 0.94 to 0.98, based on other petrels. Brooke (2004) provides estimates for related species ranging from 0.92 to 0.94. We set ^ ¼ 0:93 and ^ ¼ 6:5, and s a considered s between 0.90 and 0.97 and a between 6 and
7 to represent a plausible range.
Given the population decline, IUCN status, and age and gender bias in mortalities, a reasonable value of f is 0.1.
However, the large population size and number of breeding colonies provide a buffer against any immediate threat to the population viability, so a less conservative value of f = 0.3 may be acceptable if combined with adequate monitoring and a willingness to modify the value based on the monitoring. Because of the bias in gender and age in bycatch estimates, a modified PBR estimate was also estimated based on the number of breeding males rather than the total population. This is, to some extent, an ad hoc approach, and suggests future effort may be needed in this area. Finally, we note that managing a species that is vulnerable to bycatch from a variety of fisheries is a daunting task, both operationally and politically. Bycatch estimates are subject to deficiencies such as missing data, misidentification, bird loss prior to observation, and lack of standardisation (Uhlmann et al., 2005; Miller and Skalski,
2006). In this context, the PBR estimate is essentially an assessment tool to determine if estimated or plausible bycatch levels may have a detrimental effect, rather than a management tool for setting bycatch goals.

3.

Results

For species such as large waterfowl or seabirds with delayed fecundity and moderate to high survival (a P 3; s P 0.8), the maximum annual growth rate is less than 1.25 (Fig. 1). For large waterfowl such as geese, kmax may be near 1.2 (see examples in Niel and Lebreton, 2005), while for seabird species such as Procellariiformes, kmax is commonly less than

1 4 1 ( 2 0 0 8 ) 1 7 8 3 –1 7 9 2

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1.1, indicating that, even under optimal conditions, these populations can not grow faster than 7% (albatrosses), 10%
(petrels, shearwaters), or 20% (large waterfowl) in 1 year. This limits their ability to sustain high levels of additional mortality and their ability to quickly recover from depletion, most especially for species such as albatrosses. For seabirds such as albatrosses and petrels, especially for populations that are threatened or depleted, the maximum harvest rate may need to be 0.5% or less in order to minimize recovery time or maintain a population close to carrying capacity. For species such as greater snow geese where the population size has created problems, harvest levels equal to or greater than hmax may be required to control growth; for large waterfowl this is on the order of 10%.

3.1.

Greater snow geese

Maximum growth rates (estimate ± SE) using Eqs. (2) and
(8) for greater snow geese were estimated from the census and harvest data for the 1970–1974 (1.190 ± 0.032), 1975–
1983 (1.136 ± 0.016), and 1984–1998 (1.165 ± 0.011) periods, which combined to provide a census-based estimate of
^max ¼ 1:160 Æ 0:009. A minimal harvest rate necessary to k control the population was estimated using Eq. (9) (Table
1). Empirical estimates (±SE) from the census-harvest data suggested ^c ¼ 0:138 Æ 0:007 for the entire time period, or h ^c ¼ 0:119 Æ 0:013 for the 1975–1983 period, when high harh vest rates resulted in low growth. Similarly, natural survival estimates (±SE) for 1970–1974 (0.869 ± 0.045) and
1984–1998 (0.851 ± 0.047) were combined to estimate natural survival (0.856 ± 0.037). Incorporating uncertainty in survival rates, together with sampling from plausible values of a, allowed uncertainty in the Niel and Lebreton
(2005) estimate of kmax and the PBR estimate of hmax
(Eqs. (2) and (8) to be quantified. The census-harvest estimates (CH), these estimates (NL), and the matrix model estimates (MM) from Gauthier and Lebreton (2004) are summarized in Table 1.
All of the growth and harvest estimates fall in the same general range. When growth rates are the same, the matrix model and empirical census-based estimates for hc suggest that a somewhat higher harvest rate is necessary to maintain or reduce the population than by using ^c ¼ hmax . This could h be a result of inherent conservatism in the PBR approach or due to the desired population level being below the level which would be maintained by hmax. Finally, bounds on kmax and hc are largest when uncertainty is incorporated into
Eqs. (2) and (8). This is quite reasonable, as these estimates are based on minimal information compared to the more sophisticated approaches available.

Table 1 – Maximum growth rate (kmax) and the harvest rate (hc) required to prevent population growth in greater snow geese using matrix models (MM), and census-harvest (CH) and Niel and Lebreton (2005) (NL) growth estimates combined with the PBR harvest rate estimate
Method

kmax

Plausible range

hc (%)

Plausible range

MM
CH
NL

1.167
1.160
1.200

na
(1.148, 1.178)
(1.134, 1.323)

10.5
8.0
10.0

na
(7.4%, 8.9%)
(6.7%, 16.2%)

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3.2.

B I O L O G I C A L C O N S E RVAT I O N

Magpie geese

Assuming age at first reproduction between 2 and 3 years and survival ranging from 0.85 to 0.95, the plausible range for kmax is 1.13–1.29, with a best estimate of kmax = 1.17. Thus, if the population were known exactly and known to be increasing, it would be reasonable to set the harvest rate at hmax = 8.5%, ranging from hmax = 6.5% to hmax = 14.5%, similar to the range provided by Brook and Whitehead (2005b). However, when we protect against potential bias and include uncertainty in the population estimate, the PBR method suggests a lower harvest rate. With a population estimate of 3.5 million, a lower bound 2 million, and assuming the lower bound comes from a 95% CI, Nmin is approximately 2.8 million (Eqs. (4)–(6)). Combining this with a ‘best’ population estimate of 3.5 million, and setting f = 0.5 (as suggested by Wade (1998) when the population trend is unknown) leads to a harvest rate from Eq. (8) of ha = 3.3% (Eq. (7)), ranging from ha = 2.6% to ha = 5.7%. Thus, while initial harvest rate estimates are similar to Brook and
Whitehead (2005a,b), once variability in population estimates, protection against potential bias in population estimates or harvest levels, and the unknown trend are considered, PBR harvest rates are reduced by more than 50%. A less conservative estimate may be reasonable if there is ongoing monitoring of the species (i.e. f = 1.0 leads to ha = 6.6%).
Given current knowledge of the population, total harvests under 120,000 birds would be recommended by the
PBR method. Thus, the current harvest of at least 130,000 birds is higher than would be suggested without greater knowledge of the population, and is potentially even occurring at a rate greater than hmax. If the population were found to be increasing, the harvest could be increased to
230,000 birds. Given that current harvests are at, near, or above the PBR levels, the customized approach to the specific conditions of the species and the associated harvest methods and pressures, such as that undertaken by Brook and Whitehead (2005a), was very valuable. However, the
PBR approach is again validated as giving results similar to those from a more sophisticated analysis, and, in other contexts, could have been used to suggest that a more sophisticated analysis was needed.

3.3.

White-chinned petrels

The approximately 7 million white-chinned petrels (Brooke,
2004), and the assumption that CV b ¼ 0:5, leads to
N
Nmin = 4.6 million. Assuming a = 6.5 and s = 0.93, kmax % 1.08, with a plausible range from 1.06 to 1.10. This range is consistent with other petrels and shearwaters, while albatross populations may grow 4–7% annually in optimal conditions.
Using kmax % 1.08 and f = 0.1, the mortality rate for whitechinned petrels should be below ha = 0.27% and annual human-caused mortalities (plausible range) should be limited to 19,000 birds (14,000–24,000 birds); a less conservative approach, with f = 0.3, would allow mortalities up to 57,000 birds
(43,000–71,000 birds). Annual mortality estimates from the
1990s Patagonian toothfish fishery were well above the PBR with f = 0.1 and were possibly above the PBR with f = 0.3.
Further, when considering the PBR estimate based only on the 2.5 million breeding males (adult males constitute

1 4 1 ( 2 0 0 8 ) 1 7 8 3 –1 7 9 2

approximately 80% of the bycatch), mortality limits drop to
6800 (f = 0.1) and 20,000 (f = 0.3). When considering the gender and age bias in bycatch, it appears that annual mortalities were past those suggested by f = 0.3. While it is difficult to assess the impact on the population from these levels of mortality, it is clear that bycatch from just one fishery had the potential to cause harm to the population, suggesting that concerns about bycatch levels are justified.

4.

Discussion

The PBR method may be used both to set harvest or bycatch limits and to compare current human-caused mortalities with the PBR estimates. This can allow quick detection of potentially over-exploited species, as well as detection of species where current harvest rates are likely to be sustainable. If human-caused mortality rates from all sources total less than ha, then they should not be the primary cause for any concern. If a population is known to be in decline in these circumstances, other causes should be investigated. For mortality rates between ha and hmax, human-caused mortality may be occurring at an unsustainable rate, further investigation is warranted, improved knowledge of the population is desirable, and mitigation measures should be considered. Finally, if mortality rates are greater than hmax, it is likely that humancaused mortalities are occurring at a rate detrimental to the species and should be reduced.
Of course, estimating the number of human-caused mortalities is a daunting task in its own right, and it is important that all substantial sources of mortality are included. For example, in the white-chinned petrel example only direct mortalities from a single fishery were considered, and these estimates were imprecise. Estimating bycatch levels for even common species vulnerable to multiple fisheries is challenging and typically requires a large number of assumptions; for rare species these problems are magnified (Uhlmann et al., 2005). In addition to other sources of bycatch mortality, there may be indirect mortalities that occur through competition for food stocks with fisheries or through marine habitat change due to global climate change. Some species may be impacted by collisions with man-made objects, such as wind turbines (Everaert and Stienen, 2007), power lines (Bevanger,
1998), or motor vehicles (Forman and Alexander, 1998). Similarly, many waterfowl species are subject to direct harvest but may also be vulnerable to human-caused habitat loss or degradation. In contexts where only one source of human-caused mortality is considered, the interpretation of the PBR changes.
In these cases, PBR estimates above single-source mortality levels may suggest that those mortality levels are sustainable if there were no other human-caused mortalities. Hence, that information coupled with a declining population may suggest that effort be directed towards evaluating other sources of human-caused mortality.
The PBR mortality limits tend to be precautionary as little is assumed about the population structure, a conservative population estimate is used, and the potential for biased population estimates is generally included by setting f < 1. The selection of f is a management decision and should be done

B I O L O G I C A L C O N S E RVAT I O N

with care, balancing conservation goals, stakeholder desires, and the ability to monitor the population. Coupled with conservative estimates for survival and age at first reproduction, estimates may become overly conservative. However, this approach remains a powerful tool for making management decisions when minimal information is available and for directing resources towards species of concern. In general it is limited by the requirement of an estimate of Rmax which may be unavailable for some species. When combined with maximum growth rate estimates from Niel and Lebreton
(2005), the PBR approach may be applied to bird species with appropriate choices for a and s. With independent estimates for the components of Eq. (1), high values of a and s are conservative, and may be chosen from studies of the species of interest or of similar species. For some species the estimates of a, s, and N may not be independent. In these cases, high values of a and s may no longer be conservative. For example, if a is also used to estimate the number of unseen juvenile age classes (as may be reasonable for seabirds), kmax is still negatively related to a, but N would be positively related to a; the overall effect on the PBR is unclear.
Many bird species – especially seabirds – have similar life histories to pinnipeds and cetaceans, characterized by long life, delayed maturity, and low fecundity. Because of this, the simulation work performed by Wade (1998) in developing the rule for cetaceans and pinnipeds is especially relevant for seabirds, where maximum growth rates for species of particular interest to New Zealand managers are in the 4–10% range. However, gender and age bias in bycatch rates suggest that extensions to this method, beyond the ad hoc approach used in the case study on white-chinned petrels, need to be developed. While there are limitations to this approach, it appears to provide similar answers to more sophisticated analyses, and is a reasonable approach when there is minimal information available. The results of Milner-Gulland and Akcakaya (2001)
¸
suggest that the rule could be applied to a variety of other bird species as well, such as moderately-lived gamebirds and waterfowl, but it may be less appropriate for short-lived species such as songbirds. The PBR method also ignores differences in life stages and gender. If there is a large age or gender bias in mortalities, and no adjustments to the method are made, the results may be misleading. By itself, the method is not appropriate for very small populations or for those listed as
‘critically endangered’. That is, it should not replace other impact analyses (for example, see Inchausti and Weimerskirch,
2001, or Zador et al., 2008), but could be still be useful as an additional tool for researchers studying these populations.
More complex population models, such as matrix models, can perform well when relevant population parameters are known. Their complexity may allow a level of realism beyond that possible through the simplified model used to estimate kmax and the simple PBR decision rule. In the case studies where results from different methods were compared, more sophisticated methods yielded more precise results. Additionally, when assumptions can be made regarding missing population parameters, matrix approaches are still valuable, both in general population modelling and as a method to estimate kmax. However, the PBR method, combined with the Niel and Lebreton method for estimating kmax, requires very little information in order to arrive at a decision and is easy to cal-

1 4 1 ( 2 0 0 8 ) 1 7 8 3 –1 7 9 2

1791

culate. Finally, the PBR method, having undergone considerable simulation study, is quite robust to estimation errors
(Wade, 1998; Milner-Gulland and Akcakaya, 2001).
¸
The PBR method can be used whatever estimate of kmax is available, whether it is from Eq. (2), a matrix approach, or from census data. However, direct estimation of kmax via matrix or census methods requires a quickly growing population, where density-dependence is unlikely to be a factor. Even in cases where this occurs, estimation via matrix methods requires more information than the Niel and Lebreton method, and estimation via census trends requires long time-series.
Alternative methods, such as placing reasonable bounds on unknown parameters in matrix models, may lead to imprecise estimates of kmax, due to the number of parameters that need to be estimated.
The Niel and Lebreton (2005) estimate of kmax combined with the PBR approach is a valuable tool for management of bird species. It may be used as an initial tool to direct resources and research effort towards species where more information is needed to assess whether mortality levels are sustainable. For species where minimal information is available such as seabirds, it provides a way to assess mortality levels or harvest rates.

Acknowledgements
This work was financially supported by the New Zealand Ministry of Fisheries (contract ENV2004/05) and the Beverly Fund of the University of Otago. We are also grateful to two anonymous reviewers, whose kind suggestions improved the manuscript, and to Prof. Navjot Sodhi for managing the editorial process. REFERENCES

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