Intrinsic and realized growth rates of the greylag goose population

We used the allometric methods of Niel and Lebreton (2005) and Johnson et al. (2012) to estimate the intrinsic population growth rate (i.e. no density dependence and no anthropogenic mortality) of greylag geese. From Johnson et al. (2012), adult survival under ideal conditions for birds ranging in mass from 12 g to 8.66 kg is estimated as:

where p is the observed proportion of the population alive at the observed maximum lifespan with p ∼ beta(3.34,101.24), M is body mass in kg, α is age at first breeding and e is the error in the model relating body mass to longevity with e ∼ Normal(0,σ² = 0.087). The distribution of p is constant, and is unrelated to body mass (Johnson et al. 2012).

Using both female and male mean body masses of 3.108 kg (SD=0.274) and 3.509 (SD=0.321), respectively (Dunning Jr. 2008), and an age at first breeding of α=3 (Nilsson et al. 1997, Kampp and Preuss 2005), the median (intrinsic) survival is θ=0.889 and the 95% confidence interval is 0.785–0.943. This represents a maximum longevity of about 30 years, which agrees well with that of birds in captivity (Nigrelli 1954). We note that the use of an age at first breeding of 2 < α ≤ 3 (i.e. some portion of 2-year-olds breed; Nilsson and Kampe-Persson 2018) causes only minor differences in the value of θ.

Next, we used the values of θ=0.889(0.785–0.943) and α=3 along with Eq. 15 from Niel and Lebreton (2005) to estimate the intrinsic population growth rate as:

The median is similar to empirical values for snow geese Chen caerulescens and barnacle geese Branta leucopsis provided by Niel and Lebreton (2005).

We next estimated the realized mean growth rate, λ, using a log-linear regression model of a temporal sequence of greylag goose counts in midwinter, N^w:

The expectation of the intrinsic population growth rate arising from the log-normal distribution is thus: . For the period (2004–2012) in which national midwinter counts are available from all Range States (Heldbjerg et al. 2020), the estimated mean growth rate was λ̄ = 1.063 (1.048 - 1.079) (Fig. 1). Note that this analysis assumes that whatever the bias in national counts may be, it is relatively constant over the period 2004–2012.

We note, however, that the rate of population growth may have decreased since 2012. National counts are available from all Range States from 2004 to 2016 (i.e. four additional years) except Spain (outside Doñana) and Germany. If we use the observed growth rates in those two countries during 2004–2012 to extrapolate their respective counts through 2016, the growth rate of the flyway population was λ̄ = 1.038 (1.026 - 1.051) during 2004–2016. Counts of geese in the Netherlands and in Spain appear to be most responsible for the lower growth rate when compared to the 2004–2012 period. We use this estimate of population growth rate as it best conforms to the time period of reported abundance and offtake in the ISSMP.

Figure 1.

The intrinsic population growth rate of greylag geese as estimated using the methods of Niel and Lebreton (2005) and Johnson et al. (2012), and the realized growth rates based on national counts in the Range States of the NW/SW European population of greylag geese. Note that counts for Spain and Germany were imputed for 2013–2016. Dashed, vertical lines represent the means.

Evidence of bias in abundance and/or offtake

We first assessed the potential for bias in reported estimates of abundance and/or offtake by examining expected kill and survival rates. Kill rate is defined here as the annual rate of mortality due to all anthropogenic sources, whether hunting/culling related or not. We assumed that: 1) the population is not subject to any significant density dependence; 2) all anthropogenic mortality is due to sport hunting or to take under derogations; 3) winter mortality is primarily due to hunting/culling; and 4) offtake is additive to other sources of mortality. While all of these assumptions are unlikely to be true, we believe they represent a reasonable starting point.

There are at least two ways to estimate the realized kill rate, k̄ , independently of estimates of abundance and offtake. In the first, we use estimates of the intrinsic and realized growth rates. Under the assumptions listed above, the realized kill rate is a function of the intrinsic and realized growth rates:

such that:

We can also use Eq. 4 to infer the maximum kill rate that would not cause the population to decline as:

The second approach relies on Eq. 4, but uses intrinsic and realized survival rates, again under our four model assumptions:

such that:

We used Eq. 5, along with published values of realized survival, θ̄, from the literature (Frederiksen et al. 2004, Powolny et al. 2018) to estimate the realized kill rate of greylag geese. We then compared estimates from both methods to the kill rate implied by reported estimates of midwinter abundance and offtake in the ISSMP:

such that:

As described in Results, an examination of survival and kill rates suggested that estimates of abundance and/or offtake reported in the ISSMP are biased. Therefore, we used the estimated intrinsic and realized population growth rates to investigate the potential magnitude of that bias by using a modification of Eq. 4:

where λ̄ is the realized growth rate, λ is the intrinsic growth rate, H and N are the reported size of the offtake and the post-breeding population, respectively, and α and β are bias coefficients. If the (approximate) equality in Eq. 6 is satisfied for α = β = 1, then there is no apparent bias in estimates of abundance or offtake. We found combinations of α and β that satisfy the equality in Eq. 6 for reported values of H and N, using 5000 independent draws from the distributions for λ̄ and λ (Fig. 1).

Info-gap decision analysis

The existence of bias of unknown magnitude in greylag goose monitoring renders traditional approaches to modeling population dynamics and decision analysis unsuitable. However, in an effort to guide decision making, we explored an info-gap approach, which poses the question: ‘what level of offtake will most likely satisfy a management criterion for the largest range of uncertainty?’ In our case, the deep uncertainty concerns the true values of α and β, expressing the degree of bias in estimates of abundance and offtake, respectively. Thus, we would like to choose a management action, in this case a level of proportional change in offtake, H, that would meet some management criteria for a larger range of uncertainty in α and β than any other potential change in level of offtake.

Ultimately, the management criterion will be represented by target population sizes for one or more management units defined for greylag geese, which are derived from the different migratory behavior of geese within the flyway (Bacon et al. 2019). However, population targets are not useful as criteria in this case because it is abundance itself that is uncertain. As an alternative, we can establish a management criterion based on the predicted growth rate of the NW/SW European flyway population using Eq. 4. In other words, we can determine the nominal level of offtake that would meet a growth-rate criterion for the largest possible range in values of α and β. This approach must assume that, whatever the bias in estimates of abundance and/or offtake, it is relatively constant over time.

Population growth based on national counts during 2004–2016 was λ̄ = 1.038 (1.026 - 1.051) amid growing concern about the adverse impacts of population size. In the face of deep uncertainty about current levels of offtake and abundance, we suggest that decision makers might adopt a precautionary approach, for example, of seeking to reduce population size by 15% over the 10-year span of the ISSMP. Thus, the management goal would be an annual growth rate of . The decision maker understands (s)he is unlikely to meet the criterion of a realized growth rate λ̄ = 0.98 precisely, but would like to get as close as possible. The info-gap decision problem then is: ‘what nominal level of offtake will meet a performance criterion of λ̄-1 ≤ C , where C is some critical threshold, for as large a range in α and β as possible?’ For example, (s)he might consider C=0.02, such that 0.96 ≤ λ ≤ 1 was acceptable (i.e. population size decreasing by 4% or less per year). Accordingly, an increasing population, or a population declining more than 4% per year, would be considered unacceptable. The lower limit of 0.96 could be anything, and here we simply note that an annual λ=0.96, if realized, would reduce population size by 34% in 10 years. In June 2020, the International Working Group of the EGMP decided that a reduction of at least 20% in abundance of greylag geese was in line with the management objectives of the ISSMP. We note that the critical limit is symmetric, as expressed by λ̄-1 ≤ C . However, it need not be symmetric; for example, in the case where negative growth rates are more or less desirable than positive ones.

We first established a range of uncertainty in α and β to examine. Based on the results of our bias investigation, it is likely that estimated offtake is biased high as long as true abundance exceeds nominal abundance by a factor < 3.5. Thus, we set α ∼ uniform(0.5,3.5) and β ∼ uniform(0.2,1.0). We then examined a range of nominal values of offtake and, for each combination of α and β, predicted λ̄-1 using Eq. 4.

While the info-gap analysis relies only on the estimated intrinsic growth rate (and not on an observed growth rate), it is nonetheless sensitive to nominal values of abundance and offtake. As with the investigation of bias, we used imputed, total winter counts, but used an average of the three most recent years available (2016–2018) from the International Waterbird Counts coordinated by Wetlands International (Heldbjerg et al. 2020). We used the most up-to-date information on abundance as we were interested in identifying a prospective level of take. Unfortunately, more contemporary estimates of offtake than those reported in the ISSMP are not available, so we continued to assume that the nominal level of offtake is 450 000. Thus, nominal offtake and post-breeding abundance was assumed to be:

Evidence of bias

Using our estimates of the intrinsic growth rate, λ ≈ 1.159(1.120–1.206) and realized growth rate during 2004–2016, λ̄ = 1.038 (1.026 - 1.051), the estimate of realized kill rate was k̄ ≈ 0.10 (0.07 - 0.14). The maximum kill rate that could be tolerated without inducing a decline in greylag goose abundance was k̄ ≈ 0.14(0.11–0.17). Because the estimated growth rate of the flyway population was positive during 2004–2016, we would therefore expect the realized kill rate to be k̄ < 0.14 .

Using survival rates of adults reported by Powolny et al. (2018) of θ̄ ∊{0.81, 0.92, 0.74, 0.95, 0.85} and our median value of θ = 0.89 in Eq. 5, estimates of realized kill rates are k̄ ∊{0.09, -0.03, 0.17, -0.07, 0.04} (where ∊ means ‘is an element of’ the set denoted by {…}). Using estimates of greylag goose survival given in Frederiksen et al. (2004, Table 4) of θ̄ ∊{0.84, 0.80, 0.82, 0.85, 0.80, 0.68, 0.73}, estimates of realized kill rates are k̄ ∊{0.06, 0.10, 0.08, 0.04, 0.10, 0.24, 0.18} . For values of θ̄ > θ we get values of k̄ < 0, which are inadmissible, meaning that θ̄ is biased high or θ is biased low. We also note that if we assume θ = 0.89 is approximately correct, values of k̄ ≥ 0.17 would cause the population to decline (recall that for k > 0.14, λ < 1). If we ignore these irregularities and the fact that the annual survival rates from the literature are not independent, then the estimated mean kill rate is k̄ = 0.08 (-0.05 - 0.22), which is similar to that based on population growth rates.

Therefore, a kill rate of k̄ = 0.3 implied by estimates of abundance and offtake in the ISSMP seems doubtful at best. We can go one step further using a slight modification to Eq. 4, such that:

where γ = post-breeding age ratio of young to older birds. If k = 0.3 and we allow for no natural mortality such that θ = 1, then γ ≥ 0.48 (i.e. ≥ 32% young in fall) is needed to prevent the population from declining. While this high level of productivity has been observed in the Netherlands (Hornman et al. 2020), it has not been observed in the past decade. Age-ratio data from mainly non-migratory populations in both the Netherlands and parts of western Germany show averages of 14% and 16% young in fall, respectively (Koffijberg and Kowallik 2018, Hornman et al. 2020). And we emphasize that γ ≥ 0.48 is the minimum productivity assuming no natural mortality (θ = 1), which is clearly unrealistic given the assessments mentioned before. For comparison, the allometric estimate of intrinsic productivity is γ = 0.32(0.19–0.54), or about 24% young in the fall. Moreover, if we assume k̄ = 0.3 and the values of θ̄ reported in the literature, then survival under ideal conditions is θ > 1 for every θ except θ̄ = 0.68, which is impossible (again, assuming additive anthropogenic mortality). Thus, it seems unlikely that k̄ = 0.3 . We therefore conclude that reported abundance is biased low and/or reported offtake is biased high.

Figure 2.

Combinations of α and β that satisfy the equality in Eq. (6) for nominal values of abundance and offtake of the NW/SW European population of greylag geese that were reported in the ISSMP. The horizontal dashed line represents an unbiased reported estimate of offtake, and the vertical dashed line represents an unbiased reported estimate of goose abundance.

Using 5000 samples from the distributions for λ and λ, we solved Eq. 6 for β for a range of values of α. A plot of the resulting values of β against α can be divided into four quadrants, representing cases where: 1) H is biased low (β > 1) and N is biased high (α < 1); 2) H is biased low (β > 1) and N is biased low (α > 1); 3) H is biased high (β < 1) and N is biased low (α > 1); and 4) H is biased high (β < 1) and N is biased high (α < 1) (Fig. 2). If we were to assume that the nominal estimate of offtake is unbiased (horizontal dashed line in Fig. 2), abundance would be underestimated by a factor of about 2.5–3. On the other hand, if we assume that the nominal estimate of abundance is unbiased (vertical dashed line in Fig. 2), offtake would be overestimated by a factor of almost three. If one were to assume that actual goose abundance is unlikely to be more than about three times the nominal abundance, then inevitable conclusion is that the nominal estimate of offtake is biased high, perhaps severely so.

The conclusion that reported offtake is biased high is further supported if we consider the possibility that the intrinsic growth rate is a maximum that may not be realized in a variable environment, or that density-dependent mechanisms are acting to reduce it. Consider the following modification to Eq. 6:

where p < 1 represents a potential reduction in the intrinsic growth rate. For any values p < 1, the combinations of α and β that satisfy the equality in Eq. (7) even more strongly suggest a positive bias in reported offtake.

Info-gap results

Based on 5000 samples from the distributions for λ̄, the probabilities of meeting the management criterion of 0.96 ≤ λ ≤ 1.96 for a range of potential levels of offtake are shown in Fig. 3. Notice that all probabilities are low (< 20%), reflecting the challenge of meeting the restrictive criterion of 0.96 ≤ λ ≤ 1.00 in the face of deep uncertainty concerning the true extent of bias, α and β. A nominal level of offtake of 40% higher than that reported in the ISSMP is expected to achieve the management criterion for a wider range in α and β than any other alternative. But we emphasize that this decision would be accompanied by an 86% chance that the criterion would not be met (assuming all examined values of α and β are considered equally plausible). In other words, there would be an 86% chance that abundance could either increase or decline by more the 4% annually. Finally, we note a very broad range of changes in offtake had nearly identical (mean) probabilities of meeting the management criterion, and indeed are not statistically distinguishable from each other.

Figure 3.

Probabilities of achieving a population growth rate of 0.96 ≤ λ ≤ 1.00 for varying levels of change in reported offtake (relative to the value of 450 000 reported in the ISSMP) for NW/SW European population of greylag geese in the face of deep uncertainty about bias in estimates of abundance and offtake. Error bars represent 95% confidence limits, which account for uncertainty in the intrinsic growth rate of greylag geese.