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For British grey seals, as with many pinniped species, population monitoring is implemented by aerial surveys of pups at breeding colonies. Scaling pup counts up to population estimates requires assumptions about population structure; this is straightforward when populations are growing exponentially but not when growth slows, since it is unclear whether density dependence affects pup survival or fecundity. We present an approximate Bayesian method for fitting pup trajectories, estimating adult population size and investigating alternative biological models. The method is equivalent to fitting a density-dependent Leslie matrix model, within a Bayesian framework, but with the forms of the density-dependent effects as outputs rather than assumptions. It requires fewer assumptions than the state space models currently used and produces similar estimates. We discuss the potential and limitations of the method and suggest that this approach provides a useful tool for at least the preliminary analysis of similar datasets.

Complete censuses are not practical for most animal populations. Instead, abundances usually have to be estimated by scaling up from partial counts. This process is complicated when the components of a population differ in their detectability. Pinnipeds such as grey seals (

Grey seals are colonial breeders. Females mature at around six years of age and give birth to a single pup in the autumn. The pups are born on land and remain ashore for several weeks. This behaviour, along with their neonatal white coats, makes them relatively easy to observe. Counting the other components of these populations is much less straightforward, since while they do haul out on land, the animals spend most of their time at sea and submerged.

The species is abundant around Britain and also on the eastern seaboard of North America. There are smaller numbers of animals in the Baltic Sea and around the northern European coastline. They were heavily hunted and more recently have been seen as a serious competitor to commercial fisheries. In 1914, a pessimistic estimate that the British population of grey seals was down to 500 individuals led to the Grey Seal (Protection) Act. This gave some legal protection to the species [

While the population was growing exponentially, scaling up from pup production estimates to total population size was relatively straightforward, requiring only estimates of the proportion of females breeding and the sex ratio. However, around 1995, the previously steady growth started to slow in some regions (Figure

Grey seal pup production estimates (points) and smoothed estimates (with 95% credibility intervals) for each of the four regions.

The state space models contain observation and process components but fit them together. Instead, we use the pup production data to estimate maximum growth rates for the start of the time series, and we apply this with an assumption of the location of density dependence in the species’ lifecycle to derive the adult population sizes. Our approach separates the estimation process into three parts: first smoothing the pup production estimates to give a distribution of estimated pup production trajectories, then associating each trajectory with a set of demographic parameters, and finally applying the relevant demographic parameters to each pup production trajectory to estimate the numbers of individuals in each of the other age classes. The method presented here can be seen as a form of approximate Bayesian computation [

Since 1984, pup production at the main Scottish grey seal colonies has been monitored by series of aerial surveys carried out throughout the breeding season. Each year, between 3 and 6 flights are made over each colony, using a fixed-wing aircraft with a vertically fitted large format camera [

Equivalent counts are made directly by observers on the ground at the colonies in England and Shetland. A consistent methodology has been used to estimate the total numbers of pups in each colony and, where sufficient surveys have been completed, calculate the estimates’ precision [

We use the same prior distributions for grey seal demographic parameters as the previously published state space models [

Distributions of parameter values. The priors are taken from Newman et al. [

Symbol | Prior | Posterior | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

North Sea | Orkney | Inner Hebrides | Outer Hebrides | |||||||||

distribution | mean | sd | mean | sd | mean | sd | mean | sd | mean | sd | ||

max pup survival | Beta(14.53,6.23) | 0.7 | 0.1 | 0.7 | 0.1 | 0.7 | 0.1 | 0.7 | 0.1 | 0.7 | 0.1 | |

adult survival | Beta(22.05,1.15) | 0.95 | 0.04 | 0.92 | 0.01 | 0.96 | 0.01 | 0.95 | 0.02 | 0.91 | 0.01 | |

max fecundity | Beta(22.05,1.15) | 0.95 | 0.04 | 0.95 | 0.05 | 0.95 | 0.05 | 0.95 | 0.05 | 0.95 | 0.05 |

The population estimation was carried out in three stages: first smoothing the pup production estimates to give a distribution of estimated pup production trajectories, then associating each trajectory with a set of demographic parameters, and finally applying the relevant demographic parameters to each pup production trajectory to estimate the numbers of individuals in each of the other age classes.

We fitted generalised additive models (GAMs), with log link functions and a quasi-Poisson error structure [

Pup production in each region showed a period of approximately exponential growth (Figure

Scaling the replicate pup production trajectories up into population trajectories requires them to be combined with suitable sets of demographic parameter values. Any combination of demographic parameter values will produce a particular stable age-structure and exponential growth rate at low population densities. Given that the animals are assumed to breed first at age 6 and have fecundity and mortality rates are constant among adults, and using the notation in Table

The maximum growth rate within each estimated pup production trajectory was taken as an estimate of the low-density growth rate for that replicate population. The demographic parameter values then need to be drawn from their joint conditional probability distribution given the appropriate exponential growth rates. Explicitly calculating this distribution is not straightforward, but it can be approximated numerically by drawing from an unconditional joint probability distribution for the demographic parameters and discarding those results whose maximum growth rate falls outside a small neighbourhood of the required value. This approach is slow, because small neighbourhoods will produce high parameter rejection rates [

Two sets of incomplete deterministic age-structured matrix models of the females within each population were then constructed, with one assuming that all the density dependence was in fecundity and the other putting it all into pup survival. These differed from those in Newman et al. [

Five annual age classes were used in the model, along with a sixth that contained all the older animals. Only animals within the oldest category were considered to breed [

Equivalent calculations were made for the model with density-dependent pup survival though these used fecundity,

Two different methods were used to combine the results of the two models of each region. The first assumed that one of the two models was correct and, in the absence of information to choose between them, simply superimposed the two posterior distributions of population estimates to effectively give a single, model-averaged, overall distribution of population estimates. The second approach assumed that the truth lay somewhere between the two explicit models and placed an additional uniform prior on where the result lay between them. Two uniform random variables were, therefore, drawn to produce each of a set of estimates: one variable to identify a pair of bootstrap replicates, and the other to determine the weighting of the results of the two population models for that replicate. The result was a distribution that effectively smeared across the two directly modelled extremes. Total, rather than female only, population estimate distributions were then calculated by multiplying each replicate by a draw from a normal distribution with mean 1.73, the value used in previous analyses of this data [

Figure

Estimates (mean and 95% credibility intervals) of the total size of the grey seal populations in each region before breeding in 2007. The results for the two models are given along with those from simple (equally weighted) model averaging and applying the uniform prior across the two models. The numbers in italics are the equivalent estimates calculated from the best fitting state space models contained in the 2008 report of the UK Standing committee on Seals [

Model | 2007 Regional Population (in thousands, mean value, and 95% CIs^{1}) | ||||
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North Sea | Orkney | Inner Hebrides | Outer Hebrides | Total^{2} | |

Density dependent pup survival | 20.9 (16.4–25.7) | 46.1 (35.6–58.0) | 8.0 (5.9–10.7) | 34.3 (27.0–42.0) | 109.4 (84.8–136.4) |

Density dependent fecundity | 24.1 (19.7–29.0) | 124.6 (102.2–151.1) | 24.7 (18.3–34.2) | 69.6 (57.0–86.0) | 243.3 (190.8–277.9) |

Model averaged | 22.5 (17.1–28.3) | 85.4 (37.2–146.2) | 16.4 (6.2–32.3) | 52.1 (28.1–82.5) | 177.3 (88.5–289.2) |

Uniform prior | 22.5 (18.0–27.3) | 85.78 (45.8–131.3) | 16.3 (8.0–27.7) | 52.1 (33.1–74.3) | 176.8 (104.6–260.8) |

^{1}All the CIs include uncertainty in the population sex-ratio.

^{2}The CIs are estimated conservatively by summing those of the individual models.

Population trajectories (mean values and 95% credibility intervals) for each region. In each case, the lower (blue) set of lines show the total population estimates from the density-dependent pup survival models and the upper (black) set of lines those from the models with density-dependent fecundity.

The two models agree that there are probably slightly more than 20,000 grey seals that breed on the eastern coasts of England and Scotland (our North Sea region) and that this population is continuing to grow in a near exponential fashion. In all the other areas, the predictions diverge rapidly with the models containing density-dependent fecundity producing estimates for 2007 that are 2-3 times as large as the equivalent figures for density-dependent survival. The confidence intervals of these pairs of models do not overlap. Outside the North Sea, the precision of the population estimates would be greatly improved if it were possible to distinguish where in the grey seal life-cycle density dependence impacts most strongly. Because they do not specify the functional form of the density dependence, the models presented here can give little information on this. It is also difficult to extract this information from the more complex state space models of this system even though these do explicitly assume the form of the density dependence [

The approach presented here is Bayesian in the sense that it uses prior distributions on the density-independent demographic parameters (adult survival, and maximum pup survival, maximum female fecundity). However, it is approximate in its use of the priors on the demographic parameters and because there are no formal priors on the density-dependent components of the model or the population sizes, there are no complete likelihoods for the results. The result is a semiparametric approach to model fitting, rather than a fully parametric model. The pup production data are used to derive a maximum growth rate for the early part of the time series, and this is used, with an assumption of the location of density-dependence in the species’ lifecycle, to derive the adult population sizes together with approximate graphical representations of the density dependent effects. This strategy has some similarities to that adopted in approximate Bayesian computation (e.g., [

This methodology effectively pushes most of the uncertainty in the system into the error terms of the GAMs. These models, therefore, have lower precision than the colony-based pup production estimates and estimate each year’s expected, rather than actual, pup production. The uncertainty then passes through into the population estimates and could be expected to reduce their precision.

The age-structured population models are largely deterministic. They effectively assume that varying environmental conditions will mainly affect transitions where density dependence occurs. That is likely to be a simplification of the actual situation but not an unreasonable one given that the other processes are assumed to be relatively insensitive to the size of the population relative to its carrying capacity. Because these models do not contain a predefined form for their density-dependent components, they also avoid prescribing a distribution or pattern for the effects of environmental variation on the populations’ dynamics. Instead, such variability can be expected to simply reduce the precision of their results.

The similarity of the credibility interval widths presented here to those from the more detailed state space models suggests that the additional effects, such as demographic stochasticity and movement between areas, which are explicitly represented in those models, may have limited impact on the precision of their results in this case. This is not entirely unexpected, since demographic stochasticity should be small for such large populations, and the estimated amount of movement between colonies is also small (inspection of posterior movement parameter estimates reported in [

The uniform prior on the relative impact of density dependence on fecundity and pup survival is clear and unambiguous. It is much easier to calculate than a set of intermediate models, and this reflects the current state of ignorance as to the true balance between these factors. While it is straightforward to apply here, it might be harder to justify its combination with formal likelihood-based model selection techniques, such as Akaike’s information criterion [

Our approach could be seen as a retrograde step, since it does not attempt as complete a description of the system or utilisation of the data, as the state space models. It could also be criticised for its limited predictive and explanatory power. However, any projection of models requires extrapolation, and needs to be done cautiously. For these populations, the most obvious danger would be in the projection of density-dependent effects beyond the range of existing data, which requires a belief that their functional forms have been adequately described. It is also possible that if the state space models were modified in the light of these results, for example, by modifying them to allow adult survival to vary between areas, the precision of their estimates would improve. However, as the most appropriate analysis of datasets will always depend on their size and the availability of resources, this sort of less demanding methodology may also be appropriate for other small datasets.

Collection of aerial survey data is funded by NERC and involves many people at the Sea Mammal Research Unit and elsewhere. Additional data were collected by Natural England, the National Trust, Lincolnshire Wildlife Trust and SNH. The analysis and its interpretation have been discussed with many colleagues at St. Andrews and the members of the NERC Special Committee on Seals. The authors are grateful to them all.