less Monte Carlo variation than SIS for the same computing time; in particu- lar, standard SIS algorithms become very inefficient for random effects models (L. Thomas pers. comm.). Computing time is often an important consideration with complex Bayesian analyses, and given that our MCMC simulations took upwards of 60 hours to run, a suitable SIS implementation may have proven prohibitively time consuming. On the other hand, an advantage of SIS is that it easily handles the incorporation of new time points at the end of the series
(Newman et al. 2009), so that as new data become available each year the model
can simply be updated, whereas the MCMC algorithm would need re-running for the full time series.
A further alternative would be a classical analysis using the Kalman filter (see, e.g., Besbeas et al. 2002). Traditionally, this approach relies on the use of potentially restrictive normal approximations to discrete distributions and a linear model structure. However, analyses based on the normality assumption have been shown to be robust, at least with large population sizes (Brooks
et al. 2004). Furthermore, recent work by Besbeas et al. (2008) into methods
for initialising the Kalman filter for ecological time series, and accounting for nonlinearities, have improved and extended the usability of the Kalman filter such that it would be a viable alternative. This does not avoid the fact that the Bayesian approach naturally deals with those situations for which the Kalman filter essentially has ‘workarounds’, thereby providing a more flexible analysis framework (Millar & Meyer 2000, Jamieson & Brooks 2004). Therefore, in this case, the Bayesian approach using MCMC was the preferred choice.
6.3
Integrated population predictions
Increases in sea surface temperatures in UK coastal waters have already been correlated with reductions in seabird productivity and survival, presumably mediated through changes in prey abundance (e.g. Frederiksen et al. 2004b,
Harris et al. 2005, Sandvik et al. 2005). With UK climate-change scenarios
predicting further increases in sea surface temperature in the future (Lowe et al. 2009), serious pressures will potentially be placed on seabird populations. The frequency of extreme weather events is also expected to increase (Solomon et al. 2007), likely leading to further reductions in population growth rates through increased variability in demographic parameters (Frederiksen et al. 2008). The recent widespread breeding failures at many UK seabird colonies (e.g. Mavor
ability to reliably predict the population consequences of recent and possible future changes in demographic rates is thus vital for the proper planning of management and conservation strategies.
In Chapter 4, we implemented a modified version of the integrated popu- lation model to produce population predictions for the Isle of May guillemot colony. An advantage of the integrated approach over the standard types of pre- dictive model is the way in which the predictions simultaneously reflect both abundance and demographic data, thereby making full use of the available in- formation and, potentially, resulting in more accurate, precise and reliable pre- dictions. Also, the future projection period was essentially treated as part of the estimation model, and in this way posterior distributions for the predicted states (breeder and prebreeder population sizes) were obtained as part of the model output, rather than performing stochastic projections (Maunder et al.
2006).
Here, again, the Bayesian approach was key, and to be preferred over a clas- sical analysis. With the Bayesian analysis, we were easily able to incorporate most major sources of uncertainty into the projections, including uncertainty in the parameter and population estimates, and in the underlying demographic process. Had we been selecting between competing models, we could also have accounted for model uncertainty by using reversible jump MCMC (Green 1995,
King & Brooks 2002, King et al. 2009). But possibly the biggest advantage of
a Bayesian approach over a classical approach in the present context is in the form of the statistical output. The posterior probability distributions yielded by Bayesian analyses are simple to explain and present to managers and pol- icy makers, and automatically include the uncertainty of the estimates (Wade 2000). And the potential of posterior distributions extends beyond obtaining simple summary statistics. Once posterior samples are generated, they (or functions of them) may be queried for a large number of biologically important questions (e.g. Taylor et al. 1996, Brooks et al. 2008), or used by managers in a decision analysis to evaluate the consequences of different conservation deci- sions (Berger 1985, Taylor et al. 1996,Wade 2000). For example, we were able to obtain posterior estimates of the 10-year probabilities of population decline below a range of different thresholds, providing simple statistics for comparison between scenarios.
One source of uncertainty that could not be formally incorporated in the model framework was future uncertainty. By modelling a number of scenarios we anticipated a range of possible outcomes, but without a means to assign probabilities to these outcomes there is no way to know which, if any, is most
6.3 Integrated population predictions 127
likely. A formal way to include future uncertainty would be to use the proba- bilities associated with climate prediction scenarios, but this depends first on a model relating demographic parameters or population growth to an environ- mental variable, and second that future climate models for this variable are available. For example, Jenouvrier et al. (2009a) projected emperor penguin
Aptenodytes forsteri population responses to future sea ice changes, using In- tergovernmental Panel on Climate Change (IPCC) projections of sea ice extent. However, previous studies of the Isle of May guillemots did not identify any en- vironmental covariates that adequately explained variation in survival (Crespin
et al. 2006a, Harris et al. 2007b). Although not detailed earlier in the thesis,
we also conducted our own exploratory analysis of a range of possible envi- ronmental covariates for survival and productivity: the covariates tested were the NAO index, local and regional winter sea surface temperatures, and one- and two-year lagged versions of each of these, none of which appeared to be significantly correlated with any of the demographic parameters. However, it should be noted that these were only tested on an ad hoc basis by estimating the Spearman’s rank correlation between covariates and parameter point esti- mates, and did not take into account the sampling variability of the parameters. Thus, there may be some profit in exploring these covariates further in a more rigorous logistic regression framework, with random effects to cope with any temporal variability not explained by the covariates (see, e.g., Gimenez et al. 2008), and with the benefit of several years of additional data.
In the absence of alternative information, we restricted our analysis to as- suming that future conditions for each demographic rate mirrored either recent or long-term historical conditions. Given the recent decline in demographic performance, its probable links with environmental factors, and the predictions of future environmental change, the ‘worst-case’ scenario (i.e., productivity and survival continue at post-2000 levels) may well be the most realistic of the five scenarios tested. Under this scenario, the model predicted an expected decline in population size of 31% over 10 years. While this figure is in itself important, of greater interest, particularly for policy makers, is the variation in popula- tion trajectories, as this gives an indication of how confident we can be in our predictions. For example, the 95% credible interval of the 10-year proportional change for the above scenario ranged from an 8% to a 53% decrease, indicating that there is a lot of uncertainty in the final outcome. However, the fact that the upper limit was well below zero provides a very high degree of certainty that there will be some decline in population size over this period.