Uncertainty, robustness, and the value of information in managing an expanding Arctic goose population

Abstract

We explored the application of dynamic-optimization methods to the problem of pink-footed goose (Anser brachyrhynchus) management in western Europe. We were especially concerned with the extent to which uncertainty in population dynamics influenced an optimal management strategy, the gain in management performance that could be expected if uncertainty could be eliminated or reduced, and whether an adaptive or robust management strategy might be most appropriate in the face of uncertainty. We combined three alternative survival models with three alternative reproductive models to form a set of nine annual-cycle models for pink-footed geese. These models represent a wide range of possibilities concerning the extent to which demographic rates are density dependent or independent, and the extent to which they are influenced by spring temperatures. We calculated state-dependent harvest strategies for these models using stochastic dynamic programming and an objective function that maximized sustainable harvest, subject to a constraint on desired population size. As expected, attaining the largest mean objective value (i.e., the relative measure of management performance) depended on the ability to match a model-dependent optimal strategy with its generating model of population dynamics. The nine models suggested widely varying objective values regardless of the harvest strategy, with the density-independent models generally producing higher objective values than models with density-dependent survival. In the face of uncertainty as to which of the nine models is most appropriate, the optimal strategy assuming that both survival and reproduction were a function of goose abundance and spring temperatures maximized the expected minimum objective value (i.e., maxi–min). In contrast, the optimal strategy assuming equal model weights minimized the expected maximum loss in objective value. The expected value of eliminating model uncertainty was an increase in objective value of only 3.0%. This value represents the difference between the best that could be expected if the most appropriate model were known and the best that could be expected in the face of model uncertainty. The value of eliminating uncertainty about the survival process was substantially higher than that associated with the reproductive process, which is consistent with evidence that variation in survival is more important than variation in reproduction in relatively long-lived avian species. Comparing the expected objective value if the most appropriate model were known with that of the maxi–min robust strategy, we found the value of eliminating uncertainty to be an expected increase of 6.2% in objective value. This result underscores the conservatism of the maxi–min rule and suggests that risk-neutral managers would prefer the optimal strategy that maximizes expected value, which is also the strategy that is expected to minimize the maximum loss (i.e., a strategy based on equal model weights). The low value of information calculated for pink-footed geese suggests that a robust strategy (i.e., one in which no learning is anticipated) could be as nearly effective as an adaptive one (i.e., a strategy in which the relative credibility of models is assessed through time). Of course, an alternative explanation for the low value of information is that the set of population models we considered was too narrow to represent key uncertainties in population dynamics. Yet we know that questions about the presence of density dependence must be central to the development of a sustainable harvest strategy. And while there are potentially many environmental covariates that could help explain variation in survival or reproduction, our admission of models in which vital rates are drawn randomly from reasonable distributions represents a worst-case scenario for management. We suspect that much of the value of the various harvest strategies we calculated is derived from the fact that they are state dependent, such that appropriate harvest rates depend on population abundance and weather conditions, as well as our focus on an infinite time horizon for sustainability.