A Bayesian method for estimating moose (Alces alces) population size based on hunter observations and killed at age data

Abstract

Lots of wild species, fish and mammals, are heavy harvested through fishing and hunting. Reliable population size estimates are valuable management tools for these species. In cases where killed at age data are available, models outlined under the framework known as ”cohort analysis” or ”virtual population analysis (VPA)” are used extensively. In fish stock management several models using Bayesian techniques have been developed through the last two decades. In this study a model using a Bayesian approach for estimating moose population size is examined. The model combines principles from discrete time series analysis, where basic cohort analysis based on killed at age data constitutes the bulk, and analysis in continuous time for each hunting season based on data from hunter observations. The analysis in continuous time aims to find age- and year-specific expressions for the hunting mortality rate. In the discrete time series analysis, the hunting yield is viewed as a binomially distributed variable, with pre-harvest population size as ”number of trials” and mortality rate derived from the analysis in continuous time as ”probability parameter”. All basic principles are known from previous surveys, but the way they are assembled is, to the authors knowledge, innovative. The model performed very well when tested against simulated populations with known parameter values. For real data tests are conducted through cross-validation based on spatial subsets and by comparing results from temporal subsets. Generally the model performed well in these test. However, an issue is revealed by comparing results from different temporal subsets, since the hunters ability to spot moose seems to develop over time (years) and/or depend on moose density. This issue should not terminate the practical implementation of the model. If a satisfying solution to the issue is achieved, it might have a possible positive impact on other methods for estimating abundance of wild species based on effort, a very prevalent class of models. The real data used for testing the model, and to demonstrate some practical interpretations, are from the municipality of Ringerike in southern Norway. Killed at age data are available from 1988 till 2012 in combination with hunter observations. The estimates show a moose population size rapidly increasing in the period from 1988 till its peak in 1993 at a posterior mean population size of approximately 3900 individuals. Thereafter, in line with large hunting yields, reduced reproductivity rate and increased rate of natural mortality, the population size declined rapidly till an estimated pre-harvest population size of approximately 1700 individuals in year 2000. Thereafter the total population size has been estimated as quite stable, but with a declining trend over the last couple of years. Usually the natural (non harvest) mortality rate is assumed fixed and known when cohort analysis methods are used for estimating abundance of wild species. The model presented in this study is capable of producing reliable, and to some extent practical beneficial, posterior distributions for the natural mortality rate based on an informative prior distribution and an adequate amount of data. These posterior distributions for natural mortality rates indicate surprisingly high rates for the years around 1993.