Problematic likelihood functions from sensible population dynamics models: a case study

All International Whaling Commission assessments of the Bering-Chukchi-Beaufort Seas stock of bowhead whales (Balaena mysticetus) rely on likelihood functions derived from nearly the same data and a particular family of population dynamics models. Eleven such past bowhead assessments are compared. We note that the type of dynamics model used in all of these assessments has a strong influence on the features of the likelihood surface. We examine how the likelihood surface created by such models exhibits a narrow, cusp-shaped, flat-topped, steep-edged ridge of strong nonlinear dependency between key model parameters. We discuss how such features are generally troublesome for statistical inference and interpretation. Through examples we examine some of the implications for maximum likelihood estimation and the parametric bootstrap. Although such a dynamics model is very useful for producing realistic population trajectories, it is a poor model from which to generate likelihood functions in this case because it and the data together establish a nearly chaotic dynamical nonlinear system.