Marc Mangel, Jon Brodziak, and Gerard DiNardo

11/15/2010 09:00 AM

Applied Mathematics & Statistics

We update recent work on the scientific inference and reproductive biology of steepness in three directions. First, we show how variation in natural mortality can be included in the formula for steepness, for both a biomass dynamics and age-structure models. We do this using the delta-method, so that only the mean and covariance of natural mortality rates appear in the characterization of steepness. Second, we show how to generalize the previous methods for cases in which the stock recruitment relationship is depensatory or has an Allee effect: as spawning population falls below a certain level, per capita reproduction starts to fall, rather than approach a constant. We generally assume that the mechanism of depensation is imperfect fertilization (and thus develop a two-sex generalization of our previous work) and determine steepness in this case for both a biomass production model and an age-structured model and explore the implications of such depensatory reproduction on the response of stocks to harvesting. We briefly discuss how an increase in mortality as population size declines (as has been suggested for penguins) could also be a mechanism for depensation. Third, we describe an improved method for computing the maximum per capita reproduction in the age-structured model, and show how the equivalent for the biomass dynamics model is computed.