Accounting for Variation in Mortality and Allee Effects When

13 We update recent work on the scientific inference and reproductive biology of steepness in three 14 directions. First, we show how variation in natural mortality can be included in the formula 15 for steepness, for both a biomass dynamics and age-structure models. We do this using the 16 delta-method, so that only the mean and covariance of natural mortality rates appear in the 17 characterization of steepness. Second, we show how to generalize the previous methods for cases 18 in which the stock recruitment relationship is depensatory or has an Allee effect: as spawning 19 population falls below a certain level, per capita reproduction starts to fall, rather than approach 20