Local sensitivity equations for maximum sustainable yield reference points

Quantitative fishery harvest strategies in many countries throughout the world are based on maximum sustainable yield (MSY) reference points (RPs); in particular, the fishing mortality rate that produces MSY (i.e. F_msy) and the population biomass (B_msy) that results from fishing at F_msy. There is often great interest in the sensitivity of MSY RPs to their inputs and underlying assumptions. We perturb (i.e. ω) inputs to generate influence curves which we denote as F_msy(ω) and B_msy(ω) as functions of ω. We derive diagnostic equations for the slopes of F_msy(ω) and B_msy(ω) at the origin (i.e. no perturbation). These equations provide a conceptual understanding of how F_msy and B_msy will change in response to changes in inputs, which one does not get from a simple computational sensitivity analysis. We show that a leading factor determining the sensitivity of F_msy is the amount of curvature in the equilibrium yield curve near F_msy. Another leading factor affecting the sensitivity of B_msy is if the stock-recruitment function indicates low density-dependence in recruitment productivity. With some exceptions, we find that F_msy will increase if the body growth rate or asymptotic size increases, 2) proportions mature increase at younger ages, and 3) natural mortality rates decrease and the Ricker stock-recruitment function is used. If the fishery selectivity function of age is monotone increasing then F_msy will also usually increase if peak selectivity is shifted to an older age. Usually the signs of the slopes for B_msy and F_msy are the same.