An explicit solution for calculating optimum spawning stock size from Ricker's stock recruitment curve

Ricker’s stock recruitment model is widely used to describe the spawner-offspring relationship for fishes. After model fitting, the spawning stock size that produces the maximum sustainable yield ( S MSY ), and the harvest corresponding to it ( U MSY ), are two of the most common biological reference points of interest to fisheries managers. However, to date there has been no explicit solution for either reference point because of the transcendental nature of the equation needed to solve for them. Therefore, numerical or statistical approximations have been used for more than 30 years. Here I provide explicit formulae for calculating both S MSY and U MSY in terms of the productivity and density-dependent parameters from Ricker’s model.