Analytical expressions of sustainable benchmarks generic for all two-stage-structured models of exploited fish populations

Abstract

Of the types of fishery models that use time series of removals, fishing effort, and indices of abundance, two-stage-structured models are less prevalent, especially in developing the maximum sustainable yield (MSY) benchmarks, in part because of ignorance of how to calculate those benchmarks. To fill this gap, this study aims to derive analytical expressions for calculating MSY benchmarks associated with a continuous delay-differential model, given that the recruit stage relates to the spawning-stock biomass (SSB) according to a stock–recruit relationship (SRR) and that the SRR parameters are combined with the components of the composite (two-stage-structured or age-aggregated) yield per-recruit model CYPR14. The focus is on analytical expressions of the fishing mortality producing MSY ( FMSY), considering that the equilibrium yield comes from either the equilibrium SSB at the start of fishing regimes or the equilibrium average SSB during fishing regimes. The expressions obtained are versatile for all two-stage-structured models. Derivations employing the fished equilibrium average SSB are recommended for use because they produce precautionary MSY benchmarks and because they are consistent with their counterparts in contemporary age-structured stock assessment models.