Generalizing the Parker equation of DEPM: Incorporating the size dependence of population number and reproductive inputs to estimate spawning biomass and female population by size

Abstract

Many fisheries assessed using the daily egg production method (DEPM) have data available that inform the dependence of adult population structure and fecundity on fish body size. Yet most do not consider this size dependence and use means of female body weight and batch fecundity. Here we extend the Parker (1980) equation for DEPM spawning biomass to make explicit the size structure of females. Our generalized DEPM approach assumes female body size partitioned into weight classes. It uses data inputs of female population number proportions and batch fecundity by discrete weight class or a continuous fitted relationship of batch fecundity versus weight. We derive closed-form estimation formulas for total spawning biomass, and for two new DEPM outputs, total female population number and female population numbers by weight class. We also derive analytic formulas for the standard errors of these three population abundance estimates using the delta method. The allometric dependence of batch fecundity on female body weight is fitted by maximum likelihood with allometric error structure. In fishery management application, a principal benefit of this size-based generalization of DEPM is the added output of absolute female population numbers by weight class, which can be fitted in a fishery assessment model or can directly inform management decision making. We have posted an R package “DEPM” containing these new estimation functions and their delta method standard errors. We apply this size-based DEPM method to South Australian snapper, Chrysophrys auratus, for which large estimated decreases in spawning biomass from 2014 to 2018 informed a strong management response.