Catch equations: restoring the missing terms in the nominally generalized Baranov catch equation

Observational models for the catch of fish at age a (or size) at time t are fundamental equations in fisheries science, linking a population model with data. The well known Baranov catch equation (which assumes that fishing and natural mortalities are constant over both age and time) is a nominal basis of those most commonly used in fish stock assessment and fish population models (which assume that fishing and natural mortalities vary with both age and time). But, what should a catch equation look like, if the instantaneous rates of fishing and natural mortalities of fish of age a at time t vary with age a and time t? Without answering this question, use of those catch equations in fish stock assessment and population models renders their results uncertain. In this paper, I derive a general catch in number or in biomass equation as observational models of an age- and time-dependent model for a fish population by Taylor series expansion of, and by directly manipulating, a general catch integral, reduce it to commonly used catch equations, and compare the performance of 11 of them using data on the western king prawn Penaeus latisulcatus. I show that the nominal generalization of the Baranov catch equation misses several terms. In so doing, I derive the catch equations more accurately and restore these missing terms. Although almost all approximations overestimate the catch per recruit for older prawns, all commonly used catch equations and their extensions perform worse than theoretically sound representations of the general catch equation and their approximations. The age-specific bias of all models is <2.5, <18 and <90% for a time interval of sampling of 1, 7 and 30 days, respectively. Such large biases even for moderate values of the length of the time interval of sampling highlight a need for assessing the utility of commonly used catch equations for individual species.