Catch equations: calculating the instantaneous rate of fishing mortality from catch and back

Ecological processes are often studied by sampling. If such samples are of relatively large sizes, the effects of sampling must be taken into account before making any inferences. In fisheries science, almost all the ‘sophisticated’ process-based models and catch equations are based on the tracking of the cohorts of a fish population over time t, but a cohort-tracking equation for the sex- and age-aggregated catch remains to be derived. In this paper, I derive a general catch equation for a large class of process-based models for a fish population by tracking all the cohorts in the population within an arbitrary time interval of sampling (fishing), examine its general behaviour, and outline some of its special cases. Such a catch equation only assumes the integrability and differentiability of the quantities concerned, makes no specific biological assumptions, and applies in many contexts other than fisheries science. Its consequences in interpreting fish catch will be far-reaching, for many terms are proved to be missing in all previous catch equations, incurring a positive bias of up to 3.5 orders of magnitude and a negative bias of up to one order of magnitude in the population model implemented loosely for the southern rock lobster on an annual step. I also derive the instantaneous rate of fishing mortality as an explicit function of fish catch and other quantities, and interpret a relative index U(t) of fish abundance in biomass or number in the population extracted from its catch and effort data by use of a generalized linear model (GLM), a generalized additive model (GAM) and their like. The first and second order approximation thus obtained of the instantaneous rate of fishing mortality as a function of fish catch and other quantities provide convenient initial values for solving the nonlinear catch equations. The relative index U(t) of fish abundance in biomass or number in the population is found to have many interpretations, so that a process-based model should be used to explain any set of catch and effort data. For a single population or species, the sex- and age-aggregated catch C(t) of fish in biomass or number in the population at time t is always biased upwards for a given value of F(t); F(t) is always biased downwards for a given value of C(t). Moreover, use of the word selectivity in the phrase gear selectivity and use of the word exploitable as a modifier as in the phrase the exploitable biomass or number of the individuals in a fish population are misleading and should be abandoned. Finally, the commonly used catch equation should be abandoned in all fisheries population models because of its substantial bias.