A revisit to Pope's cohort analysis

Gulland's [Gulland, J.A., 1965. Estimation of mortality rates. Annex to Arctic Fisheries Working Group Report (meeting in Hamburg, January 1965). ICES. C.M. 1965, Doc. No. 3 (mimeographed)] virtual population analysis (VPA) is commonly used for studying the dynamics of harvested fish populations. However, it necessitates the solving of a nonlinear equation for the instantaneous rate of fishing mortality of the fish in a population. Pope [Pope, J.G., 1972. An investigation of the accuracy of Virtual Population Analysis using cohort analysis. ICNAF Res. Bull. 9, 65-74. Also available in D.H. Cushing (ed.) (1983), Key Papers on Fish Populations, p. 291-301, IRL Press, Oxford, 405 p.] eliminated this necessity in his cohort analysis by approximating its underlying age- and time-dependent population model. His approximation has since become one of the most commonly used age- and time-dependent fish population models in fisheries science. However, some of its properties are not well understood. For example, many assert that it describes the dynamics of a fish population, from which the catch of fish is taken instantaneously in the middle of the year. Such an assertion has never been proven, nor has its implied instantaneous rate of fishing mortality of the fish of a particular age at a particular time been examined, nor has its implied catch equation been derived from a general catch equation. In this paper, we prove this assertion, examine its implied instantaneous rate of fishing mortality of the fish of a particular age at a particular time, derive its implied catch equation from a general catch equation, and comment on how to structure an age- and time-dependent population model to ensure its internal consistency. This work shows that Gulland's (1965) virtual population analysis and Pope's (1972) cohort analysis lie at the opposite end of a continuous spectrum as a general model for a seasonally occurring fishery; Pope's (1972) approximation implies an infinitely large instantaneous rate of fishing mortality of the fish of a particular age at a particular time in a fishing season of zero length; and its implied catch equation has an undefined instantaneous rate of fishing mortality of the fish in a population, but a well-defined cumulative instantaneous rate of fishing mortality of the fish in the population. This work also highlights a need for a more careful treatment of the times of start and end of a fishing season in fish population models.