Virtual Population Analysis (VPA) Equations for Nonhomogeneous Populations, and a Family of Approximations Including Improvements on Pope's Cohort Analysis

Abstract

A set of "backward" virtual population analysis (VPA) equations relates catch (Ct) from continuous fishing between times t and t + 1 to population n size (Nt, Nt+1) when a portion of the stock is unavailable to fishing. The usual VPA equations become a special case where the entire stock is available (i.e. the stock is homogeneous). A close approximation to the VPA equations is Nt = Nt+1 exp(M) + CtM/(1 − exp(−M)), which has properties similar to Pope's "cohort analysis" and is somewhat more accurate in the case of a continuous fishery, especially if the natural mortality rate (M) is large. Much closer simple approximations are possible if the seasonal pattern of catches is known.