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

A set of “backward” virtual population analysis (VPA) equations relate5 catch ( C , ) from continuous fishing between times t and t + 1 to population size ( N , , N, + ,) when a portion of the stock is unavailable to frshing The usual VPA equations become a special case where the entire stock i s avaitable ( I e the stock is hornogeneous) A close approximation to the VPA equations is N , = N , , , exp(Mj + C,M/(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 i f the natural mortality rate ( M ) is large Much closer simple approximations are possible if the Seasonal pattern of catches is known