Some Procedures for use in Cohort Analysis and Other Population Simulations

Abstract

The catch equation [Formula: see text] must often be solved either forwards or backwards. Because Ft is an implicit function of catch, population size, and M, the solution must be either iterative or approximate. In this paper an improved approximation to Nt+1 is developed of the form[Formula: see text]for the forward solution and a corresponding equation for backward solution. Appropriate values of A are presented for a range of combinations of M and F; and A = 0.585 gives approximations to Nt+1 with an error of less than 1% provided that (Ft + M) is less than about 1.5. Much closer approximations are obtained if A is calculated for each case as a polynomial in M and either Ct/Nt for the forward solution or Ct−1/Nt for the backward solution. The appropriate coefficients for these polynomials are derived both by truncating Taylor's expansion and by statistical methods, and are presented here. One such polynomial for A gives errors of less than 0.02% where (F + M) is less than 1.1. Similarly good approximations can also be obtained by interpolating into a table of Nt+1/Nt against M and C/N.