Age-structured models

Background readings Lawson TA & Hilborn R (1985) Equilibrium yields and yield isopleths from a general age-structured model of harvested populations. Canadian Journal of Fisheries and Aquatic Sciences 42: Branch TA (2009) Differences in predicted catch composition between two widely used catch equation formulations. Canadian Journal of Fisheries and Aquatic Sciences 66: (Plus corrigendum, 66:1631)

Basic population processes Births (relative to number of females) Deaths Natural mortality Fishing mortality (plus vulnerability to fishing) Somatic growth Movement (immigration and emigration)

Define sequence of events For example that follows: 1.Begin year 2.Spawning 3.Fishing mortality 4.Natural mortality

Basic age-structured model sexes grouped The “plus” group Egg production Recruitment Catch weight a function of egg production e.g. Beverton-Holt Ages between 1 and n Mass at age Natural survival rate (0-1) Exploitation rate (0-1) Vulnerability (0-1) All individuals identical above the plus group age Fecundity

Definitions

Assumptions No immigration or emigration Parameters such as v, s, w, f don’t change over time Vulnerability v and size w not affected by fishing Parameters v, s, w, f are the same for all ages above n – 1

Starting conditions (t = 1) Starting recruitment Natural survival rate Exploitation rate Vulnerability Numbers in plus group age n

Plus group starting conditions (t=1)

Recruitment (t >1) Beverton-Holt stock-recruit function Recruitment Spawners (egg output) Is the curve flatter or steeper for different values of α and β? Intuition hard.

Beverton-Holt more tough questions What is unfished recruitment (R 0 ) and unfished spawning output (E 0 )? We want R 0 recruits to result in enough spawners to produce E 0 eggs, and we also want the Beverton-Holt equation to ensure that E 0 eggs will produce R 0 recruits Great interest in these parameters since these characterize the unfished population Solution: steepness parameterization

Beverton-Holt with steepness (h) E 0 or S 0 or SSB 0 R0R Spawners-per-recruit, unexploited with u t =0. See next slide. Mace, P. M. and I. J. Doonan A generalised bioeconomic simulation model for fish population dynamics. New Zealand Fisheries Assessment Research Document 88/4. Fisheries Research Centre, MAFFish, POB 297, Wellington, NZ.

SPR 0 (spawners per recruit) For u t = 0 How many spawners (eggs) would a single recruit produce in the absence of fishing?