Dynamics of Age and Size for a Stochastic Population Model

Abstract

We derive a fully stochastic model for the dynamics of age and size in a population. The key to our derivation is that the assumptions made about stochastic growth and survival, in conjunction with some approximations made in the analysis of size-selective fishing mortality, allow us to propagate through time the probability density function of size of individuals in the population by a recursive update of its parameters. The model permits a concise description of the size distribution at age based on mean log-length and variance. The stochastic process is characterized in the model in just a few parameters: two growth coefficients, one variance parameter associated with growth, instantaneous risk of natural mortality, instantaneous risk of fishing mortality, which is a separable function of size-specific selectivity and annual full recuitment fishing mortality, and initial conditions (year-class strength, initial size distribution). Estimation theory is developed for the model, and likelihood equations are derived which provide statistical principles for setting the trade-off between fitting data on length frequency, on total biomass of catch, and on age frequency. The tractability of the model permits a clear description of sampling probability distributions. A number of simpler models, such as catch-age models and delay-difference models, fit within the theoretical framework of our stochastic model.