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Improved approximations for estimation of size-transition probabilities within size-structured models

Modelling annual growth of individuals in a size-structured model requires calculation of the size-transition probabilities for moving from one size class to another. This requires evaluation of two-dimensional integrals when there is individual variability in growth. For computational simplicity, it is common to approximate the integrals by setting all individuals in a size class to the midsize of that class or by ignoring the individual variability. We develop a more accurate approximation that assumes a uniform distribution in size within each size class. The approximation is fast and hence feasible for Bayesian models in which the matrix of transition probabilities must be computed for each posterior sample. The improved accuracy of the new approximation is shown to hold over a diverse range of formulations for incremental growth. For the New Zealand Paua 5A (Haliotis iris) stock assessment model, it was found to reduce the average approximation error of the size-transition probabilities by 86% and 98% compared with the midpoint and deterministic growth approximations, respectively. Moreover, the midpoint and deterministic approximations inflated the estimated maximum sustainable yield by 6% and 8%, respectively, and the current biomass by almost 30% in comparison with the more accurate approximation.