# Some explorations of the life history ratios to describe length composition, spawning-per-recruit, and the spawning potential ratio

### Abstract

Evaluating the status of data-poor fish stocks is often limited by incomplete knowledge of the basic life history parameters: the natural mortality rate ($M$), the von Bertalanffy growth parameters ($L_\infty$ and $k$), and the length at maturity ($L_m$). A common approach to estimate these individual parameters has been to use the Beverton–Holt life history invariants, the ratios $M/k$ and $L_m/L_\infty$, especially for estimating $M$. In this study, we assumed no knowledge of the individual parameters, and explored how the information on life history strategy contained in these ratios can be applied to assessing data-poor stocks. We developed analytical models to develop a relationship between $M$/$k$ and the von Bertalanffy growth curve, and demonstrate the link between the life history ratios and yield- and spawning-per-recruit. We further developed the previously recognized relationship between $M$/$k$ and yield- and spawning-per-recruit by using information on $L_m/L_\infty$, knife-edge selectivity ($L_c/L_\infty$), and the ratio of fishing to natural mortality ($F/M$), to demonstrate the link between an exploited stock’s expected length composition, and its spawning potential ratio (SPR), an internationally recognized measurement of stock status. Variation in length-at-age and logistic selectivity patterns were incorporated in the model to demonstrate how SPR can be calculated from the observed size composition of the catch; an advance which has potential as a cost-effective method for assessing data-poor stocks. A companion paper investigates the effects of deviations in the main assumptions of the model on the application of the analytical models developed in this study as a cost-effective method for stock assessment [Hordyk, A. R., Ono, K., Valencia, S., Loneragan, N. R., and Prince, J. D. 2015. A novel length based empirical estimation method of spawning potential ratio (SPR), and tests of its performance, for small-scale, data-poor fisheries. ICES Journal of Marine Science, 72: 217–231].

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Publication
In ICES Journal of Marine Science
Date