The inverse life‐history problem,size‐dependent mortality and two extensions of results of Holt and Beverton

In 1958, Sidney Holt developed a model to determine the optimal mass at which to harvest a cohort of fish having von Bertalanffy growth and experiencing constant natural mortality. Holt and Ray Beverton then gave a life‐history interpretation to the analysis, from which Beverton developed a theory of Growth, Maturity, and Longevity (GML) that allows one to predict quantities such as age at maturity or relative size at maturity using life‐history parameters. I extend their results in two ways. First, keeping the original formulation, in which the rate of natural mortality is constant, I show how one can invert Beverton's result to determine the rate of natural mortality from life‐history data. I illustrate this inverse method with data on three species of tuna and compare the estimates with those based on tagging. Second, I extend Beverton's GML theory to include size‐dependent mortality. I explore previously published mortality models and introduce a new mortality function that has size‐independent and size‐dependent components. I show that the new size‐dependent mortality function leads to the prediction that age at maturity depends upon asymptotic size (as well as the other life‐history parameters), something that Beverton's original theory lacked. I illustrate this extension with a simple example, discuss directions for future work and conclude that nearly 60 years on these contributions of Holt and Beverton continue to lead us in new and exciting directions.