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Compensatory dynamics of fish recruitment illuminated by functional elasticities

Models of Stock Recruitment Relationships (SRRs) are often used to predict fish population dynamics. Commonly used SRRs include the Ricker, Beverton-Holt, and Cushing functional forms, which differ primarily by the degree of density dependent effects (compensation). The degree of compensation determines whether recruitment respectively decreases, saturates, or increases at high levels of spawning stock biomass. In 1982 J.G. Shepherd united these dynamics into a single model, where the degree of compensation is determined by a single parameter, however the difficulty in relating this parameter to biological data has limited its usefulness. Here we use a generalized modeling framework to show that the degree of compensation can be related directly to the functional elasticity of growth, which is a general quantity that measures the change in recruitment relative to a change in biomass, irrespective of the specific SRR. We show that the elasticity of growth can be calculated from short-term fluctuations in fish biomass, is robust to observation error, and can be used to determine general attributes of the SRR in both continuous time production models, as well as discrete time age-structured models. This framework may be particularly useful if fisheries time-series data are limited, and not conducive to determining functional relationships using traditional methods of statistical best-fit.