The variance of population and yield from an age-structured stock, with application to the North Sea herring

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Abstract
In a previous study Horwood and Shepherd (1981) developed a technique for calculating the variance of the mature component of a fish stock with a known stock and recruitment relationship and for calculating the variance of yield from the mature component of the population. This study first illustrates these techniques with reference to the North Sea herring, and second, extends the theory to allow the variance of yield to be calculated in cases when fishing occurs on additional components of the stock. The extension is not trivial since the persistence of perturbations in recruitment (such as a good year class) results in a serial correlation. This feature has been neglected in previous studies and, in the example of the North Sea herring, the neglect could cause over a 30% underestimate of variance. The application to the North Sea herring shows that as fishing mortality exceeds F = 0·4 yr−1 then the yield decreases and both the variance of yield and coefficient of variation of yield increase substantially. Maximum sustainable yields of about 106 tonnes are achieved with a range of F from 0·I to 0·4 and over this range the coefficient of variation of yield is about 10%. There is evidence to suggest that variability in recruitment is log-normally distributed and not simply additive. An approximation has been developed to accommodate this assumption and simulations show this to be accurate for practical purposes.

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