482 SCHAEFER AND BEVERTON [CHAP. 21 



with constant parameters comparable in magnitude to those estimated for 

 plaice and haddock gives catch/population curves in which the maximum catch 

 occurs at about one-third of the virgin population size. This shift of the maxi- 

 mum to the left compared with the parabola is the result partly of maintaining 

 the recruitment constant despite the reduced size of the population and partly 

 to the enhanced relative growth rate of the smaller and younger fish which 

 predominate in the stock under these conditions compared with those in the 

 virgin stock. These two factors combine to cause the rate of natural increase 

 of the stock to be rather greater at low population levels compared with high 

 ones than is implied by (17), so the maximum catch is obtained by fishing the 

 population rather more heavily and down to a smaller fraction of its virgin size. 

 If, however, the relation between stock and recruitment defined by (13) is 

 incorporated into the model, combined with a density-dependent growth, the 

 maxima of the curves are shifted to the right, and with a certain degree of 

 dependence of recruitment on stock a very close replica of a parabola is obtained. 

 With the most marked degree of change of recruitment with stock that is 

 compatible with stability, the maximum of the curve occurs even higher, at 

 about G0% of the virgin population size. The studies on which these findings 

 are based are not exhaustive and, indeed, are still in progress, but they are 

 sufficient to show that the simple parabola implied in the Schaefer approach is 

 certainly not inconsistent with the properties of the more complex and, in a 

 sense, more "realistic", of the Beverton-Holt models. Possibly, in some 

 fisheries, recruitment may be sufficiently independent of stock for the Schaefer 

 approach to tend to underestimate the extent to which the virgin stock should 

 be reduced to obtain the maximum catch — the North Sea plaice may be a case 

 in point — but that is a matter for future research. 



References 



Baranov, F. I., 1918. On the question of the biological basis of fisheries. (In Russian.) 



Izvest. nauch.-issl. Ikthiol. Inst., 1, 81-128. 

 Bertalanffy, L. von, 1934. Untersuchungen iiber die Gesetzlichkeit des Wachstums. 1 Teil. 



Allgemeine Grundlage der Theorie; mathematische und physiologische Gesetzlichkeit 



des Wachstums bei Wassertieren. Arch. Entw. Mech. Org., 131, 613-52. 

 Bertalanffy, L. von, 1938. A quantitative theory of organic growth. (Inquiries on Growth 



Laws. II). Hum. Biol., 10, 181-213. 

 Beverton, R. J. H., 1953. Some observations on the principles of fishery regulation. 



J. Cons. Explor. Mer, 19, 56-68. 

 Beverton, R. J. H., 1962. The long-term dynamics of certain North Sea fish populations. 



In The Exploitation of Natural Animal Populations. British Ecological Society, 



Symposium Number 2 (Edited by Le Cren, E.D. and Holdgate, M. W.) Oxford. 



242-259. 

 Beverton, R. J. H. and S. J. Holt, 1956. A review of methods for estimating mortality 



rates in exploited fish jaopulations, with special reference to sources of bias in catch 



sampling. Rapp. Cons. Explor. Mer, 140, Pt. I, 67-83. 

 Beverton, R. J. H. and S. J. Holt, 1957. On the dynamics of exploited fish populations. 



Fish, ftjrwrf,, Loruh, Ser. 2, 19. 



