Application of the Bertalanffy Growth Equation to Problems of Fisheries Management: A Review

Abstract

The development of large modern fisheries has led to the overexploitation, with resulting less than optimal production, of many of our fish stocks. A need to develop management techniques which would make it possible to hold stocks and exploitation rates more nearly at optimal levels has led to the development and extensive study of mathematical theory relating to fishing. An essential feature of this theory is the yield equation which relates the obtainable yield to such stock parameters as the number of recruits, rate of growth at various stages of life, natural mortality rate, and intensity of fishing. A widely used form of the yield equation is that developed by Beverton and Holt (U.K. Min. Agr. Fish. Food, Fish. Invest. 19: 1–533, 1957). In this equation the growth component is represented by the well-known Bertalanffy growth equation (Bertalanffy, Human Biol. 10: 181–213, 1938):[Formula: see text]It is not possible, however, to represent the growth of all fish satisfactorily by this equation, and where this is not possible, the Beverton and Holt yield equation also does not give satisfactory results. This situation calls for the need to develop a more generalized equation, using a more widely applicable form of the growth curve. As Bertalanffy (Helgolaender Wiss. Meeresuntersuch. 9: 5–37, 1964) has pointed out, the original equation assumes that catabolic processes are proportional to the weight of the animal, whereas anabolic processes are assumed to be proportional to the surface area or to the 2/3 power of the weight, but in practice at least the latter relationship frequently does not hold. The present paper examines the effect of different values of the exponent in the catabolic and anabolic components of the growth equation upon the shape of the growth curve and particularly upon the proportion of final size at which the point of inflection occurs. The modified yield equation embodying the generalized growth equation can be solved by means of tables of the incomplete beta-function by an extension of the method of Jones (Joint Meeting ICNAF/ICES/FAO, Lisbon, Doc. P21, 1957).