Open Access Publications from the University of California

## Fish Bulletin No. 116. Fitting A Von Bertalanffy Growth Curve By Least Squares Including Tables of Polynomials

Abstract

Historically, the growth of animals has been described by a variety of functions which relate size increase to a unit of time. These functions vary from a straight-line relationship to the more complex asymptotic-type curve. Fish growth is often represented by a function which is asymptotic to some average maximum size the fish will attain. A useful regression formula for representing these curves has three parameters, with a representing the asymptotic value of y. Stevens (1951) showed the utility of this basic curve by writing Gompertz's law, the logistic curve, and Mitscherlich's law in this form. It is also possible to write the growth curves of von Bertalanffy (1938) in this way.

The objective of this paper is to present a useful method for fitting Beverton's (1954) modification of the von Bertalanffy growth-in-length curve, but the method is adaptable to any curve which can be written in the basic form of equation. The tables in the Appendix, and the worked example should provide sufficient information for fitting a von Bertalanffy curve.

In the past, the methods used to fit a von Bertalanffy curve to observed fish length have required inefficient techniques such as guessing-by-eye (von Bertalanffy, 1938) or approximation-through-transformation (Beverton and Holt, 1957; Ricker, 1958). The method presented in this paper is not only easy to use, but is based on the efficient, well-accepted technique of least squares. Under the assumption of normality, the least squares method produces estimates which are equivalent to maximum likelihood estimates.

With computers, other iterative methods or an adaption of the described method may be used to obtain very accurate estimates of the parameters of the von Bertalanffy curve. Stevens (1951) developed an iterative procedure for fitting (1) and Nelder (1961) gives an iterative method for fitting a generalized logistic curve which includes (1) as a special case. Nelder's method does not require the x's to be equally spaced and assumes a constant variance for log y rather than for y. However, the following description and the tables in the Appendix should be useful to workers who do not have access to a computer.