A Two-Step Growth Curve: Approach to the von Bertalanffy and Gompertz Equations

Abstract

Many curves have been proposed and debated to model individual growth of marine invertebrates. Broadly, they fall into two classes, first order (e.g. von Bertalanffy) and sigmoidal (e.g. Gompertz). We provide an innovative approach which demonstrates that the growth curves are not mutually exclusive but that either may arise from a simple three-stage growth model with two steps ( k 1 and k 2 ) depending on the ratio of the growth parameters . The new approach predicts sigmoidal growth when is close to 1, but if either growth from stage A to stage B or B to C is fast relative to the other, the slower of the two steps becomes the growth limiting step and the model reduces to first order growth. The resulting curves indicate that there is a substantial difference in the estimated size at time t during the period of active growth. This novel two-step rate model generates a growth surface that allows for changes in the rate parameters over time as reflected in the new parameter n ( t ) = k 1 ( t ) - k 2 ( t ). The added degree of freedom brings about individual growth trajectories across the growth surface that is not easily mapped using conventional growth modeling techniques. This two (or more) stage growth model yields a growth surface that allows for a wide range of growth trajectories, accommodating staged growth, growth lags, as well as indeterminate growth and can help resolve debates as to which growth curves should be used to model animal growth. This flexibility can improve estimates of growth parameters used in population models influencing model outcomes and ultimately management decisions.=