TITLE:
A Two-Step Growth Curve: Approach to the von Bertalanffy and Gompertz Equations
AUTHORS:
Laura Rogers-Bennett, Donald W. Rogers
KEYWORDS:
Growth Model, Growth Surface, Rate Equation, Staged Growth, Population Models
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.6 No.5,
April
15,
2016
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 (k1 and k2) 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) = k1(t) -k2(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.=