Optimal exponent-pairs for the Bertalanffy-Pütter growth model

The Bertalanffy-Pütter growth model describes mass m at age t by means of the differential equation dm / dt = p ⋅ m a − q ⋅ m b . The special case using the Bertalanffy exponent-pair a =2/3 and b =1 is most common (it corresponds to the von Bertalanffy growth function VBGF for length in fishery literature). For data fitting using general exponents, five model parameters need to be optimized, the pair a < b of non-negative exponents, the non-negative constants p and q , and a positive initial value m 0 for the differential equation. For the case b =1 it is known that for most fish data any exponent a <1 could be used to model growth without affecting the fit to the data significantly (when the other parameters p , q , m 0 were optimized). Thereby, data fitting used the method of least squares, minimizing the sum of squared errors ( SSE ). It was conjectured that the optimization of both exponents would result in a significantly better fit of the optimal growth function to the data and thereby reduce SSE . This conjecture was tested for a data set for the mass-growth of Walleye ( Sander vitreus ), a fish from Lake Erie, USA. Compared to the Bertalanffy exponent-pair the optimal exponent-pair achieved a reduction of SSE by 10%. However, when the optimization of additional parameters was penalized, using the Akaike information criterion ( AIC ), then the optimal exponent-pair model had a higher (worse) AIC , when compared to the Bertalanffy exponent-pair. Thereby SSE and AIC are different ways to compare models. SSE is used, when predictive power is needed alone, and AIC is used, when simplicity of the model and explanatory power are needed.