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Best-fitting growth curves of the von Bertalanffy-Putter type.

ABSTRACT: INTRODUCTION:A large body of literature aims at identifying growth models that fit best to given mass-at-age data. The von Bertalanffy-Pütter differential equation is a unifying framework for the study of growth models. PROBLEM:The most common growth models used in poultry science literature fit into this framework, as these models correspond to different exponent-pairs (e.g., Brody, Gompertz, logistic, Richards, and von Bertalanffy models). Here, we search for the optimal exponent-pairs (a and b) amongst all possible exponent-pairs and expect a significantly better fit of the growth curve to concrete mass-at-age data. METHOD:Data fitting becomes more difficult, as there is a large region of nearly optimal exponent-pairs. We therefore develop a fully automated optimization method, with computation time of about 1 to 2 wk per data-set. For the proof of principle, we applied it to literature data about 217 male meat-type chickens, Athens Canadian Random Bred, that were reared under controlled conditions and weighed 28 times during a time span of 170 D. RESULTS:We compared 2 methods of data fitting, least squares using the sum of squared errors (SSE), which is common in literature, and a variant using the sum of squared log-errors SSElog. For these data, the optimal exponent-pairs were (0.43, 4.06) for SSE = 2,208.6 (31% improvement over literature values for the residual standard deviation) and (0.89, 0.93) for SSElog = 0.04599. Both optimal exponents were clearly distinct from the exponent-pairs of the common models in literature. This finding was reinforced by considering the region of nearly optimal exponents. DISCUSSION:We explain, why we recommend using SSElog for data fitting and we discuss prognosis, where data from the first 8 wk of growth would not be enough.

SUBMITTER: Kuhleitner M

PROVIDER: S-EPMC6698187 | biostudies-literature | 2019 Sep

REPOSITORIES: biostudies-literature

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Best-fitting growth curves of the von Bertalanffy-Pütter type.

Kühleitner Manfred M Brunner Norbert N Nowak Werner-Georg WG Renner-Martin Katharina K Scheicher Klaus K

Poultry science 20190901 9

<h4>Introduction</h4>A large body of literature aims at identifying growth models that fit best to given mass-at-age data. The von Bertalanffy-Pütter differential equation is a unifying framework for the study of growth models.<h4>Problem</h4>The most common growth models used in poultry science literature fit into this framework, as these models correspond to different exponent-pairs (e.g., Brody, Gompertz, logistic, Richards, and von Bertalanffy models). Here, we search for the optimal exponen ...[more]

PMID: 30895317

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Project description:The Bertalanffy-Pütter growth model describes mass m at age t by means of the differential equation dm/dt = p * m a - q * mb . The special case using the von 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). Fitting VBGF to size-at-age data requires the optimization of three model parameters (the constants p, q, and an initial value for the differential equation). For the general Bertalanffy-Pütter model, two more model parameters are optimized (the pair a < b of non-negative exponents). While this reduces bias in growth estimates, it increases model complexity and more advanced optimization methods are needed, such as the Nelder-Mead amoeba method, interior point methods, or simulated annealing. Is the improved performance worth these efforts? For the case, where the exponent b = 1 remains fixed, 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 were optimized). We hypothesized that the optimization of both exponents would result in a significantly better fit of the optimal growth function to the data and we tested this conjecture for a data set (20,166 fish) about the mass-growth of Walleye (Sander vitreus), a fish from Lake Erie, USA. To this end, we assessed the fit on a grid of 14,281 exponent-pairs (a, b) and identified the best fitting model curve on the boundary a = b of the grid (a = b = 0.686); it corresponds to the generalized Gompertz equation dm/dt = p * ma - q * ln(m) * ma . Using the Akaike information criterion for model selection, the answer to the conjecture was no: The von Bertalanffy exponent-pair model (but not the logistic model) remained parsimonious. However, the bias reduction attained by the optimal exponent-pair may be worth the tradeoff with complexity in some situations where predictive power is solely preferred. Therefore, we recommend the use of the Bertalanffy-Pütter model (and of its limit case, the generalized Gompertz model) in natural resources management (such as in fishery stock assessments), as it relies on careful quantitative assessments to recommend policies for sustainable resource usage.

Dataset Information

Best-fitting growth curves of the von Bertalanffy-Putter type.

Publications

Best-fitting growth curves of the von Bertalanffy-Pütter type.

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