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

Bertalanffy-Pütter models for the first wave of the COVID-19 outbreak

“…Thereby the curve fitted to the initial 48 data-points (red solid line) had a slightly lower asymptotic limit than the one fitted to the full data (black solid line). This observation generalizes, as for SSE the asymptotic limit computed for an initial segment of the data in general underestimates the final size ( Kühleitner et al, 2019 ). Thus, SSE could be used if all data-points came from the first wave.…”

Comparing growth patterns of three species: Similarities and differences

“…Table 2 summarizes the model parameters that minimized SSLE . The parameters for chicken are from [21]. In order to define dimensionless coordinates, asymptotic mass m max was computed as the limit of the growth curve m ( t ), when time t approaches infinity.…”

“…for chicken [2], the standard deviation of mass becomes higher for heavier animals, whence the method of least squares may not be suitable for data-fitting. Instead, as in [21] we minimized the sum of squared errors between the logarithm of the growth function and the logarithmically transformed data ( SSLE ). This defined the following function (2): , assuming model (1) with exponents a , b .…”

Growth Patterns of birds, dinosaurs and reptiles: Are differences real or apparent?

“…For chicken, results quoted from [18], the optimal model parameters (mass in gram, time in days) were a = 0.89, b = 0.93, m 0 = 32.92 g, p = 1.0952, and q = 0.7988. This translated into an asymptotic mass of 2.67 kg, an inflection-point at day 61 with890 g (33% of the asymptotic mass) and the maximal weight gain of 19.9 g/day.…”

“…For chicken, the best fitting growth model and the near-optimal models were identified in [18]. As the paper uses the same approach for the alligator and dinosaur data, the method is only sketched.…”