Optimal and near-optimal exponent-pairs for the Bertalanffy-Pütter growth model (original) (raw)
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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⋅ma−q⋅mb. 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
A Model for the Mass-Growth of Wild-Caught Fish
Open Journal of Modelling and Simulation
The paper searched for raw data about wild-caught fish, where a sigmoidal growth function described the mass growth significantly better than non-sigmoidal functions. Specifically, von Bertalanffy's sigmoidal growth function (metabolic exponent-pair a = 2/3, b = 1) was compared with unbounded linear growth and with bounded exponential growth using the Akaike information criterion. Thereby the maximum likelihood fits were compared, assuming a lognormal distribution of mass (i.e. a higher variance for heavier animals). Starting from 70+ size-at-age data, the paper focused on 15 data coming from large datasets. Of them, six data with 400-20,000 data-points were suitable for sigmoidal growth modeling. For these, a custom-made optimization tool identified the best fitting growth function from the general von Bertalanffy-Pütter class of models. This class generalizes the well-known models of Verhulst (logistic growth), Gompertz and von Bertalanffy. Whereas the best-fitting models varied widely, their exponent-pairs displayed a remarkable pattern, as their difference was close to 1/3 (example: von Bertalanffy exponent-pair). This defined a new class of models, for which the paper provided a biological motivation that relates growth to food consumption.
On the exponent in the Von Bertalanffy growth model
PeerJ, 2018
Von Bertalanffy proposed the differential equation '() = × () - × () for the description of the mass growth of animals as a function () of time . He suggested that the solution using the metabolic scaling exponent = 2/3 (Von Bertalanffy growth function VBGF) would be universal for vertebrates. Several authors questioned universality, as for certain species other models would provide a better fit. This paper reconsiders this question. Based on 60 data sets from literature (37 about fish and 23 about non-fish species) it optimizes the model parameters, in particular the exponent 0 ≤ < 1, so that the model curve achieves the best fit to the data. The main observation of the paper is the large variability in the exponent, which can vary over a very large range without affecting the fit to the data significantly, when the other parameters are also optimized. The paper explains this by differences in the data quality: variability is low for data from highly controlled experimen...
Best-fitting growth curves of the von Bertalanffy-Pütter type
Poultry Science
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...
A modified von Bertalanffy growth model dependent on temperature and body size
Mathematical biosciences, 2017
Fish growth models are widely used in fisheries as well in aquacultures and ecology. Water temperature is one of the most important factors determining the growth of fish. In the present study, we propose a growth model that includes the effect of water temperature on the growth in the von Bertalanffy growth model. Our model was applied to fit the growth data of bullhead (Cottus gobio), brown trout (Salmo trutta L.), juvenile salmon (Salmo salar), and Araucanian herring (Strangomera bentincki). The model reproduces the growth patterns of each species and fits a set of appropriate parameter values for each species. Moreover, the model reflects the seasonal growth rates quite well.
Fish and Fisheries, 2008
The common practice among researchers who study fish growth is to a priori adopt the von Bertalanffy growth model (VBGM), which is the most used and ubiquitous equation in the fisheries literature. However, in many cases VBGM is not supported by the data and many species seem to follow different growth trajectories. The information theory approach frees the researcher from the limiting concept that a 'true' growth model exists. Multi-model inference (MMI) based on information theory is proposed as a more robust alternative to study fish growth. The proposed methodology was applied to 133 sets of length-at-age data. Four candidate models were fitted to each data set: von Bertalanffy growth model (VBGM), Gompertz model, Logistic and the Power model; the three former assume asymptotic and the latter non-asymptotic growth. In each case, the 'best' model was selected by minimizing the small-sample, bias-corrected form of the Akaike information criterion (AIC c ). To quantify the plausibility of each model, the 'Akaike weight' w i of each model was calculated. Following a MMI approach, the model averaged asymptotic length L 1 for each case was estimated, by model averaging estimations of L 1 interpreting Akaike weights as a posterior probability distribution over the set of candidate models. The VBGM was not selected as the best model in 65.4% of the cases. Most often VBGM was either strongly supported by the data (with no other substantially supported model) or had very low or no support by the data. The estimation of asymptotic length was greatly model dependent; L 1 as estimated by VBGM was in every case greater than that estimated by the Gompertz model, which in turn was always greater than that estimated by the Logistic model. The percentage underestimation of the standard error of L 1 , when ignoring model selection uncertainty, was on average 18% with values as high as 91%. Ignoring model selection uncertainty may have serious implications, e.g. when comparing the growth parameters of different fish populations. Multi-model inference by model averaging, based on Akaike weights, is recommended as a simple and easy to implement method to model fish growth, for making robust parameter estimations and dealing with model selection uncertainty.
A new model for simulating growth in fish
PeerJ, 2014
A real dynamic population model calculates change in population sizes independent of time. The Beverton & Holt (B&H) model commonly used in fish assessment includes the von Bertalanffy growth function which has age or accumulated time as an independent variable. As a result the B&H model has to assume constant fish growth. However, growth in fish is highly variable depending on food availability and environmental conditions. We propose a new growth model where the length increment of fish living under constant conditions and unlimited food supply, decreases linearly with increasing fish length until it reaches zero at a maximal fish length. The model is independent of time and includes a term which accounts for the environmental variation. In the present study, the model was validated in zebrafish held at constant conditions. There was a good fit of the model to data on observed growth in Norwegian spring spawning herring, capelin from the Barents Sea, North Sea herring and in farmed coastal cod. Growth data from Walleye Pollock from the Eastern Bering Sea and blue whiting from the Norwegian Sea also fitted reasonably well to the model, whereas data from cod from the North Sea showed a good fit to the model only above a length of 70 cm. Cod from the Barents Sea did not grow according to the model. The last results can be explained by environmental factors and variable food availability in the time under study. The model implicates that the efficiency of energy conversion from food decreases as the individual animal approaches its maximal length and is postulated to represent a natural law of fish growth.
A total of 16 data sets on wild and cultivated fishes, crustaceans and molluscs were used to test and compare conventional growth curves (von Bertalanffy, Logistic, Gompertz and Richards) and a new growth model. Statistical properties for estimation of the models were evaluated and compared to determine suitability. The absolute value of the Hougaard measure of skewness of parameter estimates (h) was used as the criterion to evaluate statistical behavior of the models. For conventional curves, the cases where the estimates were severely skewed or contained considerable nonlinearity (h > 0.15) were: von Bertalanffy (93.5%), Logistic (87.5%), Gompertz (85.1%) and Richards (97.6%). Depending on the parameterization used in the new model, 87.5 to 91.6% had negligible skewness (h ≤ 0.1), indicating desirable close-to-linear behavior and better performance than conventional growth curves. The poor statistical properties for estimation of conventional growth curves call for a critical reconsideration of their indiscriminate use to model growth of fishes, crustaceans and molluscs. The new model can be reliably used to analyze growth of organisms under a wide variety of situations and to derive statistical inferences of possible relations of its parameters with ecological or management variables.
The von Bertalanffy growth function, bioenergetics, and the consumption rates of fish
Canadian Journal of Fisheries and Aquatic Sciences, 2001
The von Bertalanffy growth function (VBGF) is based on a bioenergetic expression of fish growth; therefore, size-at-age data can theoretically be used to estimate fish consumption rates. We evaluated the accuracy of VBGFderived consumption rates by performing a meta-analysis and sensitivity analysis of VBGF assumptions, and we used Bayesian parameter estimation to quantify uncertainty in these estimates. The VBGF was robust to its assumption regarding the allometry of catabolism but was highly sensitive to the assumed allometry of consumption. Consequently, the commonly used form ("specialized" VBGF), which makes a strong assumption regarding the allometric slope of consumption, often grossly underestimates (>50%) consumption. The precision of the VBGF depended on characteristics of the size-at-age data used to parameterize the model. When data indicate decelerating growth, consumption rates were estimated with good precision; we estimated a 70% probability that bluefin tuna (Thunnus thynnus) consumption rates were between 1 and 2% body mass per day. Otherwise, consumption estimates were poorly defined; yellowfin tuna (Thunnus albacares) consumption rates between 2 and 7% per day were all equally likely. We conclude that VBGF can be a useful tool for estimating fish consumption rates, but potential biases and precision of these estimates should be evaluated on a case-by-case basis. Résumé : La fonction de croissance de von Bertalanffy (VBGF) se base sur une expression de la croissance des poissons et, en conséquence, les valeurs de taille à un âge donné peuvent en théorie être utilisées pour estimer les taux de consommation. Nous avons évalué la précision des taux de consommation tirés de la VBGF en procédant à une méta-analyse et à une analyse de sensibilité des présuppositions de la VBGF et en utilisant des estimations bayésiennes des paramètres pour quantifier l'incertitude de ces taux estimés. La VBGF s'est révélée robuste en ce qui a trait à sa présupposition de l'allométrie du catabolisme, mais s'est montrée très sensible à l'allométrie supposée de la consommation. La forme couramment utilisée (la VBGF « spécialisée »), qui fait une forte présupposition au sujet de la pente allométrique de la consommation, tend donc à sous-estimer fortement (>50 %) la consommation. La précision de la VBGF dépend des caractéristiques des valeurs de taille à un âge donné utilisées pour configurer les paramètres du modèle. Lorsque les données indiquent un ralentissement de la croissance, les taux de consommation sont estimés avec une précision acceptable; nous avons ainsi déterminé une probabilité de 70 % que les taux de consommation chez le Thon rouge se situent entre 1-2 % de la masse corporelle par jour. Dans les autres cas, les estimations de consommation sont mal définies et, chez l'Albacore à nageoires jaunes, des taux de consommation de 2-7 % de la masse corporelle par jour sont équiprobables. En conclusion, la VBGF peut être un outil commode pour estimer les taux de consommation chez les poissons, mais il est nécessaire d'évaluer les sources possibles d'erreur ainsi que la précision des estimations dans chacune des situations.
Parameter estimation in a growth model for a biological population
2013
The motivating problem concerns the estimation of the growth curve of solitary corals that follow the nonlinear Von Bertalanffy Growth Function (VBGF). The most common parameterization of the VBGF for corals is based on two parameters: the ultimate length L∞ and the growth rate k. One aim was to find a more reliable method for estimating these parameters, which can capture the influence of environmental covariates. The main issue with current methods is that they force the linearization of VBGF and neglect intra-individual variability. The idea was to use the hierarchical nonlinear model which has the appealing features of taking into account the influence of collection sites, possible intra-site measurement correlation and variance heterogeneity, and that can handle the influence of environmental factors and all the reliable information that might influence coral growth. This method was used on two databases of different solitary corals i.e. Balanophyllia europaea and Leptopsammia ...