scholarly journals Optimal exponent-pairs for the Bertalanffy-Pütter growth model

2018 ◽  
Author(s):  
Katharina Renner-Martin ◽  
Norbert Brunner ◽  
Manfred Kühleitner ◽  
Werner-Georg Nowak ◽  
Klaus Scheicher
Keyword(s):  
Differential Equation ◽  
Growth Model ◽  
Explanatory Power ◽  
Information Criterion ◽  
Optimal Growth ◽  
Growth Function ◽  
Data Fitting ◽  
Model Parameters ◽  
Data Set ◽  
Von Bertalanffy Growth Function

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<b of non-negative exponents, the non-negative constants p and q, and a positive initial value m0 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, m0 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.

scholarly journals Optimal exponent-pairs for the Bertalanffy-Pütter growth model

2018 ◽  
Author(s):  
Katharina Renner-Martin ◽  
Norbert Brunner ◽  
Manfred Kühleitner ◽  
Werner-Georg Nowak ◽  
Klaus Scheicher
Keyword(s):  
Differential Equation ◽  
Growth Model ◽  
Explanatory Power ◽  
Information Criterion ◽  
Optimal Growth ◽  
Growth Function ◽  
Data Fitting ◽  
Model Parameters ◽  
Data Set ◽  
Von Bertalanffy Growth Function

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<b of non-negative exponents, the non-negative constants p and q, and a positive initial value m0 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, m0 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.

scholarly journals Optimal and near-optimal exponent-pairs for the Bertalanffy-Pütter growth model

PeerJ ◽  
2018 ◽  
Vol 6 ◽  
pp. e5973 ◽  
Author(s):  
Katharina Renner-Martin ◽  
Norbert Brunner ◽  
Manfred Kühleitner ◽  
Werner-Georg Nowak ◽  
Klaus Scheicher
Keyword(s):  
Differential Equation ◽  
Growth Model ◽  
Gompertz Model ◽  
Optimal Growth ◽  
Growth Function ◽  
Bias Reduction ◽  
Model Complexity ◽  
Model Parameters ◽  
Von Bertalanffy ◽  
Data Set

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 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.

Age, growth and reproduction of the protandrous hermaphrodite fish, Sarpa salpa, from the Portuguese continental coast

2016 ◽  
Vol 98 (2) ◽  
pp. 269-281 ◽  
Author(s):  
Rafaela Barros Paiva ◽  
Ana Neves ◽  
Vera Sequeira ◽  
Ana Rita Vieira ◽  
Maria José Costa ◽  
...  
Keyword(s):  
Information Criterion ◽  
Growth Function ◽  
Population Parameters ◽  
Von Bertalanffy Growth Function ◽  
Protandrous Hermaphrodite ◽  
Sarpa Salpa ◽  
Back Calculation ◽  
First Maturity ◽  
Available Information ◽  
Growth And Reproduction

Salema, Sarpa salpa is a commercial exploited species in the Atlantic Ocean with little available information for the essential population parameters, such as age, growth and reproduction. The present study aims to describe these parameters for S. salpa obtained off the coast of Portugal. Ages were estimated from the whole otolith readings; the minimum and the maximum ages observed were 0 and 14 years, respectively, corresponding to 5.2 and 41.4 cm of total length (TL). Whole otolith readings and back-calculation approaches were used to estimate the parameters of the von Bertalanffy growth function and the Akaike's information criterion value suggested that the second approach was the best one to describe the growth of salema: L∞ = 45.07 cm, k = 0.14 year−1 and t0 = −1.43 year. The species is a protandric hermaphrodite and the sex change process occurred between 28.6 and 40.9 cm TL. A short spawning season was identified, extending from September to November. The estimated length at first maturity for males was 24.5 cm TL, corresponding to an age of 2 years at first maturity. This species exhibited a determinate fecundity type and the relative annual fecundity varied between 462 and 2662 oocytes per gram of gutted weight.

scholarly journals Age estimation and the best growth model selection of the tentacled blenny Parablennius tentacularis (Brünnich, 1768) in the southeastern Black Sea

2021 ◽  
Vol 38 (2) ◽  
pp. 229-236
Author(s):  
Ayşe Van ◽  
Aysun Gümüş ◽  
Melek Özpiçak ◽  
Serdar Süer
Keyword(s):  
Growth Model ◽  
Black Sea ◽  
Growth Parameters ◽  
Growth Models ◽  
Information Criterion ◽  
Current Data ◽  
Logistic Growth ◽  
Data Set ◽  
Frequency Distributions ◽  
Selection Of

By the study's coverage, 522 individuals of tentacled blenny (Parablennius tentacularis (Brünnich, 1768)), were caught with the bottom trawl operations (commercial fisheries and scientific field surveys) between May 2010 and March 2012 from the southeastern Black Sea. The size distribution range of the sample varied between 4.8-10.8 cm. The difference between sex length (K-S test, Z=3.729, P=0.000) and weight frequency distributions (K-S test, Z=3.605, P=0.000) was found to be statistically significant. The length-weight relationship models were defined as isometric with W = 0.009L3.034 in male individuals and positive allometric with W = 0.006L3.226 in female individuals. Otolith and vertebra samples were compared for the selection of the most accurate hard structure that can be used to determine the age. Otolith was chosen as the most suitable hard structure. The current data set was used to predict the best growth model. For this purpose, the growth parameters were estimated with the widely used von Bertalanffy, Gompertz and Logistic growth functions. Akaike's Information Criterion (AIC), Lmak./L∞ ratio, and R2 criteria were used to select the most accurate growth models established through these functions. Model averaged parameters were calculated with multi-model inference (MMI): L'∞ = 15.091 cm, S.E. (L'∞) = 3.966, K'= 0.232 year-1, S.E. (K') = 0.122.

Age and growth estimates of the Kwangtung skate Dipturus kwangtungensis in the waters of northern Taiwan

2015 ◽  
Vol 96 (7) ◽  
pp. 1395-1402 ◽  
Author(s):  
Shoou-Jeng Joung ◽  
Chien-Chi Chen ◽  
Kwang-Ming Liu ◽  
Tzu-Chi Hsieh
Keyword(s):  
Growth Parameters ◽  
Information Criterion ◽  
Growth Function ◽  
Age And Growth ◽  
Gompertz Function ◽  
Growth Band ◽  
Von Bertalanffy Growth Function ◽  
North Eastern ◽  
Robertson Function ◽  
Northern Taiwan

The age and growth of Kwangtung skate, Dipturus kwangtungensis, in the waters off northern Taiwan were estimated from 422 specimens collected between July 2006 and July 2008 at the Tashi fishing market in north-eastern Taiwan. The sexes-combined relationship between total length (TL) and centrum diameter (D) was estimated as follows: TL = 14.11D0.888 (N = 411, r2 = 0.94, P < 0.001). Growth band pairs (comprised of translucent and opaque bands) in vertebrae were determined to form once annually, based on the centrum edge analysis. Up to 14 band pairs were found for both sexes. The von Bertalanffy growth function (VBGF), two-parameter VBGF, the Robertson function, and the Gompertz function were used to fit the observed length-at-age data. The Akaike information criterion corrected indicated that the Gompertz function best fit the observed length at age data. Sex-specific growth functions were not significantly different; the sexes-combined growth parameters were estimated as follows: asymptotic length (L∞) = 96.7 cm TL, growth coefficient (kG) = 0.144 year−1 and constant (t0) = 5.45 year (N = 364, P < 0.01).

scholarly journals Model Pertumbuhan Dan Status Sumberdaya Panulirus homarus Di Cilacap, Jawa Tengah

2019 ◽  
Vol 8 (1) ◽  
pp. 85-93
Author(s):  
Maulvi Didit Baskoro ◽  
Edi Wibowo Kushartono ◽  
Irwani Irwani
Keyword(s):  
Growth Model ◽  
Growth Parameters ◽  
Spiny Lobster ◽  
Growth Function ◽  
Fishing Effort ◽  
Maximum Sustainable Yield ◽  
Sustainable Yield ◽  
Von Bertalanffy ◽  
Von Bertalanffy Growth Function ◽  
The Status

Lobster pasir (Panulirus homarus) merupakan salah satu jenis lobster yang banyak di temukan di perairan Indonesia. Penelitian ini dilaksanakan pada bulan November 2017 sampai bulan Februari 2018 dengan lokasi pengambilan sampel di Pelabuhan Perikanan Samudra Cilacap (PPSC), di Perairan Cilacap Penelitian ini melihat bagaimana model pertumbuhan, dan status sumberdaya, mulai dari Von Bertalanffy, CPUE dan MSY. Estimasi parameter pertumbuhan lobster pasir (Panulirus homarus) dihitung dengan aplikasi Fisat II. Didapatkan hasil model pertumbuhan lobster pasir ( L∞ = 93.66 cm , K = 0.780 dan t0 -1.0950 mm). Hasil CPUE didapatkan persamaan, nilai linier sebesar  y = -0.0002 + 0.0965 x, nilai R2= 0,09443 untuk data lima tahun ke belakang terhitung dari 2012 sampai 2016 dalam upaya penangkapan (Panulirus sp.). Analisa Maximum sustainable yield (MSY) di lakukan untuk mengetahui besarnya potensi lestari Panulirus sp. dengan registrasi linier y-0.0002x + 0.965. Pendugaan MSY dan upaya penangkapan Foptimum diperoleh dengan 2412 trip dan nilai MSY 1164.031. Produksi Panulirus sp. di tahun 2016 – 2017 di Perairan Cilacap mengalami kenaikan, pada tahun 2016 terjadi kenaikan di bulan Februari, Maret, April dan Desember. Sedangakan pada tahun 2017 terjadi kenaikan di bulan Maret dan Oktober. Kenaikan ini dikarenakan musim lobster berada di bulan Oktober hingga Februari. Perubahan iklim dan penangkapan yang melebihi batas akan berpengaruh terhadap ukuran dan stok Panulirus sp. di alam. The Spiny lobster (Panulirus homarus) is one species of lobster that is widely found in Indonesian. This research was conducted on November 2017 until February 2018 the sample locations at the Cilacap (PPSC), which observed  the growth of models, and the status of resources, regretion Von Bertalanffy Growth function CPUE and MSY. The estimated growth parameters of sand lobster (Panulirus homarus) were  calculated using Fisat II application. The results of the spiny lobster  growth  model were obtained (L∞ = 93.66 cm, K = 0.780 and 0 -1.0950 mm). The CPUE results were obtained equations, linear values of y = -0.0002 + 0.0965 x for five years data from 2012 to 2016 . The Maximum Sustainable Yield (MSY) analysis was carried out to determine the magnitude of the sustainable potential of Panulirus sp. within linear  y-0.0002x + 0.965. The MSY values 1164,031 estimation and (Foptimum) fishing effort were 2.412 trips and MSY values 1164,031. The  production of Panulirus sp. start from  2016 to 2017 in the Cilacap has increased, the fact in 2016 there was an increase on February, March, April and December, while in 2017 there was an increase on March and October. So that the increase due to the lobster season being in October to February.  In addition, climate alteration and capture exceed the effect of  size and stock of Panulirus sp.

scholarly journals On the exponent in the von Bertalanffy growth model

2017 ◽  
Author(s):  
Katharina Renner-Martin ◽  
Norbert Brunner ◽  
Manfred Kühleitner ◽  
Georg Nowak ◽  
Klaus Scheicher
Keyword(s):  
Information Criterion ◽  
Growth Function ◽  
Data Sets ◽  
Scaling Exponent ◽  
Von Bertalanffy ◽  
Metabolic Scaling ◽  
Von Bertalanffy Growth Function ◽  
Definition Of ◽  
Growth Of Animals ◽  
Best Fit

Bertalanffy proposed the differential equation m´(t) = p × m (t) a –q × m (t) for the description of the mass growth of animals as a function m(t) of time t. He suggested that the solution using the metabolic scaling exponent a = 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. Using the Akaike information criterion it proposes a testable definition of ‘weak universality’ for a taxonomic group of species. (It roughly means that a model has an acceptable fit to most data sets of that group.) This definition was applied to 60 data sets from literature (37 about fish and 23 about non-fish species) and for each dataset an optimal metabolic scaling exponent 0 ≤ a opt < 1 was identified, where the model function m(t) achieved the best fit to the data. Although in general this optimal exponent differed widely from a = 2/3 of the VBGF, the VBGF was weakly universal for fish, but not for non-fish. This observation supported the conjecture that the pattern of growth for fish may be distinct. The paper discusses this conjecture.

A Comparison of von Bertalanffy and Polynomial Functions in Modelling Fish Growth Data

1992 ◽  
Vol 49 (6) ◽  
pp. 1228-1235 ◽  
Author(s):  
Y. Chen ◽  
D. A. Jackson ◽  
H. H. Harvey
Keyword(s):  
Goodness Of Fit ◽  
Growth Function ◽  
Fish Growth ◽  
Analysis Of Covariance ◽  
Polynomial Functions ◽  
Von Bertalanffy ◽  
Growth Data ◽  
Data Set ◽  
Growth Functions ◽  
Von Bertalanffy Growth Function

We compared the von Bertalanffy growth function (VBGF) and five polynomial functions (PF) in modelling fish growth for 16 populations comprising six species of freshwater fishes. Ranked results of the variance explained by each growth function indicated that VBGF described growth data better than three- and four-parameter polynomial functions. Log-transforming length and age greatly improved the goodness-of-fit of the three-parameter polynomial function. Statistical comparison of growth between populations or sexes was done using a general linear model for polynomial functions. An analysis of residual sum of squares was proposed to compare the resultant VBGFs because the nonlinear formulation of the VBGF prevented traditional analysis of covariance procedures. Fitting of different growth functions to the same growth data set yielded the same result in the intra-species growth comparisons for three species (eight populations) but different results for two species (seven populations). Where ages of the fish were less than the maximum age in the samples, dL/dt were similar for all growth functions except the parabola based on the log-transformation of length alone. The VBGF proved to be the best growth model for all 16 populations.

Age and growth of the threatened endemic skate Rioraja agassizii (Chondrichthyes, Arhynchobatidae) in the western South Atlantic

2019 ◽  
Vol 70 (1) ◽  
pp. 84 ◽  
Author(s):  
F. P. Caltabellotta ◽  
F. M. Silva ◽  
F. S. Motta ◽  
O. B. F. Gadig
Keyword(s):  
Growth Parameters ◽  
Management Strategies ◽  
Information Criterion ◽  
Growth Function ◽  
South Atlantic ◽  
Age And Growth ◽  
Biological Information ◽  
Growth Coefficient ◽  
Von Bertalanffy Growth Function ◽  
Best Fit

The Rio skate Rioraja agassizii is a threatened endemic skate species frequently caught as bycatch in the western South Atlantic. However, there is no biological information about its age and growth parameters, which would be necessary to provide science-based information for the development of management strategies for this species. The aim of the present study was to provide information about the age and growth parameters of R. agassizii. In all, 138 vertebrae from individuals ranging in size from 9.0 to 53.2-cm total length (TL) were analysed. The edge analysis indicated a trend for annual band deposition in the vertebrae. Maximum ages estimated for males and females were 6 and 10 years respectively. Akaike’s information criterion indicated that the modified two-parameter form of the von Bertalanffy growth function (using length at birth L0=9.0cm TL) provided the best fit, with derived parameters of theoretical maximum length L∞=49.6cm TL and growth coefficient k=0.27 for males and L∞=59.0cm TL and k=0.22 for females. Our results are important to understanding the resilience of this skate species to harvest, which can contribute to the development of fisheries management strategies and conservation programs.

scholarly journals On the exponent in the von Bertalanffy growth model

2017 ◽  
Author(s):  
Katharina Renner-Martin ◽  
Norbert Brunner ◽  
Manfred Kühleitner ◽  
Georg Nowak ◽  
Klaus Scheicher
Keyword(s):  
Information Criterion ◽  
Growth Function ◽  
Data Sets ◽  
Scaling Exponent ◽  
Von Bertalanffy ◽  
Metabolic Scaling ◽  
Von Bertalanffy Growth Function ◽  
Definition Of ◽  
Growth Of Animals ◽  
Best Fit

Bertalanffy proposed the differential equation m´(t) = p × m (t) a –q × m (t) for the description of the mass growth of animals as a function m(t) of time t. He suggested that the solution using the metabolic scaling exponent a = 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. Using the Akaike information criterion it proposes a testable definition of ‘weak universality’ for a taxonomic group of species. (It roughly means that a model has an acceptable fit to most data sets of that group.) This definition was applied to 60 data sets from literature (37 about fish and 23 about non-fish species) and for each dataset an optimal metabolic scaling exponent 0 ≤ a opt < 1 was identified, where the model function m(t) achieved the best fit to the data. Although in general this optimal exponent differed widely from a = 2/3 of the VBGF, the VBGF was weakly universal for fish, but not for non-fish. This observation supported the conjecture that the pattern of growth for fish may be distinct. The paper discusses this conjecture.

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