Growth curves with time-dependent explanatory variables

A class of growth models incorporating time-dependent factors and stochastic perturbations are introduced. The proposed model includes the existing growth models used in fisheries as special cases. Particular attention is given to growth of a population (in average weight or length) from which observations are taken randomly each time and the analysis of tag-recapture data. Two real data sets are used for illustration: (a) to estimate the seasonal effect and population density effect on growth of farmed prawn (Penaeus monodon) from weight data and (b) to assess the effect of tagging on growth of barramundi (Lates calcarifer)

Growth curves with explanatory variables and estimation of the effect of tagging

The von Bertalanffy growth model is extended to incorporate explanatory variables. The generalized model includes the switched growth model and the seasonal growth model as special cases, and can also be used to assess the tagging effect on growth. Distribution-free and consistent estimating functions are constructed for estimation of growth parameters from tag-recapture data in which age at release is unknown. This generalizes the work of James (1991, Biometrics 47 1519-1530) who considered the classical model and allowed for individual variability in growth. A real dataset from barramundi (Lates calcarifer) is analysed to estimate the growth parameters and possible effect of tagging on growth.

Bayesian Analysis for Nonlinear Regression Model Under Skewed Errors, with Application in Growth Curves

We have considered a Bayesian approach for the nonlinear regression model by replacing the normal distribution on the error term by some skewed distributions, which account for both skewness and heavy tails or skewness alone. The type of data considered in this paper concerns repeated measurements taken in time on a set of individuals. Such multiple observations on the same individual generally produce serially correlated outcomes. Thus, additionally, our model does allow for a correlation between observations made from the same individual. We have illustrated the procedure using a data set to study the growth curves of a clinic measurement of a group of pregnant women from an obstetrics clinic in Santiago, Chile. Parameter estimation and prediction were carried out using appropriate posterior simulation schemes based in Markov Chain Monte Carlo methods. Besides the deviance information criterion (DIC) and the conditional predictive ordinate (CPO), we suggest the use of proper scoring rules based on the posterior predictive distribution for comparing models. For our data set, all these criteria chose the skew-t model as the best model for the errors. These DIC and CPO criteria are also validated, for the model proposed here, through a simulation study. As a conclusion of this study, the DIC criterion is not trustful for this kind of complex model.

Maturation and growth curves of Macrobrachium Carcinus (Linnaeus) (Crustacea, Decapoda, Palaemonidae) from Ribeira de Iguape River, southern Brazil

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Models for the analysis of growth curves for rearing tinamous (Rhynchotus rufescens) in captivity

Growth curves models provide a visual assessment of growth as a function of time, and prediction body weight at a specific age. This study aimed at estimating tinamous growth curve using different models, and at verifying their goodness of fit. A total number 11,639 weight records from 411 birds, being 6,671 from females and 3,095 from males, was analyzed. The highest estimates of a parameter were obtained using Brody (BD), von Bertalanffy (VB), Gompertz (GP,) and Logistic function (LG). Adult females were 5.7% heavier than males. The highest estimates of b parameter were obtained in the LG, GP, BID, and VB models. The estimated k parameter values in decreasing order were obtained in LG, GP, VB, and BID models. The correlation between the parameters a and k showed heavier birds are less precocious than the lighter. The estimates of intercept, linear regression coefficient, quadratic regression coefficient, and differences between quadratic coefficient of functions and estimated ties of quadratic-quadratic-quadratic segmented polynomials (QQQSP) were: 31.1732 +/- 2.41339; 3.07898 +/- 0.13287; 0.02689 +/- 0.00152; -0.05566 +/- 0.00193; 0.02349 +/- 0.00107, and 57 and 145 days, respectively. The estimated predicted mean error values (PME) of VB, GP, BID, LG, and QQQSP models were, respectively, 0.8353; 0.01715; -0.6939; -2.2453; and -0.7544%. The coefficient of determination (RI) and least square error values (MS) showed similar results. In conclusion, the VB and the QQQSP models adequately described tinamous growth. The best model to describe tinamous growth was the Gompertz model, because it presented the highest R-2 values, easiness of convergence, lower PME, and the easiness of parameter biological interpretation.

Evaluation of water beef buffalo from birth to two years using different growth curves

The buffalo is a domestic animal species of growing world-wide importance. Research to improve genetic improvement programs is important to maintain the productivity of buffalo. The objective this research was to evaluate the growth of Brazilian buffalo to two years of age with different growth curves. Growth curves consolidate the information contained in the weight-age data into three or four biologically meaningful parameters. The data included 31,452 weights at birth and 120, 205, 365, 550 and 730 days of buffalo (n = 5,178) raised on pasture without supplementation. Logistic, Gompertz, quadratic logarithmic, and linear hyperbolic curves (designated L, G, QL, and LH, respectively) were fitted to the data by using proc NUN of SAS (SAS Institute, Inc., Cary, NC, USA). The parameters estimates for L [WT= A * (((1 + exp (-k * AGE)))**-m)] were A = 865.1 +/- 5.42; k= 0.0028 +/- 0.00002; M= 3.808 +/- 0.007; R(2) = 0.95. For G [WT= A * exp (-b * exp (-k * age)] the parameters estimates were A= 967.6 +/- 7.23; k = 0.00217 +/- 0.000015; b = -2.8152 +/- 0.00532. For QL [WT= A + b*age + k*(age*age) + m*log (age)] parameters estimates were A= 37.41 +/- 0.48; k= 0.00019 +/- 6.4E(-6); b= 0.539 +/- 0.006; m= 2.32 +/- 0.23; R(2)=0.96. For LH [WT= A + b*AGE + k*(1/AGE)] the parameters estimates were A= 23.15 +/- 0.44; k=15.16 +/- 0.66; b= 0.707 +/- 0.001; R(2)= 0.96. Each of these curves fit these data equally well and could be used for characterizing growth to two years in beef buffalo.

Critical points in logistic growth curves and treatment comparisons

Several biological phenomena have a behavior over time mathematically characterized by a strong increasing function in the early stages of development, then by a less pronounced growth, sometimes showing stability. The separation between these phases is very important to the researcher, since the maintenance of a less productive phase results in uneconomical activity. In this report we present methods of determining critical points in logistic functions that separate the early stages of growth from the asymptotic phase, with the aim of establishing a stopping critical point in the growth and on this basis determine differences in treatments. The logistic growth model is fitted to experimental data of imbibition of arariba seeds (Centrolobium tomentosum). To determine stopping critical points the following methods were used: i) accelerating growth function, ii) tangent at the inflection point, iii) segmented regression; iv) modified segmented regression; v) non-significant difference; and vi) non-significant difference by simulation. The analysis of variance of the abscissas and ordinates of the breakpoints was performed with the objective of comparing treatments and methods used to determine the critical points. The methods of segmented regression and of the tangent at the inflection point lead to early stopping points, in comparison with other methods, with proportions ordinate/asymptote lower than 0.90. The non-significant difference method by simulation had higher values of abscissas for stopping point, with an average proportion ordinate/asymptote equal to 0.986. An intermediate proportion of 0.908 was observed for the acceleration function method.

Growth Curves and Genetic Parameters in Colombian Buffaloes (Bubalus bubalis Artiodactyla, Bovidae)

Non-linear mathematical functions proposed by Brody, Gompertz, Richards, Bertalanffy and Verhulst were compared in several buffalo production systems in Colombia. Herds were located in three provinces: Antioquia, Caldas, and Cordoba. Growth was better described by the curves proposed by Brody and Gompertz. Using the datasets from herds from Caldas, heritabilities for traits such as weaning weight (WW), weight and maturity at one year of age (WY and MY, respectively), age at 50% and 75% of maturity (A50% and A75%, respectively), adult weight (beta(0)), and other characteristics, were also estimated. Direct and maternal heritabilities for WW were 0.19 and 0.12, respectively. Direct heritabilities for WY, MY, A50%, A75% and beta(0) were 0.39, 0.15, 0.09, 0.20 and 0.09, respectively. The genetic correlation for beta(0) and WY was -0.47, indicating that selection for heavy weight at one year of age will lead to lower weight at adult age. These data suggest that selection based on maturity traits can generate changes in characteristics of economic importance in beef-type buffalo farms.

Growth curves for Macrobrachium rosenbergii in semi-intensive culture in Brazil

Four 0.02-ha earthen ponds at the UNESP Aquaculture Center, Jaboticabal, São Paulo, Brazil, were stocked with newly metamorphosed Macrobrachium rosenbergii post-larvae at 1.5 animals/m2. After 8 mo, prawn density at harvest ranged from 0.3/ m2 to 0.8/m2. Growth curves were determined for each population using von Bertalanffy growth functions. Asymptotic maximum length and asymptotic maximum weight increased as final population size decreased indicating that a strong density effect on prawn growth occurs in semi-intensive culture, even when populational density varies within a small range of less than 1 animal/m2.

Growth curves and genetic parameters in colombian buffaloes (Bubalus bubalis Artiodactyla, Bovidae): Curvas de crecimiento y parámetros genéticos en búfalos (Bubalus bubalis Artiodactyla, Bovidae) en Colombia

Non-linear mathematical functions proposed by Brody, Gompertz, Richards, Bertalanffy and Verhulst were compared in several buffalo production systems in Colombia. Herds were located in three provinces: Antioquia, Caldas, and Cordoba. Growth was better described by the curves proposed by Brody and Gompertz. Using the datasets from herds from Caldas, heritabilities for traits such as weaning weight (WW), weight and maturity at one year of age (WY and MY, respectively), age at 50% and 75% of maturity (A50% and A75%, respectively), adult weight (β0), and other characteristics, were also estimated. Direct and maternal heritabilities for WW were 0.19 and 0.12, respectively. Direct heritabilities for WY, MY, A50%, A75% and β0 were 0.39, 0.15, 0.09, 0.20 and 0.09, respectively. The genetic correlation for β0 and WY was -0.47, indicating that selection for heavy weight at one year of age will lead to lower weight at adult age. These data suggest that selection based on maturity traits can generate changes in characteristics of economic importance in beef-type buffalo farms. © 2012 Universidad de Antioquia.

Growth curves for ostriches (Struthio camelus) in a Brazilian population

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Growth curves of Nile tilapia (Oreochromis niloticus) strains cultivated at different temperatures

Growth of Red, GIFT and Supreme Nile tilapia strains were evaluated. Fish were cultivated in indoor recirculation systems in 0.5 m³ tanks with controlled temperatures of 22, 28 and 30°C. Random samples of 20 fish from each strain (10 fish tank-1) were weighed at day 7, 30, 60, 90 and 120. Exponential model (y=AeKx) and Gompertz model (y = Aexp(-Be-Kx)) were fitted and the estimates parameters were obtained by Weighted Least Squares. At 22°C, Red, GIFT and Supreme strain presented similar growth and fit of exponential model. GIFT and Supreme strain presented higher growth rate at 30°C of cultivation when compared to Red strain. Temperature influences weight and age at the inflection point. The temperature of cultivation influences the growth description of Red, GIFT and Supreme tilapia strains. It changes the age and weight at inflection point and the qualities of growth model fits, changing the variation of the batch.