World Aquaculture 13 ventricosa) (J. Agardh, 1887) in aquaria habitats. Journal of the Marine Biological Association. Penha-Lopes, G., A. L. Rhyne, J. Lin and L. Narciso. 2005. The larval rearing of the marine ornamental crab, Mithraculus forceps (A. Milne Edwards, 1875) (Decapoda: Brachyura: Majidae). Aquaculture Research 36: 1313-1321. Penha-Lopes, G., A. L. Rhyne, J. Lin and L. Narciso. 2006. Effects of temperature, stocking density and diet on the growth and survival of juvenile Mithraculus forceps (A. Milne Edwards, 1875) (Decapoda: Brachyura: Majidae). Aquaculture Research 37: 398-408. Penha-Lopes, G., J. Figueiredo and L. Narciso. 2007. Modelling survival and growth of Mithraculus forceps’ larvae and juveniles (A. Milne Edwards, 1875) (Decapoda: Brachyura: Majidae) in aquaculture. Aquaculture 264: 285–296. Pinheiro, J. C. and D. M. Bates. 2000. Mixed-Effects Models in S and S-PLUS —Statistics and Computing. Springer-Verlag New York. Rhyne, A. L., G., Penha-Lopes and J. Lin. 2005. Growth, development, and survival of larval Mithraculus sculptus (Lamark) and Mithraculus forceps (A. Milne Edwards) (Decapoda: Brachyura: Majidae): economically important marine ornamental crabs. Aquaculture 245: 183-191. Sidebar Response curve y (x)= a + bx +cx2 + dx3 (Equation 1) Response curve is a linear (polynomial) model (Equation 1, where x is the variable, like stocking density or prey density) that allows us to predict survival or growth for the range of a variable, including values that were not tested, allowing us to find the optimum value for a certain variable. Response curves can only be applied to quantitative variables. To identify the polynomial effect: cubic, quadratic or linear, that adjusts better, we used the orthogonal polynomials method. These models were fit to the observed data with computer regression models in Statistica 7.0. The optimum value is estimated as the x value that produces the higher y (Figures 2, 3 and 4). Linear and Non-Linear Models The linear (Equation 4) and non-linear models, as in asymptotic and logistic models, (Equations 2 and 3), that allow us to predict results through time only for the tested levels of a variable (treatments), can be applied to both quantitative and qualitative variables and allow us to statistically test if parameters are significantly different between the different levels of the variable. Levels of a variable are the tested values of that variable, for instance 10, 20, 40 and 80 larvae/L are levels of the variable stocking density. The data are fit to the data using libraries “lme” and “nlme” developed by Pinheiro and Bates (2000) in software R. The program begins by estimating each parameter of the model (by maximum likelihood) for each one of the levels of the variable. The effect of the variable on each one of the parameters is tested using analysis of variance, incorporated in the development of the model. If a certain parameter is not significantly different between the different levels of the variable, the model will use the same value for that parameter for all levels of the variable, but if the parameter is significantly different between the different levels of the variable, the models will use a different value for the parameter for each level of the variable (Pinheiro and Bates 2000 for further details). Asymptotic model (Equation 2), The asymptotic model (Equation 2, where x is time) has three parameters: Φ1 is the asymptote as xg∞and represents the final survival to juvenile ( percent); Φ2 is the logarithm of the rate constant, corresponding to a half-life of t0.5 = log2/exp( Φ2) which gives an idea on the synchrony of metamorphosis (greater values indicate greater synchrony); and Φ3 is the value of x at which y = 0 indicating the time just before the first larvae is expected to metamorphose to juvenile or minimum larval duration (Figures 2 and 3). Logistic model (Equation 3) The logistic model (Equation 3, where x is Time) has four parameters: Φ1 is the horizontal asymptote as xg∞ and represents the initial percent survival; Φ2 is the horizontal asymptote as xg +∞ and represents the theoretical maximum survival for the treatment (level of the variable); Φ3 is the x value at the curve inflection point which response is midway between the asymptotes and gives us an idea of the period of greater mortality during juvenile culture; Φ4 is a scale parameter distance on the x-axis and gives us an idea of the mortality synchronism through development, a lower Φ4 indicates higher mortality synchronism (Figure 4). Linear (polynomial) Models y (x)= a + bx +cx2 + dx3 (Equation 4) The linear model (Equation 4, where x is Time) has four parameters: a, b, c and d. Unlike in the non-linear models, these parameters have no biological meaning (Figures 4 and 5).
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