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 araribá 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

Recommendation of soil fertility levels for willow in the southern highlands of Santa Catarina

The species Salix x rubens is being grown on the Southern Plateau of Santa Catarina since the 1940s, but so far the soil fertility requirements of the crop have not been assessed. This study is the first to evaluate the production profile of willow plantations in this region, based on the modified method of Summer & Farina (1986), for the recommendation of fertility levels for willow. By this method, based on the law of Minimum and of Maximum for willow production for the conditions on the Southern Plateau of Santa Catarina, the following ranges could be recommended: pH: 5.0-6.5; P: 12-89 mg dm-3; Mg: 3.2-7.5 mg; Zn: 5.0-8.3 mg dm-3; Cu: 0.8-4.6 mg dm-3; and Mn; 20-164 mg dm-3. The Ca/Mg ratio should be between 1.2 and 2.9. For K and Ca only the lower (sufficiency level), but not the upper threshold (excess) was established, with respectively 114 mg dm-3 and 5.3 cmol c dm-3. It was also possible to determine the upper threshold for Al and the Al/Ca ratio, i.e., 1.7 cmol c dm-3 and 0.28, respectively. For maximum yields, the clay in the soil surface layer should be below 320 g dm-3

Interactive Nonlinear Multiobjective Optimization Methods

An overview of interactive methods for solving nonlinear multiobjective optimization problems is given. In interactive methods, the decision maker progressively provides preference information so that the most satisfactory Pareto optimal solution can be found for her or his. The basic features of several methods are introduced and some theoretical results are provided. In addition, references to modifications and applications as well as to other methods are indicated. As the role of the decision maker is very important in interactive methods, methods presented are classified according to the type of preference information that the decision maker is assumed to provide.peerReviewe