Nonautonomous difference equations with applications

This work is divided in two parts. In the first part we develop the theory of discrete nonautonomous dynamical systems. In particular, we investigate skew-product dynamical system, periodicity, stability, center manifold, and bifurcation. In the second part we present some concrete models that are used in ecology/biology and economics. In addition to developing the mathematical theory of these models, we use simulations to construct graphs that illustrate and describe the dynamics of the models. One of the main contributions of this dissertation is the study of the stability of some concrete nonlinear maps using the center manifold theory. Moreover, the second contribution is the study of bifurcation, and in particular the construction of bifurcation diagrams in the parameter space of the autonomous Ricker competition model. Since the dynamics of the Ricker competition model is similar to the logistic competition model, we believe that there exists a certain class of two-dimensional maps with which we can generalize our results. Finally, using the Brouwer’s fixed point theorem and the construction of a compact invariant and convex subset of the space, we present a proof of the existence of a positive periodic solution of the nonautonomous Ricker competition model.

Simultaneous analysis of free amino acids and biogenic amines in honey and wine samples using in loop orthophthalaldeyde derivatization procedure

This work presents a RP-HPLC method for the simultaneous quantification of free amino acids and biogenic amines in liquid food matrices and the results of the application to honey and wine samples obtained from different production processes and geographic origins. The developed methodology is based on a pre-column derivatization with o-phthaldialdehyde carried out in the sample injection loop. The compounds were separated in a Nova-Pack RP-C18 column (150 mm × 3.9 mm, 4 μm) at 35 °C. The mobile phase used was a mixture of phase A: 10 mM sodium phosphate buffer (pH 7.3), methanol and tetrahydrofuran (91:8:1); and phase B: methanol and phosphate buffer (80:20), with a flow rate of 1.0 ml/min. Fluorescence detection was used at an excitation wavelength of 335 nm and an emission wavelength of 440 nm. The separation and quantification of 19 amino acids and 6 amines was carried out in a single run as their OPA/MCE derivatives elute within 80 min, ensuring a reproducible quantification. The method showed to be adequate for the purpose, with an average RSD of 2% for the different amino acids; detection limits varying between 0.71 mg/l (Asn) and 8.26 mg/l (Lys) and recovery rates between 63.0% (Cad) and 98.0% (Asp). The amino acids present at the highest concentration in honey and wine samples were phenylalanine and arginine, respectively. Only residual levels of biogenic amines were detected in the analysed samples.