Modeling a biological systems, is a cyclic process which involves constructing a model from current theory and beliefs and then validating that model against the data. If the data does not match, qualitatively or quantitatively then there may be a problem with either our beliefs or the current theory. At the same time directly finding a model from the existing data would make generalizing results difficult. A considerable difficultly in this process is how to specify the model in the first place. There is a need to be practice which accounts for the growing use of mathematical and statistical methods. However, as a systems becomes more complex, standard mathematical approaches may not be sufficient. In the field of ecology, the standard techniques involve discrete maps, and continuous models such as ODE's. The intent of this work is to present the mathematics necessary to study hybrids of these two models, then consider two case studies. In first case we con sider a coral reef with continuous change, except in the presence of hurricanes. The results of the data are compared quantitatively and qualitatively with simulation results. For the second case we consider a model for rabies with a periodic birth pulse. Here the analysis is qualitative as we demonstrate the existence of a strange attractor by looking at the intersections of the stable and unstable manifold for the saddle point generating the attractor. For both cases studies the introduction of a discrete event into a continuous system is done via a Dirac Distribution or Measure. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2015. / FAU Electronic Theses and Dissertations Collection

Cette thèse porte d'une part sur l'étude de la stabilité du biexciton dans le nanotube de carbone et, d'autre part, sur le contrôle de l'émission du nanotube par le couplage des nanotubes de carbone à des antennes plasmoniques. La technique de spectroscopie optique non-linéaire de saturation d'absorption, appliquée aux nanotubes de carbone, nous a permis d'effectuer la première observation du biexciton dans cette nanostructure. Plus précisément, deux raies d'absorptions induites sont observées et attribuées au trion et à l'exciton par des études en températures et en puissance du laser de pompe. La mise en oeuvre d'une configuration à trois faisceaux basée sur un schéma en double-pompe permet de confirmer la photo-génération du biexciton en tant qu'excitation élémentaire du nanotube de carbone. Le biexciton est observé avec une énergie de liaison de 107 ± 1 meV pour la chiralité (9,7) et présente un profil asymétrique de Fano. Une première estimation de la dynamique de recombinaisondu biexciton est donnée par description quantitative du processus de Fano. Le modèle est basé sur le formalisme de la susceptibilité non-linéaire du troisième ordre χ(3) incluant le couplage coulombien entre le biexciton et le continuum des paires électron-trou de la première singularité de Van Hove. Le facteur de Fano est évaluée à q = 5 et conduit à l'estimation du taux de recombinaison Auger du biexciton B ∈ [0.1; 1] μm·ps−1 . Le rendement radiatif du biexciton est ainsi estimé à 10−6 .Dans le but d'étudier les nanotubes en cavité plasmonique, une expérience de micro-photoluminescence et une expérience de spectroscopie en champ sombre, sont développés dans le domaine spectrale des télécommunications (1.3 μm et 1.55 μm). La caractérisation de divers types échantillons de nanotube et des antennes plasmoniques sont présentés. Des résultats préliminaires sur un échantillon de nanotubes associés dans une configuration patch à des antennes plasmoniques montrent une corrélation entre la position des antennes et les zones luminescentes, ainsi qu'un changement de l'allure des spectres de photoluminescence. Ces premiers résultats constituent une transition dans la démarche de notre projet. L'étape de calibration des expériences est en phase de finition et l'étude des propriétés physique des nanotubes en cavité plasmonique représente désormais une activité opérationnelle au sein de notre équipe. / This thesis focus on both the biexciton's stability in carbon nanotubes, and the control of the nanotube emission through its coupling to plasmonic antenna.We report the first observation of biexciton in carbon nanotubes by means of spectral holeburning nonlinear optical spectroscopy. More precisely, two induced absorption lines are detected and assigned to trion and biexciton after investigation of their temperature and pump power dependences. An additional proof of the detection of the biexciton, as an elementary excitation of carbonnanotubes, is given in a three-beam configuration based on a two-pump scheme. The biexciton of the (9,7) chirality is observed with a binding energy of 107 ± 1 meV and shows an asymetric Fano lineshape. A first estimation of the biexciton's recombinaison dynamics is given by the quantitative analysis of the nonlinear signal. Our analytic model is formulated in the framework of the chi(3) nonlinear response, including coulomb interaction between biexcitons and free electron-hole pairs lying in the first Van Hove singularity. A Fano factor of about q = 5 is determined, which drives us to the estimation of biexciton's Auger recombinaison rate B ∈ [0.1; 1] μm · ps−1 . The Biexciton's radiative yield is then estimated of the order of 10−6 .In order to study nanotubes in plasmonic cavities, we developed micro-photoluminescence and dark-field spectroscopy experiments in the optical fiber telecommunication wavelengths (1.3 μm and 1.55 μm). Caracterisation of nanotube samples and plasmonic antenna are presented. Preliminary results on nanotubes inserted in a patch antenna have shown correlation between antenna's position and the spatial distribution of luminescence. Moreover, a change in the carbon nanotube's photoluminescence profile is observed. These results appear to be a turning point in our work. The calibration of our experiment is at its end and studies of optical properties of carbon nanotubes coupled to plasmonic antenna are now on stream in our team.

<p>We consider existence and stability of breather solutions to discrete nonlinear Schrodinger (dNLS) and discrete Klein-Gordon (dKG) equations near the anti-continuum limit, the limit of the zero coupling constant. For sufficiently small coupling, discrete breathers can be uniquely extended from the anti-continuum limit where they consist of periodic oscillations on excited sites separated by "holes" (sites at rest).</p> <p>In the anti-continuum limit, the dNLS equation linearized about its discrete breather has a spectrum consisting of the zero eigenvalue of finite multiplicity and purely imaginary eigenvalues of infinite multiplicities. Splitting of the zero eigenvalue into stable and unstable eigenvalues near the anti-continuum limit was examined in the literature earlier. The eigenvalues of infinite multiplicity split into bands of continuous spectrum, which, as observed in numerical experiments, may in turn produce internal modes, additional eigenvalues on the imaginary axis. Using resolvent analysis and perturbation methods, we prove that no internal modes bifurcate from the continuous spectrum of the dNLS equation with small coupling.</p> <p>Linear stability of small-amplitude discrete breathers in the weakly-coupled KG lattice was considered in a number of papers. Most of these papers, however, do not consider stability of discrete breathers which have "holes" in the anti-continuum limit. We use perturbation methods for Floquet multipliers and analysis of tail-to-tail interactions between excited sites to develop a general criterion on linear stability of multi-site breathers in the KG lattice near the anti-continuum limit. Our criterion is not restricted to small-amplitude oscillations and it allows discrete breathers to have "holes" in the anti-continuum limit.</p> / Doctor of Philosophy (PhD)