Thèse numérisée par la Division de la gestion de documents et des archives de l'Université de Montréal.

The first part is a study of an ecological model with one herbivore and $N$ plants. The system has a new type of functional response due to the speculation that the plants compete with each other and have different levels of toxin which inhibit the herbivore's ability to eat up to a certain amount. We first derive the model mathematically and then investigate, both analytically and numerically, the possible dynamics for this model, including the bifurcation and chaos. We also discuss the conditions under which all the species can coexist. The second part is a study in the normal form theory. In particular, we study the relations between the normal forms and the first integrals in analytic vector fields. We are able to generalize one of Poincare's classical results on the nonexistence of first integrals in an autonomous system. Then in the space of 2n-dimensional analytic autonomous systems with exactly n resonances and n functionally independent first integrals, we obtain some results related to the convergence and generic divergence of the normalizations. Lastly we give a new proof of the necessary and sufficient conditions for a planar Hamiltonian system to have an isochronous center.

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This report explores several applications of iteration and the various connections that can be made to different areas of mathematics. The ties iteration has to the Wada Property, bifurcation diagram, root finding, and applications in geometry are all investigated. Finally, a rationale for incorporating iteration into secondary mathematics courses to support a more robust curriculum is discussed. / text

We analyze the double Hopf bifurcations which occur in two geophysical fluid dynamics models: (1) a two-layer quasigeostrophic potential vorticity model with forcing and (2) a mathematical model of the differentially heated rotating annulus experiment. The bifurcations occur at the transition between axisymmetric steady solutions and non-axisymmetric travelling waves. For both models, the results indicate that, close to the transition, there are regions in parameter space where there are multiple stable waves. Hysteresis of these waves is predicted. For each model, center manifold reduction and normal form theory are used to deduce the local behaviour of the full system of partial differential equations from a low-dimensional system of ordinary differential equations. In each case, it is not possible to compute the relevant eigenvalues and eigenfunctions analytically. Therefore, the linear part of the equations is discretized and the eigenvalues and eigenfunctions are approximated from the resulting matrix eigenvalue problem. However, the projection onto the center manifold and reduction to normal form can be done analytically. Thus, a combination of analytical and numerical methods are used to obtain numerical approximations of the normal form coefficients, from which the dynamics are deduced. The first model differs from those previously studied with bifurcation analysis since it supports a steady solution which varies nonlinearly with latitude. The results indicate that the forcing does not qualitatively change the behaviour. However, the form of the bifurcating solution is affected. The second model uses the Navier-Stokes equations in the Boussinesq approximation, in cylindrical geometry. In addition to the double Hopf bifurcation analysis, a detailed axisymmetric to non-axisymmetric transition curve is produced from the computed eigenvalues. A quantitative comparison with experimental data finds that the computed transition curve, critical wave numbers and drift rates of the bifurcating waves are reasonably accurate. This indicates that the analysis, as well as the approximations which are made, are valid.

Thèse numérisée par la Division de la gestion de documents et des archives de l'Université de Montréal

The thesis is based on the use of mathematical theories and techniques to gain qualitative and quantitative insight into the transmission dynamics of hepatitis C virus (HCV) in an IDU (injecting drug user) population. A deterministic model, which stratifies the IDU population into eight mutually-exclusive compartments (based on epidemiological status), is considered. Rigorous qualitative analysis of the model establishes, for the first time, the presence of the phenomenon of backward bifurcation in HCV transmission dynamics. Three routes (or causes) to such a dynamic phenomenon have been established. Furthermore, five main parameters that play a dominant role on the transmission dynamics of the disease have been identified. Numerical simulations of the model show that the re-infection of recovered individuals has marginal effect on the HCV burden (as measured in terms of the cumulative incidence and prevalence of the disease) in the IDU community.

Thesis (M.Sc.) - Carleton University, 2006. / Includes bibliographical references (p. 88-93). Also available in electronic format on the Internet.

Thesis (Ph. D.)--Rutgers University, 2007. / "Graduate Program in Mechanics." Includes bibliographical references (p. 106-110).

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