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This thesis explores several aspects of the correspondence between classes of linear ordinary differential equations (ODEs) in the complex plane and certain quantum integrable models (IMs), also known as the ODE/IM correspondence. First, we enlarge the set of ordinary differential equations that enter the correspondence. Differential equations satisfied by Wronskians between solutions of specific ODEs are obtained and are associated to nodes of particular Dynkin diagrams. In the second part of the thesis we generalise the correspondence to encompass massive IMs. Starting from an integrable nonlinear partial differential equation corresponding to the classical A2(l) affine Toda field theory (ATFT), we expand the set of integrable models that enter the correspondence. This establishes an ODE/IM correspondence for a massive IM. We then extend the results to the An-1 (1) ATFTs and the particular example of D3 (l) ATFT.

We consider certain kinds of homoclinic bifurcations in three-dimensional vector fields. These global bifurcations are characterized by the existence of a homo clinic orbit that converges to a saddle equilibrium in both forward and backward time. If the equilibrium has a complex pair of (stable) eigenvalues, it is a saddle-focus, and one speaks of a Shilnikov homoclinic orbit. In this case, the homoclinic orbit converges towards the equilibrium in a spiralling fashion. On the other hand, if the saddle equilibrium has two real (stable) eigenvalues, then the homoclinic orbit converges generically to the saddle along the direction given by the weak stable eigenvector. The possible unfoldings of a codimension-one homoclinic bifurcation depend on the sign of the saddle quantity: when it is negative, breaking the homoclinic orbit results in a single stable periodic orbit from a saddle-focus homoclinic orbit; one speaks of a simple Shilnikov bifurcation. However, when the saddle quantity is positive, then the mere existence of a Shilnikov homoclinic orbit induces complicated dynamics, and one speaks of a chaotic Shilnikov bifurcation. For a homoclinic orbit to a real saddle, on the other hand, always a single periodic orbit bifurcates, which is attracting when the saddle quantity is negative and of saddle type when it is positive. In this thesis we show how the global three-dimensional phase space is organized near certain homoclinic bifurcations by the two-dimensional global stable manifolds of equilibria and periodic orbits. To this end, we consider a model of a laser with optical injection that contains Shilnikov homoclinic orbits and a model by Sandstede that features different kinds of homoclinic bifurca- tions to a saddle. We find that, in the simple Shilnikov case, the stable manifold ofthe saddle-focus forms the basin boundary of the bifurcating stable periodic orbit. On the other hand, in the chaotic case, the stable manifold of the equilibrium is the accessible set of a chaotic saddle that contains countably many periodic orbits of saddle type. In the case of a homoclinic bifurcation to a saddle, the stable manifold of the saddle is either an orientable or nonorientable two-dimensional surface. A change of orientability occurs at two kinds of codimension-two homoclinic bifurcations, called inclination flip and orbit flip bifurcations. At either of these flip bifurcation points, the stable manifold is neither orientable nor nonorientable, but just at the transition between both states. We show how this transition occurs for the case of negative saddle quantity, and how the basin of attraction of the stable periodic orbit is organized in different ways by the stable manifold of the saddle depending on the (non)orientability of the bifurcation. Finally, we show how the stable manifold rearranges both itself and the overall dynamics in phase space near the codimension-two transition from a saddle to saddle-focus homoclinic bifurcation that occurs at a so-called Belyakov point.

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This thesis is devoted to the qualitative analysis of solutions of partial differential N equations and delay partial differential equations with applications to population biology. The first part deals with the problem of finding the length scales for the Navier-Stokes system on a rotating sphere and for a class of generalized reaction-diffusion system on a planar domain. Since the reaction-diffusion system under investigation has many biological and physical applications, it is crucial to be able to prove a positivity preserving property for solutions of this system. Motivated by its applications, the question of asymptotic positivity of solutions, as well as positivity for all time for a reaction-diffusion model is investigated. The presence of the fourth-order derivative in the equation makes the application of the maximum principle impossible. It will be shown that with the help of the ladder method, a positivity preserving property for this type of system can be proved. In all calculations, the application of interpolation inequalities of the Gagliardo-Nirenberg type with explicit and sharp constants gives the best possible results, and all calculations contain only known constants. Next, nonlinear analysis of the Extended Burgers-Huxley equation on a planar domain with periodic boundary conditions is performed. The geometric singular perturbation theory is then used to prove persistence of the travelling wave solutions in the case when a small perturbation parameter multiplies the fourth- order derivative. The second part of this thesis considers partial differential equations with time delay. We propose and study two mathematical models of stage-structured population. First, we study a nonlocal time-delayed reaction-diffusion population model on an infinite one-dimensional spatial domain. Depending on the model parameters, a non-trivial uniform equilibrium state may exist. We prove a comparison theorem for our equation for the case when the birth function is monotone, and then we use this result to establish nonlinear stability of the non-trivial uniform equilibrium state when it exists. A certain class of non-monotone birth functions relevant to certain species is also considered, namely, birth functions that are increasing at low densities but decreasing at high densities. In this case we prove that solutions still converge to the non-trivial equilibrium, provided the birth function is increasing at the equilibrium level. Then we derive a stage-structured model for a single species on a finite one-dimensional lattice. There is no migration into or from the lattice. The resulting system of equations, to be solved for the total adult population on each patch, is a system of delay equations involving the maturation delay for the species, and the delay term is nonlocal involving the population on all patches. We prove that the model has a positivity preserving property. The main theorem of the paper is a comparison principle for the case when the birth function is increasing. Using this theorem we prove that, when the model admits a positive equilibrium, the positive equilibrium is a global attractor. Then we establish a comparison principle that works for very general birth functions, and then we use this theorem to prove convergence theorems in the case when the birth function qualitatively resembles one used in the Nicholson's blowflies equation. We conclude by solving system numerically, using DDE tool in MATLAB. The thesis is concluded by a discussion of some open problems.