Mathematical Epidemiology and Ecology (Supervisor: Stephen Gourley)

Dr Gourley can offer supervision in various topics in mathematical epidemiology and ecology. Supervision is
available in:

•  theory and applications of delay differential equations, 
•  mathematical modelling of invasive insects and weeds, 
•  mathematical modelling of insect-borne diseases such as dengue, malaria, West Nile virus and bluetongue, 
•  modelling of diseases spread by migratory birds, and,
•  modelling species dynamics at multiple scales, for example where insect larvae and adults perceive space in a different  way and at different scales.

Projects can be devised in these areas. All projects will require a sound background in ordinary and partial differential equations. A strong background in real analysis is a big advantage.

Numerical weather prediction: multi-scale variational data assimilation and high-resolution models (Supervisor: Prof. Ian Roulstone)

Assessing risk from extremes of weather and climate, evaluating sustainable energy resources, and generally improving our ability to adapt to a changing climate, all require detailed knowledge of how weather patterns will affect our lives in the future. Reliable weather forecasts are currently issued for 3 to 4 days ahead at continental and global scales, though more detailed local/regional information from higher-resolution weather prediction models requires a radical new approach (because relatively rapid, unstable nonlinear processes become more important).

This project will contribute to the development of high-resolution numerical weather prediction systems (which are recognized as an important part of evaluating the impact of climate change at local or regional scales) and their use to generate meaningful information for end users e.g. where forecasts are used as part of a flood warning system. Success will lead to immediate benefits for all users of forecasts (e.g. Environment Agencies, Aviation Authorities (e.g. CAA), etc.).

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Construction of invariant sets for nonautonomous ODEs (Supervisor: Dr Jonathan H.B. Deane)

Invariant sets (subsets of the phase plane in which solutions of an ODE remain for all time) are important because they delineate qualitatively different sorts of behaviour displayed by solutions of an ODE, for instance separating solutions that remain bounded for all time from ones which blow up in finite time. Proving that a given set is invariant generally requires the proof of an inequality on the boundary of the set. Even in the case of a second-order non-autonomous ODE, this essentially planar method yields a subset of the actual invariant set. An investigation into how to optimise this procedure to obtain best possible constructions, and possibly also how to automate the construction using computer algebra or a low-level computer language, is the purpose of this PhD project.

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Patterns in Surface Chemistry (Supervisor: Dr Rebecca Hoyle)

Regular patterns arise naturally in many physical, chemical and biological systems - from hexagonal convection cells on the surface of the sun to stripes on a zebra's back. Constantly changing irregular patterns of carbon monoxide (CO) and oxygen are seen during CO oxidation on platinum crystals in the [100] orientation. Recently a reaction-diffusion model is developed to reproduce this pattern formation and created numerical simulations that show patterns made up of moving CO and oxygen fronts. Possible PhD projects in this area include: extending the model to include the formation of subsurface oxygen at higher pressures or developing a similar model for the NO + NH3 reaction on Pt{100}. These interdisciplinary projects are great opportunities for Maths graduates to apply their skills in a new area, or for Chemistry graduates with good maths and computing skills to move into theory.

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Chaotic Advection in Large Airways (Supervisor: Prof Peter Hydon)

Babies who are born very prematurely usually need mechanical help to breathe. One particularly effective type of mechanical ventilator uses High Frequency Ventilation (HFV), in which a low volume of air is delivered to the lung at a frequency of 10-15 Hz. However the volume of air used is so small that the conventional theory of breathing does not explain how adequate gas transport can be achieved. It has recently been shown that a type of stirring called`chaotic advection' greatly enhances transport in small airways. The purpose of this project is to find out the extent to which chaotic advection operates in the large airways, and thus to develop strategies for obtaining the best possible transport at various frequencies. This project combines mathematical biology, dynamical systems theory and computational fluid dynamics. It is of direct relevance to medicine, and the student will have the opportunity to work with some of the UK's leading medical specialists in HFV.

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Spiral waves and defect interaction (Supervisor: David Lloyd and Bjorn Sandstede)

Spiral waves have been found in many experiments such as the famous Belousov-Zhabotinsky reaction and the oxidation of carbon-monoxide on platinum surfaces. In the former reaction, period doubling of spiral waves has been observed, and a numerical simulation of a model problem is shown to the left. This bifurcation has been analysed in 2D. Of interest would be an analytic and numerical study of this phenomenon in 3D where the line defect that connects the core to the bottom boundary becomes a two-dimensional surface, which itself may exhibit a interested dynamics. More generally, the various line defects visible in the simulation may interact with each other, and it would be of interest to study the time and length scales involved in this interaction from a analytical and numerical perspective.

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Theory and numerics of reversing symmetry breaking bifurcations of Hamiltonian relative periodic orbits (Supervisor: Dr Claudia Wulff)

Relative periodic orbits (RPOs) of symmetric dynamical systems are periodic orbits for the symmetry reduced system. Both the theory and numerics of periodic orbits of general dynamical systems is well-developed. But additional structure such as time-reversing and preserving symmetries and symplecticity changes the generic behaviour of dynamical systems dramatically. Recently there has been a lot of progress in the development of a bifurcation theory for symmetric and Hamiltonian systems, but the theory is far from being complete. In particular the bifurcation theory of reversible symmetric periodic and relative periodic orbits, is still in its beginning, as is the numerical analysis of symmmetry breaking bifurcations of reversible periodic orbits.
The topic of this project is to analyze reversing symmetry breaking bifurcations of RPOs theoretically and to derive and implement numerical methods for their detection and computation within the package SYMPERCON of Wulff, Schebesch, Schilder. The results will be applied to various symmetric Hamiltonian systems, in particular to N-body systems.

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Mathematics of Storytelling (Supervisor: Dr Rebecca Hoyle)

How do oral histories, tales that encode some part of a community's history or shared culture, spread and persist? Can we model this mathematically, perhaps using an agent-based approach, where we create individuals, give them attributes and behaviours and link them together in an evolving social network? I'd like to find out, perhaps using the evolution of children's nursery rhymes as an example. This project would involve researching the history and geographical distribution of nursery rhymes and attempting to build a model that can reproduce a similar pattern of spread. It would suit a Maths or Computing Science graduate with an interest in social science and some programming skills who is comfortable with an open-ended and exploratory approach in the initial stages.

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Faraday waves (Supervisor: Dr Anne Skeldon)

Patterns can be made to form on the surface of a container of fluid by shaking the container up and down. The shaking has to be at the right frequency and right amplitude for any patterns to form - if you move a container of fluid up and down very slowly then the surface of the fluid will remain flat. There are a wealth of experimental results showing a variety of different possible patterns. On the theoretical front, arguments based on symmetries and the way different patterns interact has led to some knowledge of the mechanisms that cause particular patterns to become dominant. Although these have been tested on mathematical models describing the fluid, there are still significant gaps in our understanding. The aim of this project is to investigate further the theory underlying pattern selection in the Faraday and related problems. This will require using a variety of different dynamical systems techniques both theoretical and computational.

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Fast ODE solvers using analytical continuation (Supervisor: Dr Jonathan H.B. Deane)

Fast, accurate methods for solving nonlinear ODEs with polynomial nonlinearity are important in many applications. One suitable method is based on Taylor series, and is also known as the cell-to-cell mapping technique. Roughly speaking, the solution of the ODE is expanded in a power series around a point t=t₀, and a suitably modified ratio test applied to the high-order coefficients of the series. The test gives an estimate of the radius of convergence of the series, R (among other things), and so we can compute the solution accurately at, say, t=t₀+½ R. This is effectively numerically-implemented analytical continuation. In practice, R can be quite large and so the ODE can be solved in correspondingly large time steps. In some recent work on the varactor equation, an increase in speed by a factor of about 10-50 was obtained using this method compared to, for instance, Runge-Kutta methods. The method appears to be promising but a great deal remains unknown about the assumptions on which it is based, its performance and the circumstances under which it fails.

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Molecular Motors (Supervisor: Dr Rebecca Hoyle)

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Molecular motors are proteins that transform chemical energy into mechanical work on a molecular level, generating forces and leading to motion. We are studying myosin V, a motor involved in intracellular transport in animal cells. It has two heads that bind to an actin filament and a long neck that attaches to its cargo, such as vesicles and organelles. The myosin molecule walks hand-over-hand along the actin track via the coordinated binding and release of its heads. We have used energetics to model the interaction of external load and intramolecular strain with the ATP hydrolysis cycle that drives the stepping action, and performed a detailed quantitative fit to experimental data. Possible PhD projects include: applying the same methodology to a variety of other molecular motors to determine how well the established models compare with experimental data, and how the evolved physical characteristics of the motors relate to their biological function. This is interdisciplinary work in an exciting and fast-moving area of biophysics. An enthusiasm for learning about biophysics and communicating with experimentalists is essential. This project would suit a graduate in Applied Maths or Physics, or possibly a Biology graduate with strong quantitative skills. Programming skills are needed to adapt existing codes.

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