Mathematics & StatisticsAvailable PhD projects

This project addresses a major imaging problem from biomedical research, by working with a range of image modalities including standard histological analysis, multiplexed biomarker analysis with diverse applications across oncology, inflammatory, cardiovascular and metabolic diseases.

Imaging science is a relatively new branch of applied mathematics for emerging applications in almost all areas of cutting-edge research. The field is growing rapidly as new technologies are constantly driving the development.

This imaging project is an opportunity to undertake one of our new and exciting cross-disciplinary projects lying at the interface of mathematics and cancer medicine. The candidate does not need to have any knowledge in cancer diseases or medicine but will need strong mathematical knowledge through a degree in maths, or in computer science / physics / engineering with essential maths components, as well as good programming skills.

This project addresses a major imaging problem from biomedical research, by working with a range of image modalities including standard histological analysis, multiplexed biomarker analysis with diverse applications across oncology, inflammatory, cardiovascular and metabolic diseases.  We will work with both public datasets and also real-life datasets from the collaborating company; visiting the company and being able to communicate and work with non-academics are expected for the project (for the latter some training will be given).  Advanced geometry by way of PDEs or energy minimization functionals has been widely used in mathematical imaging. However, the aim of the project is to design vision-language frameworks, which uses mathematical models and deep learning methods to analyze image of cells and tissues and transfer flexibly to both vision-language understanding and generation tasks and describe the complexity in the tissue in the form of sentences.

To do this, the successful candidate will first be trained to understand current data analysis pipelines to co-register, collate and interrogate information across different data types and length-scales (resolution-wise). This is about multimodal imaging and involves variational models and deep learning algorithms for such multiscale problems. Then he or she will investigate how large language models (LLMs) work for these biomedical images by way of interpretation.  LLMs have exhibited exceptional ability in language understanding, generation, interaction, and reasoning. Here language is used as a generic interface to connect vision-based AI models to solve AI tasks in the biomedical context.

The new method is expected to consider spatial location, cell multiple interactions and neighborhood information, fundamentally different from traditional clustering methods such as K-means. From this project, a student can get trained in advanced mathematics, imaging models, biomedical knowledge, deep learning algorithms and novel natural language models. The achieved approach will serve as a key step toward advanced artificial intelligence in computational pathology and acceleration of new drugs development.

Overall, this studentship is attractive because the topic under study is interesting and modern and a candidate has the flexibility to focus on all or just one of the key components:  maths, AI and NLP in terms of leading research. Apart from the collaboration opportunity with a top UK company, there is also a collaborating possibility with Universities of Glasgow and Liverpool which the primary supervisor is strongly associated with.

This is a fully funded scholarship by the University (covering both tuition fees and stipend for a UK student for 3.5 years) but the work will involve collaboration with the pharmaceutical company AZ (Astra-Zeneca).   There is a strong possibility to add an industrial top payment on stipend beyond the level set equal to an UKRI studentship.

Applicants should have, or be expecting to obtain in the near future, a first class or good 2.1 honours degree (or equivalent) in mathematics or in a closely related discipline with a high mathematical content.

This imaging project is an opportunity to undertake one of our new and exciting cross-disciplinary projects lying at the interface of mathematics and cancer medicine. 

Imaging science is a relatively new branch of applied mathematics for emerging applications in almost all areas of cutting-edge research. The field is growing rapidly as new technologies are constantly driving the development.

This imaging project is an opportunity to undertake one of our new and exciting cross-disciplinary projects lying at the interface of mathematics and cancer medicine. The candidate does not need to have any knowledge in cancer diseases or medicine but will need strong mathematical knowledge through a degree in maths, or in computer science / physics / engineering with essential maths components, as well as good programming skills.

This multidisciplinary project aims to address a key step in designing new drugs and testing their efficacy is a comprehensive understanding of disease mechanisms at the single-cell and tissue context level. Working with teams from Astra-Zeneca, we investigate a range of biomedical imaging modalities and study cellular heterogeneity and the intricate interactions in disease microenvironments to chart cellular heterogeneity, complex tissue structures, and dynamic changes during diseases progression.

In terms of research and development, it addresses a set of outstanding challenges in both mathematical and AI fields. Working with applied mathematicians from Strathclyde and computing scientists from Queen’s University of Belfast, along with AI and data analysis teams from AZ, the successful student will get a thorough training in Mathematical Imaging (e.g variational models and diffeomorphic maps), Artificial Intelligence (neural networks and transfer learning),  Biomedical Research (Multimodal imaging, transcriptomics and H&E) and gain practical problems solving skills that are highly values in both Academia and Industries. Through the supervisors’ contacts and networks, the student will also benefit from and apply his or her skills to discussions and potential collaborators with other Universities including Universities of Birmingham and Liverpool and NHS hospitals such Clatterbridge Cancer Centre.  

Overall, this studentship is attractive because the topic under study is interesting and modern and a candidate has the flexibility to focus on all or just one of the key components:  maths, AI and biomedical analysis in terms of leading research. Apart from the collaboration opportunity with a top UK company, there is a ready secondary supervisor (Prof Styles) at QUB with complementary expertise and  also a large collaborating network.

Classification of teas (types, quality grades, region of origin etc) has been examined by many researchers, using both chemical composition and more recently digital image analysis techniques to extract features from the image that are useful for classification.

Factors affecting the success of classification include the choice of features, the classifier and the imaging modality. Building on previous work at Strathclyde in collaboration with the EEE department, this project will allow the student to examine the choice of any of these to achieve optimal results. Intending students should have a strong statistical background and excellent computer skills, and be competent/be able to quickly become competent in the use of both R and Matlab.

Second supervisors: George Gettinby, Magnus Peterson

Worldwide losses of honey bee colonies have attracted considerable media attention in recent years and a huge amount of research. Researchers at Strathclyde have experience since 2006 of carrying out a series of surveys of beekeepers in Scotland (http://personal.strath.ac.uk/a.j.gray/) and now have 5 years of data arising from these surveys. These data have been used to estimate colony loss rates in Scotland and to provide a picture of beekeepers’ experience and management practices.

This project will examine the data in more detail than has been done so far, and is likely to involve data modelling and multivariate methods to identify risk factors. Part of the project will involve establishing the spatial distribution of various bee diseases.

This project will build on links with the Scottish Beekeepers’ Association and membership of COLOSS, a network linking honey bee researchers in Europe and beyond. Intending students should have a strong statistical background and excellent computer skills, and be competent/be able to quickly become competent in the use of R.

Introduction
In this project we shall look at how stochastic models can be used to describe how infectious diseases spread. We will start off by looking at one of the simplest epidemic models, the SIS (susceptible-infected-susceptible) model. In this model a typical individual starts off susceptible, at some stage catches the disease and after a short infectious period becomes susceptible again.

These models are used for diseases such as pneumococcus amongst children and sexually transmitted diseases such as gonorrhea amongst adults (Bailey, 1975). Previous work has already looked at introducing stochastic noise into this model via the disease transmission term (Gray et al. 2011). This is called environmental stochasticity which means introducing the random effects of the environment into how the disease spreads. This results in a stochastic differential equation (SDE) model which we have analysed. We have derived an expression for a key epidemiological parameter, the basic reproduction number.

In the deterministic model this is defined as the expected number of secondary cases caused by a single newly-infected individual entering the disease-free population at equilibrium. The basic reproduction number is different in the stochastic model than the deterministic one, but in both cases it determines whether the disease dies out or persists. In the stochastic SIS SDE model we have shown the existence of a stationary distribution and that the disease will persist if the basic reproduction number exceeds one and die out if it is less than one.

SDE Models with Environmental Stochasticity
We have also looked at other SDE models for environmental stochasticity. One of these took a simple deterministic model for the effect of condom use on the spread of HIV amongst a homosexual population and introduced environmental stochasticity into the disease transmission term. Again we found that a key parameter was the basic reproduction number which determined the behaviour of the system.

As before this was different in the deterministic model than the stochastic one. Indeed it was possible for stochastic noise to stabilise the system and cause an epidemic which would have taken off in the deterministic model to die out in the stochastic model (Dalal et al., 2007). Similar effects were observed in a model for the internal viral dynamics of HIV within an HIV-infected individual (Dalal et al., 2008).

Demographic Stochasticity
The real world is stochastic, not deterministic, and it is difficult to predict with certainty what will happen. Another way to introduce stochasticity into epidemic models is demographic stochasticity. If we take the simple homogeneously mixing SIS epidemic model with births and deaths in the population we can derive a stochastic model to describe this situation by defining p(i, j, t) to be the probability that at time t there are exactly i susceptible and j infected individuals and deriving the differential equations satisfied by these probabilities.

Then we shall look at how stochastic differential equations can be used to approximate the above set of equations for p(i, j, t). This is called demographic stochasticity and arises from the fact that we are trying to approximate a deterministic process by a stochastic one (Allen, 2007). Although the reasons for demographic and environmental stochasticity are quite different the SDEs which describe the progress of the disease are similar. The first project which we shall look at is analysis of the SIS epidemic model with demographic stochasticity along the lines of our analysis of the SIS epidemic model with environmental stochasticity.

Further Work
After this we intend to look at other classical epidemiological models, in particular the SIR (susceptible-infected-removed) model in which an individual starts off susceptible, at some stage he or she catches the disease and after a short infectious period he or she becomes permanently immune. These models are used for common childhood diseases such as measles, mumps and rubella (Anderson and May, 1991). We would look at introducing both environmental and demographic stochasticity into this model.

Other epidemiological models which could be analysed include the SIRS (susceptible-infected-removed-susceptible) epidemic model, which is similar to the SIR epidemic model, except that immunity is not permanent, the SEIS (susceptible-exposed-infected-susceptible) model which is similar to the SIS model, but includes an exposed or latent class, and the SEIR (susceptible-exposedinfected- removed) model, which similarly extends the SIR model.

We would also aim to look at other population dynamic models such as the Lotka-Volterra predator-prey model. There is also the possibility of developing methods for parameter estimation in all of these epidemiological and population dynamic models, and we have started work on this with another Ph. D. student (J. Pan).

References
1. E. Allen, Modelling with Itˆo Stochastic Differential Equations, Springer-Verlag, 2007.
2. R.M. Anderson and R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.
3. N.T.J. Bailey, The Mathematical Theory of Infectious Disease and its Applications, Second Edition, Griffin, 1975.
4. A.J. Gray, D. Greenhalgh, L. Hu, X.
5. N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for AIDS and condom-use. J. Math. Anal. Appl. 325, 36-53, 2007.
6. N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal HIV dynamics. J. Math. Anal. Appl. 341, 1084-1101, 2008.

Differential equation models are commonly used to model infectious diseases. The population is divided up into compartments and the flow of individuals through the various classes such as susceptible, infected and removed is modelled using a set of ordinary differential equations (Anderson and May, 1991, Bailey, 1975). A basic epidemiological parameter is the basic reproduction number. This is defined as the expected number of secondary cases produced by a single newly infected case entering a disease-free population at equilibrium (Diekman and Heesterbeek). Typically the disease takes off if R0 > 1 and dies out if R0≤1.

However media awareness campaigns are often used to influence behaviour and if successful can alter the behaviour of the population. This is an area which has not been studied much until recently. The student would survey the existing literature on media awareness models in the literature and with the supervisor formulate mathematical models using differential equations for the effect of behavioural change on disease incidence. These would be examined using both analytical methods and computer simulation with parameters drawn from real data where appropriate. The mathematical techniques used would be differential equations, equilibrium and stability analyses and computer simulation.

References
1. R.M. Anderson and R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.
2. N.T.J. Bailey, The Mathematical Theory of Infectious Disease and its Applications, Second Edition, Griffin, 1975.
3. O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and computation of the basic reproduction number R0 for infectious diseases in heterogeneous populations. J. Math. Biol. 28, 365-382.
4. A. K. Misra, A. Sharma and J. B. Shukla, Modelling and analysis of effects of awareness programs by media on the spread of infectious diseases. Math. Comp. Modelling 53, 1221- 1228.
5. A. K. Misra, A. Sharma, V. Singh, Effect of awareness programs in controlling the prevalence of an epidemic with time delay, J. Biol. Systems, 19(2), 389-402,

In recent years there has been much work on reaction-diffusion equations in which the diffusion mechanism is not the usual Fickian one. Examples are integro-differential equations, porous media type equations, pseudodifferential equations, p-Laplacian type equations and prescribed curvature type (saturating flux) equations.

The motivation for this work comes from material science and mathematical ecology. However, there are applied contexts where these diffusion mechanisms have never been considered. One is in the area of combustion and the other is in the area of regularised conservation laws and shock propagation. This project, which would build on the work I did through the years on integrodifferential models and recently with M. Burns on the prescribed curvature equations, will use PDE, asymptotic, and topological methods to explore the dynamics of blowup and of shock propagation in canonical examples of reaction equations and nonlinear scalar conservation laws regularised by non-Fickian diffusion terms.

References:

[1] M. Burns and M. Grinfeld, Steady state solutions of a bistable quasilinear equation with saturating flux, European J. Appl. Math. 22 (2011), 317-331.

[2] M. Burns and M. Grinfeld, Steady state solutions of a bistable quasilinear equation with saturating flux, European J. Appl. Math. 22 (2011), 317-331.

Recently, a new class of model has been developed to describe, for example, phase separation in materials such as binary alloys. These take the form of integrodifferential equations. Coarsening, that is, creation of large scale patterns in such models is poorly understood.

There are partial results [1, 2] that use the maximum principle, while for most interesting problems such a tool is not available. This will be a mixture of analytic and numerical work and will need tools of functional analysis and semigroup theory.

References:

[1] D. B. Duncan, M. Grinfeld, and I. Stoleriu, Coarsening in an integro-differential model of phase transitions, Euro. J. Appl. Math. 11 (2000), 561-572.

[2]  V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Euro. J. Appl. Math. 17 (2006), 221-232.

The Strathclyde Centre for Doctoral Training (SCDT) in "Data-driven uncertainty-aware multiphysics simulations" (StrathDRUMS) is a new, multi-disciplinary centre of the University of Strathclyde, which will carry out cutting-edge research in data-driven modelling and uncertainty quantification for multiphysics engineering systems.

StrathDRUMS will train the next generation of specialists to apply non-deterministic model updating, digital twin techniques, and advanced uncertainty treatments to real-world challenges in civil and aerospace engineering. In our research we aim to study complex systems such as aeroplanes, spacecrafts, buildings and bridges using rigorous mathematical concepts, formulations and computational methods.

We are pleased to announce an available funded 3.5 year PhD studentship project within the centre on “High-dimensional computations with applications to uncertainty quantification for multiphysics engineering systems”, supervised by Dr Yoshihito Kazashi in the Department of Mathematics and Statistics and co-supervised by Dr Sifeng Bi (Mechanical & Aerospace Engineering), Dr Marco de Angelis (Civil & Environmental Engineering) and Dr Michele Ruggeri (Mathematics and Statistics). In engineering applications, various types of uncertainty are modelled with a large number of parameters.

However, working with such models yields high-dimensional problems, such as high-dimensional integration to compute expectations and high-dimensional approximation to construct surrogate models, that are computationally very challenging. The successful candidate will develop and analyse numerical techniques in computational uncertainty quantification. They will focus specifically on cutting-edge numerical analysis of high-dimensional integrals (that outperform Monte Carlo methods), function approximation, probability density function estimation as well as optimal solvers of partial differential equations that take uncertainty into account.

Although the student will be based in the Department of Mathematics and Statistics at the University of Strathclyde, they will be benefit from supervision by the wider SCDT team. According to the student interest and background, the research will be aligned and supported by one of StrathDRUMS partners, which include National Manufacturing Institute Scotland (NMIS), National Physical Laboratory (NPL), and UK Atomic Energy Authority (UKAEA). The student will thus be integrated within a vibrant and active multi-disciplinary research community with in-house training opportunities available across multiple faculties.

In addition to undertaking cutting-edge research, the student will be registered for the Postgraduate Certificate in Researcher Development (PGCert), which is a supplementary qualification to develop core skills, networks, and career prospects.

The successful candidate will be expected to conduct high-quality research in the areas of computational uncertainty quantification, participate in relevant training activities and events provided by StrathDRUMS, disseminate research findings through publications and presentations, contribute to the wider research community through engagement and collaboration with other researchers.

We encourage applications from UK-based students for this position. However, we also welcome strong international students to apply, provided they are able to cover the difference between the home and international tuition fees on their own.

The studentship should start on 1 October 2023. The studentship will fund the annual Home tuition fees and a tax-free stipend for 3.5 years. The stipend rates are announced annually by UKRI. To give an idea, for the 2022/23 academic year, the annual UKRI stipend is £17,668.

Applicants should have, or be expecting to obtain soon, a first class or good 2.1 honours degree (or equivalent) in mathematics or in a closely related discipline with a high mathematical content.  Excellent written and verbal communication skills, analytical and problem-solving skills, ability to work independently and as part of a team are essential. Programming skills and some knowledge of analytical and numerical methods in uncertainty quantification are desirable.

To apply, candidates should send their CV, transcript, and cover letter to y.kazashi@strath.ac.uk. Review of applications starts on 15 June 2023. Applications will be accepted until the position is filled.

Chemical modifications on DNA play important roles in the normal functioning of cells, and changes to these modifications are a hallmark of disease. This PhD project aims to combine novel mathematical models with data from state-of-the-art DNA sequencing technologies to gain a greater understanding of the biological processes underpinning the addition and removal of such modifications to DNA.

DNA methylation occurs when chemical methyl groups attach to DNA and this influences how cells interpret the underlying DNA sequence. Changes in DNA methylation can have harmful consequences, with unusual methylation patterns being associated with disease. A greater understanding of the processes involved in the development and maintenance of methylation patterns will help in understanding the currently unknown causes and consequences of the unusual patterns associated with disease.

The aim of this project is to use mathematical modelling, statistical techniques and data analysis to study DNA methylation systems. In particular, we will consider models describing methylated clusters within DNA that can undergo six processes: birth, death, growth, decay, coagulation and fragmentation. Combined with experimental data from cutting-edge sequencing technologies, these models will be used to extract currently unknown information regarding DNA methylation and demethylation processes.

Throughout the project, the student will have the opportunity to develop skills in fields such as mathematical modelling, statistical analysis, Bayesian inference and data science. There will also be opportunities for the student to interact and visit biological experts in the field of DNA methylation (Dr Duncan Sproul, Institute of Genetics and Cancer, University of Edinburgh).

While prior biological experience is not necessary, applicants should have a keen interest in learning about biological systems and in applying mathematical and statistical techniques to biological problems. The project will also involve data analysis and computational work, meaning that experience using a programming language such as R, Matlab, Mathematica or Python would be advantageous, though not necessary.

Funding:  Funding for a student meeting the EPSRC definition of a home student is secured.  This EPSRC DTP studentship will cover the payment of fees to the university and a yearly UKRI minimum stipend (£19,237 for academic year 2024-2025). In addition, the student will have access to a Research Training Support Grant worth a total of £5,250 over the 42-month studentship.

Eligibility: Applicants should have, or expect to obtain by the start date of the project (01/10/24), a first class or 2:1 honours degree (or equivalent) in mathematics, statistics or a closely related discipline. Funding for a student meeting the EPSRC definition of a home student is secured. In the event of an outstanding international student, additional funds will be sought.

Primary Supervisor: Dr Lyndsay Kerr

Additional Supervisor/s: Professor John Mackenzie

Contact Details: Informal enquiries should be addressed to Dr Lyndsay Kerr (lyndsay.kerr@strath.ac.uk)

Applications:  Applications can be made here.

Phenology is the study of seasonal biological events, such as migration, egg laying, or flowering. The iconic ‘match-mismatch hypothesis’ (Cushing, 1974, 1975) predicts that changes in phenology may affect synchronicity with energy sources and so impact fitness. Examples of this may be failure to migrate in time to exploit a food source elsewhere, or to lay eggs to synchronise hatching with seasonally available food. As species rely on different environmental cues to time these events, it is possible that climate change will disrupt important ecological connections with cascading consequences at the level of the ecosystem.

Although originally conceived in the context of marine biology, the match-mismatch hypothesis has since been embraced as a general concept in ecology. As evidence of phenological shifts in response to changing climate mounts (Parmesan & Yohe, 2003), there has been a surge in publications in the ecological literature reviewing and re-evaluating the hypothesis (Kharouba and Wolkovich 2023; Samplonius et al. 2021). In general, the conclusion is that the conditions under which phenological asynchrony leads to effects on fitness depends on the ecosystem context – in particular the extent to which a consumer species or group is bottom-up or top-down regulated.

The central question for this studentship is: under what circumstances will the widely observed climate-related shifts in phenology lead to notable consequences at the level of the ecosystem?

While ecosystem models already include many aspects of known ecology and trophic coupling, the processes governing phenology and the sensitivity of the system to atch-mismatch effects are glaringly missing. Ideally, phenological characteristics should be an emergent property of such models. This is the case for phytoplankton and lower trophic levels, but not for mid- and higher trophic levels. The proposed PhD project aims to spearhead a step change in ecosystem modelling by representing these processes.

If successful in your application, you will complete a programme of doctoral research:

Systematically reviewing the recent surge in literature on the evidence for match-mismatch effects on fitness in marine ecosystems, compared to terrestrial and freshwater systems.

Analysing high resolution plankton time series data sets from the Scottish Coastal Observatory, and other sources, to diagnose environmental cues driving phenology.

Building strategic (exploratory) population dynamics models of resource-consumer-predator systems to test hypotheses about the sensitivity of match-mismatch effects on fitness to the nature of trophic coupling.

Parameterising phenological processes for mid-trophic levels, guilds of benthos and fish, in an existing end-to-end ecosystem model (StrathE2E).

Investigating the sensitivity of spawning, recruitment, and migration phenology to environmental cues in the ecosystem and compare the ecosystem’s sensitivity to phenological trends across different regional implementations of StrathE2E throughout the Atlantic Ocean.

Further information:  To apply please visit https://www.strath.ac.uk/studywithus/postgraduateresearchphdopportunities/science/mathematicsstatistics/marinematch-mismatch/.

You will be registered as a student in the department of Mathematics and Statistics at the University of Strathclyde, where Dr Laverick and Prof Heath are based. You will also be supervised by Prof Diele from Edinburgh Napier University.

The Marine Directorate of the Scottish Government will support this project, providing an excellent opportunity for you to work with a non-academic partner. You will use data from the Scottish Coastal Observatory (SCObs) and present to the UK Pelagic Habitat Working Group to support DEFRA-funded work on the impacts of changes in the pelagic habitat on ecosystem services.

Training:

You will:

Have access to training funds, valued at £16,425 for the current academic year, which can be spent to support professional development, attend meetings and conferences, and the acquisition of technical skills.

Complete the Strathclyde PGCert in Researcher Professional Development (website).

Receive training in coding, systematic review, time series analysis, and the mathematical modelling of populations and ecosystems.

About SUPER: 

SUPER is multi-institutional, cross-disciplinary, and interdisciplinary; helping to foster a new generation of students equipped to take on diverse careers and to manage our natural environments more sustainably. SUPER brings together the research strengths of the Universities of Aberdeen, Edinburgh Napier, Heriot-Watt, Highlands and Islands, St Andrews, Stirling, Strathclyde, and the West of Scotland. All institutional partners are members of the Marine Alliance for Science and Technology for Scotland (MASTS), whose research and training collaborations address cutting-edge scientific challenges across the Natural Environment Research Council (NERC; part of the UKRI) remit. Underpinning these research partners, providing additional training and projects, are stakeholder organisations including industry and governmental bodies.

SUPER DTP students will be part of a cohort that will develop together, forming a nurturing network, supported by bespoke events. The SUPER cohort will pursue research, engage in training, and learn from each other as part of a large multi-disciplinary group. Members are offered unparalleled opportunities to understand societal and environmental challenges and to deliver science for the benefit of wider society, with international implications.

Background reading:

Edwards, M. & Richardson, A.J. (2004) Impact of climate change on marine pelagic phenology and trophic mismatch. Nature, 430(7002), 881–884.

Atkinson, A. et al. 2015. Questioning the role of phenology shifts and trophic mismatching in a planktonic food web. Progress in  Oceanography 137, 498–512.

Simmonds, E.G. et al. (2020). Phenological asynchrony: a ticking time-bomb for seemingly stable populations? Ecology Letters 23:12, 1766-1775. https://doi.org/10.1111/ele.13603

Flynn, J.K., Speirs, D.C., Heath, M.R. & Mitra, A. (2021). Subtle differences in the representation of consumer dynamics have large effects in marine food web models. Frontiers in Marine Sc ience, 8:638892. doi: 10.3389/fmars.2021.638892

Heath, M.R., Speirs, D.C., Thurlbeck, I, & Wilson, R.J. (2021) StrathE2E: an R package for modelling the dynamics of marine food webs and fisheries. Methods in Ecology and Evolution 12, 280-287. https://doi.org/10.1111/2041-210X.13510

Cushing, D.H. (1974) The natural regulation of fish populations. In: Harden Jones, F.R. (Ed.) Sea fisheries research. London, UK: Elek Science, pp. 399–412.

Cushing, D.H. (1975) Marine ecology and fisheries. Cambridge, UK: Cambridge University Press.

Parmesan, C. & Yohe, G. (2003). A globally coherent fingerprint of climate change impacts across natural systems. Nature 421, 37-42

Samplonius, J.M. et al. (2021). Strengthening the evidence base for temperature-mediated phenological asynchrony and its impacts. Nature Ecology and Evolution 5, 155–164. 

Kharouba, H.M. & Wolkovich, E.M. (2023) Lack of evidence for the match-mismatch hypothesis across terrestrial trophic interactions. Ecology Letters, 26, 955–964

Nematic liquid crystals are partially ordered materials that are intermediate between solid and liquid phases of matter. Nematics have widespread applications in science and technology, notably the thriving liquid crystal display industry. This project focuses on the mathematical modelling of nematic configurations in prototype systems in an interdisciplinary framework, including the analysis of the associated systems of nonlinear partial differential equations and numerical computations of the solution landscapes. The project will also focus on the potential applications of the theoretical and numerical results to materials technologies.

A Leverhulme Trust-funded 36-month PhD position is available, to work with Professor Apala Majumdar(apala.majumdar@strath.ac.uk). Please contact her as soon as possible if you are interested. The funding covers home tuition fees and a living stipend, along with opportunities to visit international collaborators at Luxembourg and Verona. There might be opportunities for overseas candidates too.

Up to 2002, most of the existing strong convergence theory for numerical methods requires the coefficients of the SDEs to be globally Lipschitz continuous [1]. However, most SDE models in real life do not obey the global Lipschitz condition. It was in this spirit that Higham, Mao and Stuart in 2002 published a very influential paper [2] (Google citation 319) which opened a new chapter in the study of numerical solutions of SDEs---to study the strong convergence question for numerical approximations under the local Lipschitz condition.

Since the classical explicit Euler-Maruyama (EM) method has its simple algebraic structure, cheap computational cost and acceptable convergence rate under the global Lipschitz condition, it has been attracting lots of attention.

Although it was showed that the strong divergence in finite time of the EM method for SDEs under the local Lipschitz condition, some modified EM methods have recently been developed these SDEs. For example, the tamed EM method was developed in 2012 to approximate SDEs with one-sided Lipschitz drift coefficient and the linear growth diffusion coefficient. The stopped EM method was developed in 2013. Recently, Mao [3] initiated a significantly new method, called the truncated EM method, for the nonlinear SDEs. The aim of this PhD is to develop the truncated EM method. The detailed objectives are:

(1) To study the strong convergence of the truncated EM method in finite-time for SDEs under the generalised Khasminskii condition and its convergence rate.

(2) To use the truncated EM method to investigate the stability of the nonlinear SDEs. Namely to study if the numerical method is stochastically stable when the underlying SDE is stochastically stable and to study if we can infer that the underlying SDE is stochastically stable when the numerical method is stochastically stable for small stepsize.

A PhD studentship might be available for the project.

References:

[1] Mao X., Stochastic Differential Equations and Applications, 2nd Edtion, Elsevier, 2007.

[2] Higham D., Mao X., Stuart A., Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40(3) (2003), 1041--1063.

[3] Mao X., The truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 290 (2015), 370--384.

Cholesteric liquid crystals are chiral systems which possess a spontaneously formed helical structure with the pitch in micron range which is important for various applications in optics and nanophotonics. In recent years the interest has shifted in the direction of lyotropic cholesterics which are the solutions of various chiral macromolecules, viruses or chiral nanocrystals. These systems are important for biology (for example, cholesteric states of DNA) and also provide some very useful natural anisotropic chiral materials.

Among the most interesting resources to explore are cellulose and chitin, key biopolymers in the plant and animal world, respectively. Both have excellent mechanical properties and  can be extracted as nanorods with high degree of crystallinity.Both are also chiral. Molecular chirality of such nanorods is amplified into a helically modulated long-range ordered cholesteric liquid crystal phase when they are suspended in water.

The aim of this project is to develop a molecular-statistical theory of chirality transfer in cholecteric nanorod phase, determined by steric and electrostatic chiral interactions, and quantitatively describe the variation of helix pitch as a function of rod length, concentration, dispersity and temperature. The theory will be built upon the previous results, obtained for different cholesteric liquid crystals ((see some references to our work below).

The project will include a collaboration with two experimental group at the University of Luxembourg and the University of Stuttgart.  These groups have an enormous expertise in the field of lyotropic liquid crystals.

[1] Honorato-Rios, C., Lehr, C., Sch¨utz, C., Sanctuary, R., Osipov, M. A., Baller, J. and Lagerwall, J. P.F.., Fractionation of cellulose nanocrystals: enhancing liquid crystal ordering with- out promoting gelation, Asia Materials, 10, 455–465 (2018).

[2]. Dawin, Ute C., Osipov, Mikhail A. and Giesselmann, F. Electrolyte effects on the chiral induction and on its temperature dependence in a chiral nematic lyotropic liquid crystal J of Phys. Chem. B, 114 (32). 10327-10336 (2010)

[3] A. V. Emelyanenko, M. A. Osipov and D. A. Dunmur, ]  Molecular theory of helical sense inversions in chiral nematic liquid crystals Phys. Rev. E, 62, 2340 (2000)

Elastic constants of nematic liquid crystals, which describe the energy associated with orientational deformation of such anisotropic fluids, are among the most important parameters for various applications of liquid crystal materials. The elastic constants of nematic liquid crystals have been well investigated both experimentally and theoretically in the past. During the past decade a number of novel liquid crystals materials with unconventional molecular structure have been investigated and it has been found that these systems are characterised by the anomalous values and behaviour of the elastic constants. In particular, it has been shown that in the nematic phase exhibited by the so-called V-shaped bent-core liquid crystals two of the three elastic constants decrease nearly to zero with the decreasing temperature. This behaviour is still very poorly understood.

It should be noted that bent-core liquid crystals attract a very significant attention at present because they also exhibit a number of unusual novel phases with a nanoscale helical structure. It is now generally accepted that a transition into these unusual phases may be driven by the dramatic reduction of the elastic constants.

The aim of this project is the generalise the existing molecular-statistical theory of elasticity of nematic liquid crystals to the case of bent-core nematics, composed of biaxial and polar molecules, using the preliminary results obtained in recent years (see, for example, the references to some of our recent papers given below). Another aim is to explain the existing experimental data on the temperature variation of the elastic constants of bent-core nematic liquid crystals.

The project will include a collaboration with the experimental group at the University of Leeds and the theoretical group from Russian Academy of Sciences.  These collaborations are very important for  the success of the project.

[1] M. A. Osipov and G. Pajak, Effect of polar intermolecular interactions on the elastic constants of bent-core nematics and the origin of the twist-bend phase, The European Physical Journal E 39, 45 (2016).

[2] M. A. Osipov and G. Pajak, Polar interactions between bent-core molecules as a stabilising factor for inhomogeneous nematic phases with spontaneous bend deformations, Liquid Crystals 44, 58 (2016).

[3] S. Srigengan, M. Nagaraj, A. Ferrarini, R. Mandle, S.J. Cowling, M.A. Osipov, G. Pająk, J.W. Goodby and H.F. Gleeson, J. Mater. Chem. C, 2013, 6, 980

We wish to develop innovative methods for modelling high-dimensional time series. Practical time series data, including both continuous-valued and discrete-valued data such as climate record data, medical data, and financial and economic data, are used for empirical analysis.

Models for forecasting multivariate conditional mean and multivariate conditional variance (volatility) are concerned. Techniques for dimension reduction, such as dynamic factor analysis, are used.

The estimation of models for panel data analysis and the option valuation with co-integrated asset prices is discussed.

Nowadays people often meet problems in forecasting a functional. A functional may be a curve, a spatial process, or a graph/image. In contrast to conventional time series analysis, in which observations are scalars or vectors, we observe a functional at each time point; for example, daily mean-variance efficient frontiers of portfolios, yield curves, annual production charts and annual weather record charts.

Our goal is to develop new models, methodology and associated theory under a general framework of Functional Time Series Analysis for modelling complex dynamic phenomena. We intend to build functional time series models and to do forecasting.

When the true economic system consists of many equations, or our economic observations have a very high dimension, one may meet the ``curse of dimensionality" problem. We try to impose a common factor structure to reduce dimension for the parametric and nonparametric stability analysis of a large system. Replacing unobservable common factors by principle components in parametric and nonparametric estimation will be justified.

In contrast to conventional factor models which focuses on reducing dimensions and modeling conditional first moment, the proposed project devotes attention to dimensional reduction and statistical inference for conditional second moments (covariance matrices). The direct motivation lies in the increasing need to model and explain risk and uncertainty of a large economic system.

The other distinctive point is that the proposed project considers factor models for high frequency data. A key application is the analysis of high dimensional and high-frequency financial time series, although the potential uses are much wider.

Many fish populations worldwide have been heavily exploited and there is accumulating evidence from both observational and theoretical studies that this harvesting can induce evolutionary changes. Such responses can affect the stock sustainability and catch quality, and so there is a recognized need for new management strategies that minimise these risks. Most results suggest that high mortality on larger fish favours early maturation.

However, recent theoretical work has shown that trade-offs between growth and maturation can lead to more complex evolutionary responses. Surprisingly, harvesting large fish can select for either late or early maturation depending on the effect of maturation on growth rate. To date most theoretical studies have used evolutionary invasion analyses on simple age-based discrete-time models or on continuous-time coupled ODE representations of size structure.

In common with many generic models of fish population dynamics, population control occurs by unspecified density-dependence at settlement. While these simplifications carry the advantage of analytical tractability, the analysis assumes steady state populations. This, together with the stylised life-histories precludes comparing model results with field data on secular changes in size-distributions an sexual maturity.

The work in this project will develop a new generation of testable model for fisheries-induced adaptive changes with the potential to inform future management decisions. This will involve developing a consumer-resource model which a length-structured fish population feeding on a dynamic biomass spectrum.

Differently-sized fish will compete for food by exploiting overlapping parts of the food size spectrum. The population will be partitioned by length at maturity, and this will be the heritable trait under selection. The model will be used to explore how changes in mortality and food abundance affect the evolutionarily stable distribution of maturation lengths.

Comparisons with survey data on North Sea demersal fish will be used to assess whether the historical harvest rates are sufficient to explain growth rate changes as an evolutionary response. Finally the evolutionarily stable optimal harvesting strategies will be identified.

In recent years there has been an explosive growth of interest in the behaviour and control of fluids at small (typically sub-millimetre) scales motivated by a range of novel applications including ink-jet printing and lab-on-a-chip technologies.

Much of the most exciting current research concerns the interaction between fluids and both rigid and flexible structures at small scales, and so the aim of the present project is to use a judicious combination of asymptotic methods and judiciously chosen numerical calculations to bring new insight into the behaviour of a variety of novel fluid-structure interaction problems in microfluidics.

In the last decade or so there has been an explosion of interest in droplet evaporation, driven by new technological applications as diverse as crop spraying, printing, cooling technologies such as heat pipes, and DNA micro-array analysis.

One particularly interesting aspect of this problem which has thus far received relatively little attention is that of fluids whose surface tension exhibits a local minimum with temperature, known as self-rewetting fluids, a property that can have a profound effect on the dynamics of droplets on heated substrates.

The aim of the project is to build on the existing literature on conventional surface-tension-gradient driven spreading and droplet drying (see, for example, the references to some of our recent work on these problems given below) to bring new physical insight into this challenging scientific problem, and hence to harness the novel properties of self-rewetting droplets in a range of applications.

The project will be a collaboration colleagues at the University of Edinburgh who will be undertaking a parallel series of experimental investigations on this problem which will be key to the successful outcome of the project.

Dunn, G.J., Wilson, S.K., Duffy, B.R., David, S., Sefiane, K. “The strong influence of substrate conductivity on droplet evaporation” J. Fluid Mech. 623 329-351 (2009)

Dunn, G.J., Duffy, B.R., Wilson, S.K., Holland, D. “Quasi-steady spreading of a thin ridge of fluid with temperature-dependent surface tension on a heated or cooled substrate” Q. Jl. Mech. appl. Math. 62 (4) 365-402 (2009)