I'm always on the look-out for good PhD students to supervise. You can apply for both funding and places through the Sheffield School of Maths and Stats, where there is information on funding requirements and eligibility. Please feel free to contact me for informal enquiries. Projects could be in any area of science related to my work. Typically, I am looking for students from mathematical backgrounds who have a demonstrable interest in ecology. However, for the right project I would also consider a biology graduate with exceptional analytic skills. Below are a couple of example projects, to give you an idea of my interests.

Spatial predator-prey dynamics in the landscape of fear

 
    The mathematical study of predator-prey dynamics is one of the oldest subjects in mathematical biology. In a spatial setting, understanding predator-prey interactions is important for a variety of ecological issues, such as maintaining biodiversity and establishing sound conservation principles. Mathematical models have played a vital role in this understanding for many decades. However, many questions remain unanswered.

    This project will examine the effect of the `landscape of fear' on spatial population dynamics. As well as affecting the number of animals in a population, the existence of predators is a spatial deterrent, meaning that prey will avoid areas of the landscape where they believe predators are likely to be present. Meanwhile, predators will be actively seeking-out areas of the landscape where they expect to find prey. Recent empirical research, from both ecology and neuroscience, suggests that animals build up a cognitive map of their environment which they use to navigate the terrain, seeking-out areas that may be beneficial to them and avoiding those that will not.

    The candidate will use mathematical models to understand (a) how the landscape of fear emerges, (b) what it might look like under different ecologically-motivated scenarios, and (c) how it affects the demographic dynamics and ultimately the survival of each species. Ecologically, this will be motivated by various study systems that range from caribou-wolf interactions to orca-narwhal systems. However, we will always seek out general lessons that may be applicable across taxa.

    Mathematically, we will study systems of reaction-diffusion-taxis equations, where the `reaction' term models birth and death of animals, the `diffusion' term models aspects of movement that are either unknown or not explicitly modelled, and `taxis' is movement away from undesirable areas towards more desirable ones. These will be coupled through ordinary differential equations which model the extent to which a part of space is considered desirable or undesirable by the species at any point in time. The candidate will make use of a variety of tools from spatial pattern-formation analysis, building on recent studies of cross-diffusive systems, which have many mathematical similarities to the systems studied here.

Modelling disease spread through populations of territorial animals and the effect of culling

 
    (Co-supervised by Dr Alex Best from the School of Maths and Stats at the University of Sheffield)

    The culling of animals in an attempt to slow or halt disease spread is a controversial method, yet frequently adopted by government agencies in the UK and beyond. In particular, the recent badger culls in the UK have generated significant controversy and attention. The central argument is that, by removing a proportion of the wild animals (e.g. badgers) that transmit a disease, this disease will be less likely to spread to humans and/or livestock. However, scientific evidence typically points towards it being an ineffective strategy.

    Indeed, it has been hypothesised that culling of territorial animals can make disease spread worse, by something called a `perturbation effect', whereby culling causes territorial and social structures to break down. This may result in animals moving longer distances and therefore having a greater probability of encountering other animals. Although this is unlikely to happen if the culling is extreme (the mathematical limit being extinction), it suggests that moderate culling efforts may result in increased disease spread.

    The aim of this project is to test this hypothesis from a mathematical modelling perspective. The candidate will build upon recent models of territory formation, constructed by the supervisor and collaborators, to (a) incorporate first the effect of disease spread under different classical disease-model scenario, then (b) examine the effect of culling strategies on the spreading speed. The initial approach will be to use a stochastic individual-based modelling approach to examine these questions. Then the candidate will construct partial differential equation approximations of this model, making use of travelling wave theory to gain an analytic understanding of the effect of culling. The paradigmatic system will be European badgers (Meles meles), but we will aim to make the models as general, and generalisable, as possible.