I am always on the lookout for good PhD students to supervise. Please contact me if you are interested in this possibility. You can apply for both funding and places through the Sheffield School of Maths and Stats, where there is information on funding. Projects could be in any area of science related to my work, and don't need to be confined to the specific projects below. However, here are a few ideas...

 

Fight, Flee, Feed, or Reproduce? Determining when and why animals perform their necessary tasks

 
    For animals to survive and pass their genes on to the next generation, they must balance their time between a variety of tasks. These are broadly characterised by the "Four F's", but specific tasks differ amongst species, populations, and seasons. For example, a male fox with young cubs may need to split his time between territorial defence (Fight) and finding food for himself and his family (Feed). Prior to having cubs, the requirement for finding food may be lower, allowing more time for seeking a mate (Reproduce) and the inevitable conflict that arises from this (Fight, Flee). As another example, small birds may constantly be needing to either feed or escape predation, eliciting switching behaviours throughout the day.

    The task of this project is to develop generic techniques for detecting such behavioural switches from data on the animals' movement with respect to its landscape. So if an animal is defending its territory, you might expect to find it moving towards, or along, the territory border. When it switches to a foraging mode, it would be likely move towards food-rich areas. Statistical techniques to detect such behavioural switches are likely to require combining Markov chain models with mathematical functions describing selection of resources, in a coherent and efficient way. The student will develop such techniques and use them to investigate strategies used by a variety of different animals, which could range from foxes in UK cities to Amazonian birds to Canadian caribou.

    After this initial stage, the project could go in a variety of directions depending on the student's interests. One direction would be to use the resulting models to determine optimal behavioural-switching strategies. Are the animals behaving optimally as evolutionary theory might suggest? If not, why not? Another would be to investigate extensions and generalisations of the statistical techniques for use in broader contexts, beyond ecology.

 

The mathematics of territorial pattern formation

 
    Territoriality is a widespread phenomenon in the animal kingdom. Various species of birds, mammals, reptiles, fish, and more, seek to claim parts of the terrain for exclusive use, either by themselves or their group. At one scale of description, territoriality can be viewed as a set of behavioural decisions, such as scent-marking or visual cues, that serve to exclude other animals of the same species (known as conspecifics). On a much larger spatio-temporal scale, territoriality is observed as patterns of space use: separate areas that roughly tessellate the terrain.

    This two-scale definition begs the question: can we prove mathematically that certain behavioural mechanisms give rise to territorial patterns? Further, what sort of mechanisms are necessary and/or sufficient for producing these patterns?

    Many questions about pattern formation have been successfully studied from the perspective of both individual-based models and their partial differential equation approximations. The last two decades have seen the problem of territory formation successfully attacked using such techniques. However, they have hitherto mainly been concerned with scent-marking processes. It remains an open question to prove mathematically whether processes other than scent marking may be sufficient for territory formation.

    The initial purpose of this project will be to model various biologically observed territorial behaviours, other than scent-marking, and investigate the conditions under which territorial patterns may emerge. These could include territorial `battles' or display events, or sharing and competing for resources. The project could go in a variety of directions from here, depending on the interests of the candidate. One could be to explore related concept of `home range' patterns, where animals spend most of their time in confined areas, but without deliberate exclusion of others. Another could involve extending techniques to spatial pattern formation in other competitive interactions, such as predator-prey. A third could examine the evolutionary benefits of territoriality: under what circumstances should we expect territorial behaviour to evolve?