## Potential student projects

### Marine ecosystems and fishing

Multi-species communities are often described by food webs, which focus on species as the key variable determining what an individual eats. But in marine ecosystems, fish can grow by several orders of magnitude during their lives, and their diet changes as they do so. This means that body size can be more important than species identity in describing predator-prey interactions. Size-specturm models focus on body size as a key variable and keep track of how biomass flows from prey to predator through mortality and growth. These models take the form of a system of nonlinear integro-partial differential equations. This project will use size-spectrum models to investigate the dynamics of multi-species communities. These models will also be used to investigate the effect of different fishing strategies on key outcomes such as fisheries yield and impact on ecosystem function and community structure.### Random walks of building blocks

Populations of cells - sometimes called the building blocks of life - can be modelled by random walks, where cells take a sequence of steps in randomly chosen directions. If the cells all move independently, the population density follows the heat equation. But in reality, cells interact with one another via physical contacts and chemical signals. Their movement can also be impeded by obstacles in the form of extracellular material. These interactions can be included in random walk models. This project will investigate the behaviour of the cells in these models and how it affects the diffusive characteristics of the population as a whole.### The wisdom and madness of crowds

Groups of people can often come up with better solutions to problems than any one individual on their own could. But groups can also get carried away by the spread of bad ideas, e.g. the global financial crisis, or the spread of fake news. These processes can be represented by social networks: people and the connections between them - see this interactive tutorial by Nicky Crase. This project is about mathematical models of the spread of an idea or behaviour through a social network. The aim is to investigate how the network structure (e.g. clustering, degree distribution, small-world) affects the speed of transmission through the network and the success or failure of the contagion to take over the network.### Collective cell behaviour

Collective cell behaviour is the driving force behind many physiological processes, including embryonic development, tissue repair and tumour growth. Experiments on collective cell behaviour typically collect data at the level of the population rather than the individual cell. We'd like to be able to translate data from observing populations of cells into knowledge about how individual cells work and how they interact with their neighbours. This project will approach this problem using approximate Bayesian computation (ABC). At its simplest, this involves sampling model parameters from a prior distribution and simulating cell behaviour. If the model output is "close" to the experimental data, the parameter values are accepted as part of the posterior distribution, otherwise they are rejected. This will be used to estimate quantities such as cell proliferation and movement rates and the strength of interactions with neighbouring cells.### Spatial moment dynamics for invasive populations

Spatial moments are a way of describing the spatial distribution of a population that accounts for correlations between individual locations, e.g. whether individuals tend to occur together in clusters or tend to be spaced apart at regular intervals. Classical models based on average population density cannot distinguish between these and other cases, but they can crucially affect population dynamics. This project will use spatial moment dynamics to model an invasive population and investigate the two-way interplay between spatial structure and macroscopic outcomes such as population size and invasion speed. These models use a combination of individual-based stochastic models and an integro-partial differential equation for the second spatial moment coupled with an appropriate moment closure approximation.
Michael Plank, Room 614 Erskine Building, School of Mathematics and Statistics, University of Canterbury, Christchurch 8140, New Zealand

Tel: +64 3 3692462

Email: michael.plank@canterbury.ac.nz