Student projects

Below is a list of abstract of projects available for potential students. Please contact me if you are interested in working on any of them.

Travelling waves and spirals in models of non-transitive competition

Co-supervised with Hinke Osinga. Suitable for PhD students.

Interactions between plant and animal species have long been known to cause oscillations, or cyclical behaviour, in species population densities. We are interested in particular in so-called intransitive competition between species. Here, each species has both advantageous and disadvantageous traits, and no single species dominates. This type of competition, exemplified by the game of Rock-Paper-Scissors, is thought to increase biodiversity, as no single species is the “best” competitor. Mathematical models of such competition can contain objects called heteroclinic networks, close to which the dynamics of the system are intermittent. The three species, spatially-homogeneous model (“well-mixed” Rock-Paper-Scissors) is well understood, but the addition of spatial dependencies, anisotropies, or additional species results in further complexity. For instance, allowing the populations to diffuse in space can lead to travelling waves, spiral waves and complex dynamics in a partial differential equation model.

The PhD research will focus on the analysis of travelling waves and spiral waves in small competition networks with spatial dynamics. The ideal PhD student will be able to combine cutting-edge mathematical analysis with advanced numerical techniques in order to make predictions about the solutions, such as wavespeeds and wavelengths of travelling waves, or stability of different sizes of spiral waves.

Further details on how to apply for this position can be found here.

Chaotic switching in heteroclinic networks

Suitable for MSc students.

Heteroclinic networks are special solutions to dynamical systems consisting of a set of states (frequently equilibria) connected by trajectories. Near an attracting heteroclinic network, the dynamics are typically intermittent, with the trajectory spending long periods of time near the states in the network, with very rapid transitions between them. The order in which the states of the network are visited may follow a periodic sequence, or it may be irregular or chaotic, even when the governing system is deterministic.

This project will investigate the behaviour and properties of trajectories which approach heteroclinic networks in an irregular fashion, sometimes known as chaotic switching. Research questions may include: (i) what are the statistical properties of the sequence of states visited, and are they stationary in time? (ii) can such trajectories appear to be Markov, and under what circumstances? (iii) is the sequence of states visited ergodic? (i.e. do statistical properties of the sequences of states behave the same in time as over different initial conditions?).

Evolutionary algorithms and Turing Machines

Co-supervised with Matthew Egbert (Computer Science). Suitable for BSc Hons or MSc students.

In recent work, we have developed a method to implement any specified Turing machine (TM) using a continuous-time recurrent neural network (CTRNN). The resulting dynamics are attracted to an invariant set which shares many properties with a heteroclinic network. The aim of this project is to investigate what happens when the parameters of such a system are allowed to evolve via a genetic algorithm. In which cases is the pre-designed architecture a local optimum to solving some particular task? Does the evolved system lose the “heteroclinic-like” structures in the designed system?

This project would involve (i) setting up the designed Turing machine for solving a particular task (e.g. a ‘copy’ routine); (ii) using a genetic algorithm and appropriate fitness function to evolve the parameters of the system in (i) to determine if it is a local optimum; (iii) setting up a generic CTRNN controller to solve the same task, again evolving the parameters to solve the same problem; (iv) comparing the outputs of (i), (ii) and (iii) — including deciding how best to do this comparison.

Does including delay in CTRNN-based Evolutionary Robotics facilitate the evolution of complex control architectures?

Co-supervised with Matthew Egbert (Computer Science). Suitable for MSc students.

Evolutionary Robotics (ER) typically define control architectures as a system of ODEs (e.g. CTRNN). In previously published work we have shown that delay can play a functional role, i.e. can be used by evolved strategies as part of a dynamical solution to a cognitive task. In this project we will do comparative study, to try to establish (i) if delay based controllers (CTRNN + delay) DDE systems can solve classic ER tasks (e.g. Beer's categorical perception and relational categorization studies); (ii) if so, how do the evolved solutions work? and (iii) is the evolutionary search for functional solutions faster or slower than classic evolutionary robotics CTRNN/ODE methods?

Models of interactions between artificial intelligence and the environment

Co-supervised with William Godsoe (Lincoln University). Suitable for BSc Hons or MSc students.

Decision making for environmental issues is increasingly being performed by computer generated predictions from artificial intelligence (AI) algorithms. Traditionally these algorithms have been framed narrowly as neutral, black-box statistical methods, and predictions are often applied to living things, such as crops, in forestry, or to pests or diseases for their management. There are increasing concerns that this narrow focus is leading to perverse outcomes, in part due to un-modelled feedback loops between the AI and the environment it is being used in.

This project will involve the development and study of a toy mathematical model of the interactions between an AI and a simple ecological system. The ecological model could be set up with various levels of complexity, such as ODEs, PDEs or agent-based models, depending on the background and interests of the student.

Higher order interactions in ecological modelling

Suitable for BSc Hons students.

Interactions between species, such as predator-prey or competition, are often modelled using simple Lotka-Volterra type differential equations. Such models are an idealisation, and ecologists are interested in the effect of higher-order interactions. This project will use dynamical systems methods to investigate and classify the qualitative effects of higher-order interactions in two- and three-species models.