Research

I have research interests in two distinct areas: applied dynamical systems, and mathematical models of animal behaviour. Current projects are outlined below.

Dynamical systems

Heteroclinic networks as models for neural processes

I am interested the stability and bifurcations of heteroclinic cycles and heteroclinic networks in non-linear dynamical systems. Such cycles are generically of high codimension but in the presence of symmetry they can be found to be robust, that is, they exist in an open region of parameter space. Recently I have been working with Peter Ashwin at the University of Exeter in the UK investigating methods for using heteroclinic networks as models of neural processes.

Heteroclinic cycles in spatially extended systems

Scissors cut Paper, Paper wraps Rock, Rock blunts Scissors: the simple game of Rock–Paper–Scissors provides an appealing model for cyclic dominance between competing populations or strategies in evolutionary game theory and biology. The model has been invoked to explain the repeated growth and decay of three competing strains of microbial organisms and of three colour-morphs of side-blotched lizards. In a mathematical model of three competing species which allows for spatial distribution and mobility, waves of Rock can invade regions of Scissors, only to be invaded by Paper in turn; these waves can be organised into spirals, with roughly equal populations of the three species at the core of each spiral, and each species dominating in turn in the spiral arms. In collaboration with Alastair Rucklidge at the University of Leeds in the UK, we are trying to further our understanding of the spiral patterns in the spatially-extended model, in particular, to understand what determines the wavelength, rotation speed and stability of the spiral waves. If successful, we predict that the methods of analysis developed for the Rock-Paper-Scissors example could be amenable to a wide variety of physical models, including heart ventricular fibrillation, excitable chemical systems and reaction-diffusion systems.

Delay equations and feedback control

The stabilisation of unstable periodic orbits using feedback control has attracted the attention of many authors over a number of years. The time-delayed feedback method of Pyragas has been of particular interest. I have collaborated with Mary Silber (University of Chicago) and Bernd Krauskopf on a number of aspects of Pyragas control, and also with Bernd have investigated the effect of time-delays on models of climate.

Mathematical models of animal behaviour

The Multi-Scale Straightness Index (MSSI)

Together with Todd Dennis (Biology, Auckland), I developed a new method for analyzing animal tracking data. This method is simple to implement and utilizes data on a range of spatial and temporal scales, and can be used to classify movement data. For an online version (in development) of the script to compute the MSSI, please see the MSSI page.

Models of pigeon navigation

How migrating animals find their way over long distances remains one of the great, unanswered questions facing biologists today. Despite intensive research for over 60 years, there has been no convincing explanation of the mechanisms animals use for determining their position relative to a target location. My research in this area combines ideas from both mathematics and behavioural ecology, and with colleagues from the School of Biology Sciences, I hope to develop a new mathematical model of animal navigation. In 2011, I was awarded Marsden Fast Start grant for this research. Due to their ease of handling and willingness to home, homing pigeons have long been the experimental model for the study of animal navigation. Our research involves the development of geometric techniques used to explain an observed 'orientation error' in the initial homing directions of pigeons. We will then develop a predictive mathematical model for how animals navigate over long distances. These results will be applicable to a wide variety of migratory species. We expect that our results will explain how birds such as godwits can fly non-stop from Alaska to New Zealand, which requires locating a target only 2-3 degrees wide when migration begins.