Applications are invited for an 23-month Postdoctoral Fellowship at the University of Auckland for the project *Spiral waves and heteroclinic cycles in Rock–Paper–Scissors*. This research is funded by a Faculty Research Development Fund grant and will be supervised by Associate Professor Claire Postlethwaite and Professor Hinke Osinga in the Mathematics Department. Professor Alastair Rucklidge (University of Leeds, UK) will collaborate on the project. The Fellowship is available from January 2018; the exact start date will be negotiated with the successful applicant.

**Project description**

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.

The proposed project is 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. Computations will be done using bifurcation theory techniques and advanced numerical analysis. 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.

The succesful applicant will hold a PhD in Mathematics or Applied Mathematics and will ideally have demonstrable skills in the following:

- Dynamical systems theory;
- Numerical analysis, particularly numerical continuation and boundary value problems;
- Simulation of partial differential equations;
- Programming skills in Matlab, Fortran or similar.

Informal enquiries are welcome by email, to either Claire Postlethwaite

Applications should include a cover letter, CV, and names and email addresses of three academic referees who can be contacted in support of your application. Applications should be sent by email to the above email addresses; those received before 31 October 2017 will receive full consideration.

Two PhD positions are available from 1 March 2017 on the Marsden grant *Understanding the geometry of dynamics: invariant manifolds and their interactions*. Interested candidates are encouraged to apply.

The scholarship provides student fees and support for a PhD student to investigate the geometry of dynamical systems, which will be supervised by Professors Bernd Krauskopf and Hinke Osinga. Specifically, the goal is to explain novel kinds of chaotic dynamics and intricate oscillatory patterns formed by a mix of large and small amplitudes. Such dynamics are found, for example, in cell models, pulsating lasers, chemical reactions, and bursting neurons. The emphasis is on generic phenomena that capture the essence of the dynamical behaviour. The student will consider representative model systems and investigate how the phase space is organised by smooth surfaces known as invariant manifolds: their interactions are the key to understanding the overall behaviour in this context.

Students from all nationalities with a suitable higher degree in Mathematics or Applied Mathematics are eligible for these PhD positions. We are particularly interested in students with demonstrable skills in some of the following:

- dynamical systems theory;
- numerical analysis with emphasis on bifurcation theory and path following;
- differential geometry;
- topology;
- computer graphics and visualisation;
- programming in Fortran or C-related language (Python);

Informal inquiries are welcome by email.

**Contact:** Professor Hinke Osinga:

Hinke Osinga

Last modified: Mon Oct 2 10:18:14 BST 2017