Wave propagation in excitable media with evolving boundaries

This project will identify existence and stability conditions for travelling waves in nonlinear PDEs with time-dependent domains. Travelling waves are a common modality for transporting signals in biological systems. In many scenarios, such as those observed in developmental biology or tumour growth, the domain over which the transport takes place evolves over time, either due to domain growth, or to re-arrangement of the tissue. Whilst there exists a number of studies of pattern formation on growing domains, there is paucity of results on how temporal evolution of the domain shape affects wave propagation.

This project aims to address this gap by finding conditions under which waves can be established in nonlinear PDE models with evolving domain shapes. Travelling pulse solutions in PDE models may be understood as homoclinic connections, a perspective that facilitates the construction of dispersion curves that link wave properties, such as speed, to properties of the underlying dynamics. We have recently computed stability conditions for travelling pulses in non-locally coupled, excitable PDE models posed over infinite one-dimensional domains and now seek to expand these results to a finite but evolving domain.

The project will identify existence and stability conditions for propagating solutions under the approximation that propagation takes place on a faster timescale than the domain growth, consistent with many biological systems. This assumption will facilitate a mixed-timescale analysis of the system so that the changes to the profile and speed of the wave as the domain evolves can be understood by studying the system on the slower timescale. The mathematical analysis in this project will be linked to the developing zebrafish embryo (in collaboration with Dr Steffen Scholpp, Bioscience; a prototypical system in developmental biology that is commonly used as an exemplar of a system with non-local signalling.