Spiral Waves and Target Patterns

Many biological, chemical, and physical processes exhibit spiral waves and target patterns. Spiral waves occur, for instance, in autocatalytic chemical reactions, during cellular signaling of amoebae, and in cardiac tissue where they are often the cause of cardiac arrhythmias and the precursor to ventricular fibrillation and cardiac arrest. Similarly, target patterns appear prominently in autocatalytic reactions and are also often found in systems with small spatial inhomogeneities.

This project focuses on understanding when spiral waves and target patterns arise, under which conditions they can be observed, and what mechanisms cause them to change their shape or disappear. The investigator will develop analytical, computational, and geometric tools to answer these questions and apply the findings to models of autocatalytic chemical reactions and wave propagation in cardiac tissue. Graduate and undergraduate students will be engaged in these research activities.

This project focuses on understanding existence, multiplicity, stability, and bifurcations of spiral waves and target patterns through the development of spatial-dynamics methodologies that can be applied to a broad range of reaction-diffusion systems. The investigator will establish multiplicity results for one-dimensional spiral waves and target patterns in reaction-diffusion systems using spatial dynamics and apply these results to the Brusselator model.

The second project centers on transverse instabilities of spiral waves and target patterns. Transverse instabilities do not contribute to the spectra of planar structures, though they do cause linear instabilities. The aim is to investigate how these instabilities manifest themselves on large bounded disks.

The third project focuses on making the numerical computation of spiral spectra more robust by developing preconditioners that rely on appropriate exponential weights to reduce the norm of the resolvent independently of the size of the domain. In the last project, singular perturbation methods will be developed to understand the limits of profiles and spectra of planar spiral waves and target patterns when some of the diffusion coefficients in the model approach zero.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.