Spatiotemporal dynamics and localized patterns after subcritical bifurcation

Develop a complete analysis of the spatiotemporal dynamics of the amplitude equations derived for bulk-surface reaction-diffusion systems in a ball, including identifying and characterizing analogs of localized patterns that may arise following subcritical Turing bifurcations.

Background

Beyond steady-state branches and primary bifurcation behavior, the paper highlights the potential for rich spatiotemporal phenomena in the amplitude equations, including localized structures analogous to those known in planar reaction-diffusion settings when bifurcations are subcritical.

The authors indicate that a complete paper of such dynamics—encompassing pattern selection, metastability, and possible localized states—remains to be carried out, motivating focused investigation of these O(3)-equivariant normal forms in the bulk-surface context.