Stripe or ring patterns have been observed in many numerical simulations of singularly perturbed reaction-diffusion systems. They occur for activator-inhibitor systems such as the well-known Gierer-Meinhardt model of biological morphogenesis, for certain hybrid chemotaxis reaction-diffusion systems modeling fish skin patterns on growing domains, and for the Gray-Scott system of theoretical chemistry. In many instances a stripe or ring pattern is unstable to a breakup instability, which leads to the disintegration of the stripe or ring into a sequence of spots. In other cases, a stripe is de-stabilized by a transverse or zigzag instability, leading to a wriggled stripe. In certain cases, this wriggled stripe is the precursor to a complicated space-filling labyrinthian pattern. Most previous studies of this phenomenon are based on a weakly nonlinear theory near some spatially uniform steady-state. In contrast, the stripes that we consider are typically localized along some planar curves and have cross-sections that deviate substantially from the background state. The cross-sections are either homoclinic or front-back transition solutions of certain ODE systems. Our analysis of stripe stability involves a combination of singular perturbation theory, the spectral theory of nonlocal eigenvalue problems, and numerical computations. The instabilities of these localized stripes are illustrated for various reaction-diffusion models, and some general results are given for the occurrence of spot-generating instabilities, zigzag instabilities, and labyrinthian patterns.

This is joint work with Theodore Kolokolnikov, Wentao Sun, and Juncheng Wei.