Explicitly solvable nonlocal eigenvalue problems and the stability of localized stripes in reaction-diffusion systems

The transverse stability of localized stripe patterns for certain singularly perturbed two-component reaction-diffusion (RD) systems in the asymptotic limit of a large diffusivity ratio is analyzed. In this semi-strong interaction regime, the cross-sectional profile of the stripe is well-approximated by a homoclinic pulse solution of the corresponding 1-D problem. The linear instability of such homoclinic stripes to transverse perturbations is well known from numerical simulations to be a key mechanism for the creation of localized spot patterns. However, in general, owing to the difficulty in analyzing the associated nonlocal and nonself-adjoint spectral problem governing stripe stability for these systems, it has not previously been possible to provide an explicit analytical characterization of these instabilities, including determining the growth rate and the most unstable mode within the band of unstable transverse wave numbers. Our focus is to show that such an explicit characterization of the transverse instability of a homoclinic stripe is possible for a subclass of RD system for which the analysis of the underlying spectral problem reduces to the study of a rather simple algebraic equation in the eigenvalue parameter. Although our simplified theory for stripe stability can be applied to a class of RD system, it is illustrated only for homoclinic stripe and ring solutions for a subclass of the Giererâ Meinhardt model and for a three-component RD system modeling patterns of criminal activity in urban crime.