Explicitly Solvable Nonlocal Eigenvalue Problems and the Stability of Localized Stripes in Reaction‐Diffusion Systems

Abstract

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 crosssectional 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 non self-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 wavenumbers. 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 Gierer-Meinhardt model and for a three-component RD system modeling patterns of criminal activity in urban crime.

Existence, Stability, and Dynamics of Ring and Near-Ring Solutions to the Saturated Gierer--Meinhardt Model in the Semistrong Regime

“…We seek a homoclinic orbit solution of this problem that tends to the fixed point v 0 = 0 as |η| → ∞. A general result for the existence of such a homoclinic solution was first proved in Theorem 5 of [4] (see also [25]). The result is as follows: Lemma 2.1.…”

“…[17]), and for ring solutions with a particular exponent set (cf. [25]), that a certain class of eigenfunctions can lead to breakup instabilities for which the numerical solutions to (2.13) no longer characterize the slow dynamics of an interface. As such, it is important to investigate when instabilities of this nature could arise.…”

“…For the two-dimensional GM model, straight stripe patterns in rectangular domains have been studied (cf. [17], [25], [10]). In higher dimensions, the existence of steady-state ring solutions in N -dimensional radially symmetric domains was investigated in [29], without any stability analysis.…”

“…In certain special cases, the analysis of the NLEP can be reduced to the much simpler problem of analyzing an explicit transcendental equation for the eigenvalue parameter (cf. [25], [28]). …”

“…In higher dimensions, the existence of steady-state ring solutions in N -dimensional radially symmetric domains was investigated in [29], without any stability analysis. Ring solutions were analyzed in [25], [20], [21] and the linear stability analysis of such solutions to a breakup instability, which triggers the disintegration of a ring into localized spots, relied on studying the spectrum of a nonlocal eigenvalue problem (NLEP). In certain special cases, the analysis of the NLEP can be reduced to the much simpler problem of analyzing an explicit transcendental equation for the eigenvalue parameter (cf.…”