Stabilizing a homoclinic stripe

Abstract

For a large class of reaction-diffusion systems with large diffusivity ratio, it is well known that a two-dimensional stripe (whose cross-section is a one-dimensional homoclinic spike) is unstable and breaks up into spots. Here, we study two effects that can stabilize such a homoclinic stripe. First, we consider the addition of anisotropy to the model. For the Schnakenberg model, we show that (an infinite) stripe can be stabilized if the fast-diffusing variable (substrate) is sufficiently anisotropic. Two types of instability thresholds are derived: zigzag (or bending) and break-up instabilities. The instability boundaries subdivide parameter space into three distinct zones: stable stripe, unstable stripe due to bending and unstable due to break-up instability. Numerical experiments indicate that the break-up instability is supercritical leading to a 'spotted-stripe' solution. Finally, we perform a similar analysis for the Klausmeier model of vegetation patterns on a steep hill, and examine transition from spots to stripes.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'.

Keywords: pattern formation; reaction–diffusion systems; stability of patterns.

Figure 1.

Top: Boundaries for zigzag and break-up instabilities in the d–b parameter plane with ε = 0.025, (x, y)∈ [ − 1, 1] × [0, 1], A = 1. Note a good agreement between asymptotics and numerics. Rows A–C show snapshots of full numerical simulations of (1.1) in different parameter regions. Parameters A are in the stable region. Parameters B: stripe exhibits a break-up instability. Parameter C: The stripe is stable with respect to break-up instability, but unstable with respect to a zigzag instability. As a result, the stripe starts to bend (note the slow time scale). Eventually, this is followed by a break-up of a bended stripe. (Online version in colour.)

Figure 2.

Simulation of (5.1) with ε = 0.05, A = L = 1, c = 10, and with d as follows: (a) d = 1.8ε; (b) d = 1.3ε; (c) d = 1.2ε; (d) d = 1.2ε; (e) d = 1.3ε. Initial conditions are close to those shown in the first column.

Figure 3.

f(μ) versus μ.

Figure 4.

Simulation of (1.1) on a square 2 × 2 domain with ε = 0.025, A = 1 and with d and b as indicated. For each value of (d, b), the eventual steady state is shown (at t = 10⁷). As anisotropy is decreased, first, the stripe bifurates into a spotted stripe, then the spotted stripe breaks up resulting in a more uniform spot distribution.