More than Skew: Asymmetric Wave Propagation in a Reaction-Diffusion-Convection System

Abstract

Convection-induced instability in reaction-diffusion systems produces complicated patterns of oscillations behind propagating wavefronts. We transform the system twice: into lambda-omega form, then into polar variables. We find analytical estimates for the wavefront speed which we confirm numerically. Our previous work examined a simpler system [E. H. Flach, S. Schnell, and J. Norbury, Phys. Rev. E 76, 036216 (2007)]; the onset of instability is qualitatively different in numerical solutions of this system. We modify our estimates and connect the two different behaviours. Our estimate explains how the Turing instability fits with pattern found in reaction-diffusion-convection systems. Our results can have important applications to the pattern formation analysis of biological systems.

Keywords: Schnakenberg; convection; limit cycle; reaction-diffusion; travelling wave.

Figure 1

Pattern found for a diffusion system with convection and limit-cycle reaction kinetics (1). The initial disturbance propagates and becomes pronounced, forming a regular pattern with aligned oscillations. The propagation is linear, forming a V-shape. This is a numerical solution using NAG D03PCF, plotting species u with μ = 1.1, D₁ = 0.01, D₂ = 0.01, ρ = 1. The reactants are initially at steady state: (u, v) = (1/μ, μ), with a small disturbance at x = 0. The boundaries are held at zero derivative: u_x, v_x = 0.

Figure 2

Phase space for the limit-cycle reaction (3), given by numerical solution. The phase curve spirals out from the steady state to meet the limit cycle (broad loop). The Schnakenberg (a) system is highly elliptical, even at the start of the trajectory, by the steady state. The transformed Schnakenberg (b) shows a more circular behaviour initially, but the limit cycle shape remains more complicated. The parameter here is μ = 0.95. The solution was found using Matlab ode15s.

Figure 2

Phase space for the limit-cycle reaction (3), given by numerical solution. The phase curve spirals out from the steady state to meet the limit cycle (broad loop). The Schnakenberg (a) system is highly elliptical, even at the start of the trajectory, by the steady state. The transformed Schnakenberg (b) shows a more circular behaviour initially, but the limit cycle shape remains more complicated. The parameter here is μ = 0.95. The solution was found using Matlab ode15s.

Figure 3

Numerical data compared to analytical estimates. The left wave speed is close to the low parameter estimate for all parameter values. In contrast, the right wave speed is far from the low parameter estimate, and fairly close to the high parameter estimate. While the behaviour corresponds to the analysis, the behaviour of θ is different to expectations. In the left wavefront speed, the darker surface is the low parameter estimate: $\gamma /2-\sqrt{1+\epsilon }\sqrt{1-{\mu }^2}$; The lighter surface is the high parameter estimate: $\left(1-p\right)\gamma /2-\sqrt{(1+p+\left(1-p\right)\epsilon }\sqrt{1-{\mu }^2}$. In the right wavefront speed, the lighter surface is the high parameter estimate: $\left(1+p\right)\gamma /2+\sqrt{(1-p+\left(1+p\right)\epsilon }\sqrt{1-{\mu }^2}$. The similar dark surface is a less extremal estimate, with p cos ϕ = 1, giving the surface as $\gamma +\sqrt{2\epsilon }\sqrt{1-{\mu }^2}$; the mid-toned, shortened surface is the low parameter estimate: $\gamma /2+\sqrt{1+\epsilon }\sqrt{1-{\mu }^2}$. The parameter is μ = 0.95, giving $p\approx \sqrt{2}$. The points are data read from numerical solutions of the original Schnakenberg reaction-diffusion-convection equation (2) using NAG D03PCF. In the individual runs, the reactants are initially at steady state: (u, v) = (0, 0), with a small disturbance at x = 0. The boundaries are held at zero derivative: u_x = 0, v_x = 0.

Figure 3

Numerical data compared to analytical estimates. The left wave speed is close to the low parameter estimate for all parameter values. In contrast, the right wave speed is far from the low parameter estimate, and fairly close to the high parameter estimate. While the behaviour corresponds to the analysis, the behaviour of θ is different to expectations. In the left wavefront speed, the darker surface is the low parameter estimate: $\gamma /2-\sqrt{1+\epsilon }\sqrt{1-{\mu }^2}$; The lighter surface is the high parameter estimate: $\left(1-p\right)\gamma /2-\sqrt{(1+p+\left(1-p\right)\epsilon }\sqrt{1-{\mu }^2}$. In the right wavefront speed, the lighter surface is the high parameter estimate: $\left(1+p\right)\gamma /2+\sqrt{(1-p+\left(1+p\right)\epsilon }\sqrt{1-{\mu }^2}$. The similar dark surface is a less extremal estimate, with p cos ϕ = 1, giving the surface as $\gamma +\sqrt{2\epsilon }\sqrt{1-{\mu }^2}$; the mid-toned, shortened surface is the low parameter estimate: $\gamma /2+\sqrt{1+\epsilon }\sqrt{1-{\mu }^2}$. The parameter is μ = 0.95, giving $p\approx \sqrt{2}$. The points are data read from numerical solutions of the original Schnakenberg reaction-diffusion-convection equation (2) using NAG D03PCF. In the individual runs, the reactants are initially at steady state: (u, v) = (0, 0), with a small disturbance at x = 0. The boundaries are held at zero derivative: u_x = 0, v_x = 0.

Figure 4

The θ behaviour (colour map) shows a broad propagation of pattern. However, the instability (r > 0, z-axis) does not propagate so widely. We see that the instability is only present for half of the underlying behaviour. The left wavefront is actually at the centre of the pattern, the line traced out from the initial disturbance. The parameters here are ε = 1.0, γ = 5.0, μ = 0.85. This is a numerical solution of the original Schnakenberg reaction-diffusion-convection equation (2) using NAG D03PCF. The reactants are initially at steady state: (u, v) = (0, 0), with a small disturbance at x = 0. The boundaries are held at zero derivative: u_x = 0, v_x = 0.