Turing pattern formation in the Brusselator system with nonlinear diffusion

Abstract

In this work we investigate the effect of density-dependent nonlinear diffusion on pattern formation in the Brusselator system. Through linear stability analysis of the basic solution we determine the Turing and the oscillatory instability boundaries. A comparison with the classical linear diffusion shows how nonlinear diffusion favors the occurrence of Turing pattern formation. We study the process of pattern formation both in one-dimensional and two-dimensional spatial domains. Through a weakly nonlinear multiple scales analysis we derive the equations for the amplitude of the stationary patterns. The analysis of the amplitude equations shows the occurrence of a number of different phenomena, including stable supercritical and subcritical Turing patterns with multiple branches of stable solutions leading to hysteresis. Moreover, we consider traveling patterning waves: When the domain size is large, the pattern forms sequentially and traveling wave fronts are the precursors to patterning. We derive the Ginzburg-Landau equation and describe the traveling front enveloping a pattern which invades the domain. We show the emergence of radially symmetric target patterns, and, through a matching procedure, we construct the outer amplitude equation and the inner core solution.

Wavefront invasion for a chemotaxis model of Multiple Sclerosis

“…Following the approach based on the multiple scales method adopted by [5,9,10], we set a small control parameter η 2 = (χ −χ c )/χ c , which gives the dimensionless distance from the bifurcation value of χ . Upon translation of the equilibrium P * to the origin, the system (1) can be written as:…”

Hopf bifurcation in a fractional diffusion food-limited models with feedback control

“…It was also shown that subdiffusion suppresses the formation of Turing pattern [14]. In [16,17], Turing patterns were induced by the anomalous diffusion both in Brusselator chemical system and Boissonade chemical system. Additionally, in systems with Lévy flights, the emergence of spiral waves and chemical turbulence from the nonlinear dynamics of oscillating reaction-diffusion patterns was investigated in [18].…”

“…In higher dimension, the Laplacian is replaced by the operator ∇ γ = −(− ) γ /2 . Pattern formation in reaction-diffusion systems with anomalous diffusion has received considerable attention [12][13][14][15][16][17]. For instance, it was shown that the Lévy flights type superdiffusion induces the formation of Turing pattern [15].…”

Turing pattern dynamics and adaptive discretization for a super-diffusive Lotka-Volterra model

“…At molecular level, classical diffusion arises as the result of standard Brownian motion, and it is typically characterized by the dependence of the mean square displacement of a randomly walking particle on time (Δx) 2 ∼ t. Apart from classical (or normal) diffusion, molecules may undergo anomalous diffusion effects (as discussed in e.g. Bouchard and Georges 1990;Klafter 2000, 2004;Sokolov et al 2002;Golovin et al 2008;Gambino et al 2013). These phenomena (in contrast to normal diffusion) are rather characterized by the more general dependence…”

“…Pattern formation in reaction diffusion systems with anomalous diffusion has recently received considerable attention (Gafiychuk and Datsko 2006;Henry et al 2005;Henry and Wearne 2002;Langlands et al 2007;Weiss 2003;Golovin et al 2008;Gambino et al 2013). For instance, it was shown that sub-diffusion suppresses the formation of Turing patterns (Weiss 2003).…”