Weakly nonlinear stability analyses of one-dimensional Turing pattern formation in activator-inhibitor/immobilizer model systems

Abstract

The development of one-dimensional Turing patterns characteristic of the chlorite-iodide-malonic acid/starch reaction as well as analogous Brussellator/immobilizer and Schnackenberg/immobilizer model systems is investigated by means of a weakly nonlinear stability analysis applied to the appropriately scaled governing equations. Then the theoretical predictions deduced from these pattern formation studies are compared with experimental evidence relevant to the Turing diffusive instabilities under examination in order to explain more fully the transition to such stationary symmetrybreaking spatial structures when the temperature or pool species concentrations vary.

Weakly nonlinear analysis of a model of signal transduction pathway involving membrane based recptors

“…According to Stephenson and Wollkind [7], (I) and (II) represent the homogeneous and striped states, respectively, while (III) can be identified with a rhombic pattern possessing a characteristic angle φ. Here, a 1 = 0.1, a 2 = 0.1, a 3 = 0.5, a 4 = 0.1, a 5 = 0.1, a 6 = 0.1, b 1 = 0.1, b 2 = 0.1, µ 1 = 0.9 and q 2 c = 1.…”

“…(2.9) of the form used by Stephenson and Wollkind in[7],ṽ (x, y, t) ∼ A(t) cos(q c x)ṽ 1010 + B(t) cos(q c z)ṽ 0101 + A 2 (t)[ṽ 2000 + ṽ 2020 cos(2q c x)] + A(t)B(t)[ṽ 1111 cos(q c (x + z)) + ṽ 111(−1) cos(q c (x − z))] + B 2 (t)[ṽ 0200 + ṽ 0202 cos(2q c z)] + A 3 (t)[ṽ 3010 cos(q c x) + ṽ 3030 cos(3q c x)] + A 2 (t)B(t)[ṽ 2101 cos(q c z) + ṽ 2121 cos(q c (2x + z)) + ṽ 212(−1) cos(q c (2x − z))] + A(t)B 2 (t)[ṽ 1210 cos(q c x) + ṽ 1212 cos(q c (x + 2z)) + ṽ 121(−2) cos(q c (x − 2z))] + B 3 (t)[ṽ 0301 cos(q c z) + ṽ 0303 cos(3q c z)] (3.1) where ṽ (x, y, t) ≡ i(x, y, t)…”

Domains of attraction of walking and running attractors are context dependent: illustration for locomotion on tilted floors

“…In general, we will consider amplitude equations that have been discussed frequently in the context of pattern formation. The amplitude equations that will be presented below describe pattern formation in fluid layers under Benard instability [22,23,24,25,26,27,28,29,30], pattern formation in chemical reaction-diffusion systems [31], pattern formation in metal solidification processes [32] and the emergence of patterns in diffusive signaling processes in cells [33,34]. Moreover, the amplitude equations can be transformed into the form of Lotka-Volterra equations [35].…”

Spatial Turing-type Pattern Formation in a Model of Signal Transduction Involving Membrane-based Receptors Coupled by G Proteins

“…In the next section, we shall carry out a weakly non-linear stability analysis on (2.14)–(2.15) in order to show the existence of rhombic and hexagonal planform solutions to our model following the technique discussed by Wollkind et al(1994) and reviewed by Stephenson and Wollkind (1995).…”

“…Please see the work of Stephenson and Wollkind, (1995) for more detail of the technique. On substituting this solution (3.7) into (3.5), we obtain a sequence of vector systems, each of which corresponds to one of these terms.…”